Discovering Mathematics: Algebra, Geometry, Advanced Algebra (2014, 2015, 2017)

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Report for High School

Alignment Summary

The instructional materials for the Discovering series meet expectations for alignment to the CCSSM for high school. For focus and coherence, the series showed strengths in the following areas: attending to the full intent of the mathematical content contained in the standards, attending to the mathematical modeling process, spending the majority of time on the content from CCSSM widely applicable as prerequisites (WAPs), engaging students 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. For rigor and the mathematical practices, the series showed strengths in the following areas: supporting the intentional development of students' conceptual understanding, opportunities for students to develop procedural skills, utilizing mathematical concepts and skills in engaging applications, displaying a balance among the three aspects of rigor, supporting the intentional development of reasoning and explaining, and supporting the intentional development of seeing structure and generalizing.

High School

Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Alignment (Gateway 1 & 2)

Meets Expectations

Gateway 3

Usability

Usability (Gateway 3)

Meets Expectations

Overview of Gateway 1

Focus & Coherence

Gateway 1

v1.0

Gateway 1

Focus & Coherence

Criterion 1.1: Focus & Coherence

Meets Expectations

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 Discovering series meet the expectations for Focus and Coherence. The instructional materials meet the expectations for: attending to the full intent of the mathematical content contained in the High School Standards, attending to the mathematical modeling process, spending the majority of time on the CCSSM widely applicable as prerequisites (WAPs), 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. The instructional materials partially meet the expectations for letting students fully learn each non-plus standard and making meaningful connections in a single course and throughout the series.

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 the Discovering series meet the expectations for attending to the full intent of the mathematical content contained in the High School Standards for all students. Overall, the instructional materials address most of the non-plus standards, however, there are a few instances where all aspects of the non-plus standards are not addressed across the courses of the series.

The following are examples of standards that are fully addressed:

The standards from F-BF are developed starting in the Discovering Algebra course with a study of recursive sequences and writing explicit expressions to represent the sequences. Work with recursive sequences is then expanded to working with exponential equations. In Discovering Algebra, the work in this domain concludes with students identifying and exploring the various function transformations. Work with the F-BF domain continues in the Discovering Advanced Algebra course with a further study of recursive sequences and transformations and a study of building inverse functions.
F-IF.7: The materials introduce time-distance graphs in Discovering Algebra Lesson 3.4 by discussing the intercept, periods of non-movement, and speeding up and slowing down. The standard is further addressed through Discovering Advanced Algebra as students examine maximum, minimum, and zero values of quadratic and polynomial functions as well asymptotes and end behavior of rational functions.
A-CED.3: In Discovering Advanced Algebra Lesson 2.5, students determine appropriate and reasonable constraints for application problems involving profit from two types of birdbaths. Students also determine if solutions are reasonable in various problems.
G-CO.7: In Discovering Geometry Lesson 4.4, students create various triangles using constructions and GeoGebra when given select criteria (such as, one side and two angles must be the same) to generate congruence “shortcuts” for triangles such as ASA, AAS, SAS, and SSS. Students also reason and investigate as to why the shortcut “SSA” does not exist.

The following standards are partially addressed:

G-SRT.1a: In Discovering Geometry, Coordinate Geometry 7, students are questioned if “...the corresponding sides are parallel? Explain.” when examining a dilated set of triangles. However, the materials do not address whether a line passing through the center of dilation remains unchanged in either Coordinate Geometry 7 or Discovering Geometry Lesson 7.1.
G-GPE.5: In Discovering Geometry, Coordinate Geometry 5, Example A, students are directed to find the equation of a perpendicular line through a point to find perpendicular bisectors. In Discovering Geometry, Coordinate Geometry 11, students are provided with the parallel slope property and perpendicular slope property and are given problems in which to use them in proofs. However, the materials do not contain a proof of these two properties.
S-IC.5: This standard is not identified in the Discovering Algebra, Discovering Geometry, or the Discovering Advanced Algebra correlation documents. However, examination of the materials reveal that students compare treatments in Discovering Advanced Algebra Lesson 9.1, but at no time do the materials contain simulations to decide if differences between parameters are significant.

Standard S-IC.6 is not addressed within the three courses of the series. The correlation document for Discovering Advanced Algebra suggests this standard is addressed in Lesson 9.4. Upon examination of the materials, no indication can be found where reports that were either publisher-created or student- generated based on data are to be evaluated.

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 Discovering series meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The full intent of the modeling process is used to address nearly all of the modeling standards by the instructional materials of the series. Throughout the series, there are a number of lessons and activities that contain a variety of components of the modeling process described in the CCSSM. Each course also contains a separate document titled Modeling Tasks, which contains a preface that accurately describes the modeling cycle and provides an example modeling problem with a possible solution.

While the Modeling Tasks contain the same preface for each course, the problems provided are different for each course. The Modeling Tasks do not indicate standards or topics, and most of the problems included in the Modeling Tasks allow students the opportunity to engage in all aspects of the modeling cycle. The Modeling Tasks address nearly all of the standards identified as modeling standards. The tasks are not explicitly aligned to standards or to lessons in the materials, so students are able to complete the task by referencing all of the mathematics they have learned, rather than being guided by the lesson or chapter in which the task appears. The Modeling Tasks also include a rubric that can be used to score the modeling problems.

Some examples of Modeling Tasks that address modeling standards and the full intent of the modeling process include:

In Discovering Algebra, Get Your Hamburgers Here!, students determine how much to charge for hamburgers in order to maximize their profits. Students are provided some assumptions based on a survey given to current customers. Students make sense of the problem by creating a model to compute a variety of hamburger prices. Students use their model to interpret and validate their solutions. Students may need to adjust their model as needed to determine the best price for the hamburgers. Students must report their solutions by providing an explanation of their results.
In Discovering Algebra, Who Doesn’t Love Honey?, students determine whether they can isolate half of the bees while they are still healthy in order to produce enough honey for the science fair. Students must make assumptions about how many bees the infection began with, formulate a way to determine the time when half of the bees would be infected, and determine if they would be able to save half the bees based on when the infection was discovered.
In Discovering Geometry, Let’s Go Camping!, students design a tent that fits a set of constraints. Students create a sketch of their designs; however students are not provided direction in regards to the design or formulas used to solve the problem. Students create a model of their tents and create formulas to compute if the surface area and volume are within the given constraints. Students interpret and validate their solutions. Students prepare a report to present to their family which includes a list of pros and cons of each design.
In Discovering Geometry, Country Boy Gas Solution, students determine new dimensions of a given propane gas tank in order to double its volume. Students formulate a model to find the volume of the described tank. Students determine how the tank can change and how the changes will affect the volume. Students use their model to check dimensions to meet the demands of the problem. Students report what the new dimensions are. In this task, the sample answers consider changing either the diameter or the length of the tank; students may also find a way to change both dimensions which would create a tank with double the volume.
In Discovering Advanced Algebra, A Heavy Fish Tale, students develop a plan to create a method to determine the largest fish caught. Students create a model to determine the largest fish caught and calculate repeatedly to determine if the model is accurate (interpretation, validation).
In Discovering Advanced Algebra, What Will You Have with Your Coffee?, students are given information about running a concession stand in a local hospital. Based on the given information and making assumptions about the number of people they would be serving and how many items they could sell, students present a sales strategy for the business. Students create a model for each product and use the model to compare the costs and profits for the products. Based on the results, students modify their sales strategy for the business.

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 Discovering series, when used as designed, meet expectations for spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

In Discovering Algebra, students spend a majority of their time working with WAPs from the Number and Quantity, Algebra, Functions, and Statistics and Probability conceptual categories. For example:

In Lesson 8.4, students factor quadratic equations to determine zeros using the Zero Product Property (A-SSE.3a). Students also factor quadratic equations from general form to vertex and/or factored form. Students check their work by confirming the locations of zeros as a result of factoring via the graphing calculator.
In Lesson 4.1, students use "secret codes" in which each input of the code has only one output to introduce functions and vocabulary such as domain and range (F-IF.1). In Lesson 7.1, students encounter function notation and its relationship to the graph.

The Discovering Geometry course focuses on the widely acceptable prerequisites in the Geometry conceptual category. For example:

In Lesson 12.2, students solve word problems using trigonometric functions (G-SRT.8). In Example 1 of the lesson, students use the angle of elevation to determine the distance a sailboat is located from a lighthouse.
In Lessons 1.1, 1.2, and 1.3, students learn and use precise definitions for the terms such as line segment, angle, parallel lines, and perpendicular lines (G-CO.1). Students practice labeling each of these and answer questions about each of them.

During Discovering Advanced Algebra, students spend a majority of their time working with widely acceptable prerequisites from Number and Quantity, Statistics and Probability, Algebra, and Functions:

In Lesson 4.6, students graph logarithmic functions by looking at how different transformations change each function (F-IF.7e). In Lesson 7.5, students graph trigonometric functions by looking at how different transformations change the period, midline, and amplitude.
In Lesson 9.1, students compare and contrast various types of studies such as experimental, observational, and surveys. Students also make predictions based on sample data from a population in Exercise 7 (S-IC.1).

Indicator 1B.ii

The materials, when used as designed, allow students to fully learn each standard.

The instructional materials reviewed for the Discovering series, when used as designed, partially meet expectations for letting students 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. However, in cases where the standards expect students to prove, derive, or develop a concept, the materials often provide students with the proofs, derivations, and concept developments.

For the following standards, the materials partially meet the expectation for allowing students to fully learn each standard. These examples represent standards which are present but did not allow students to fully learn the standard:

A-SSE.4: In Discovering Advanced Algebra, Lesson 4.8, students are guided through a series of steps to derive the formula for the sum of a finite geometric series. Students are not deriving the formula themselves, which is the expectation of the standard.
A-APR.1: In Discovering Algebra, Chapter 8 and Discovering Advanced Algebra, Chapter 6, students add, subtract, and multiply polynomials. However, students have limited opportunities to develop understanding that polynomials are “closed” under these operations.
A-REI.5: In Discovering Advanced Algebra, Lesson 2.2, students solve systems of equations using elimination and verify results with a calculator. However, students do not prove that replacing one equation with the sum of that equation and a multiple of the other equation produces a system with the same solutions.
F-IF.3: In Discovering Advanced Algebra, Lesson 1.1, students work with sequences. However, the materials do not refer to sequences as functions, whose domain is a subset of the integers.
F-IF.7b: In Discovering Advanced Algebra, Lesson 4.3, Exercise 6b, students graph cube root functions. The materials provide a limited number of problems for students to graph cube root functions.
F-IF.9: In Discovering Algebra, Lesson 3.3, Exercise #4, students compare properties of two functions represented in different ways (tables and graphs). Beyond Exercise #4, there is a limited number of opportunities for students to compare properties of two functions.
F-TF.2: In Discovering Advanced Algebra, Lessons 7.3 and 7.5, students interpret radian measures of angles using the unit circle. However, students do not explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers.
G-C.5: In Discovering Geometry, Lessons 8.4 and 9.6, students solve problems by finding the area of a sector or the length of an arc. While students derive the formula for arc length in Lesson 9.6, students do not derive the formula for the area of a sector in either lesson.

Indicator 1C

The materials require students to engage in mathematics at a level of sophistication appropriate to high school.

The instructional materials reviewed for the Discovering series, when used as designed, meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials provide students with opportunities to engage in real-world problems throughout the series. Students engage in problems that use number values that represent real-life values--fractions, decimals, and integers. The context of most of the scenarios are relevant to high school students.

Examples of where the materials require students to engage in mathematics at a level of sophistication appropriate to high school include:

In Discovering Algebra, Lesson 4.1, students apply ratios and proportions from middle school to finding the rate of change of a function. Students write the rate of change in terms of unit rates that use compound units.
In Discovering Algebra, Lesson 8.1, students solve quadratic equations that have various types of real numbers including terminating decimals, irrational numbers, and integers. Students also work with age-appropriate contexts including the height of a falling baseball, the height of a model rocket, and the height of an arrow being shot from ground level.
In Discovering Advanced Algebra, Lesson 4.1, students use exponential equations that model both growth and decay. Students use various types of real numbers including decimals and integers and age-appropriate context such as population growth, growth of plants, and the price of a used automobile.
In Discovering Advanced Algebra, Lesson 8.2, students apply key takeaways of basic statistics and probability from middle school to find the probability of compound events such as the probability that two students will be successful, and the probability of getting a number of questions correct in a row on a true/false test when just guessing.
In Discovering Geometry, Lesson 11.4, students use rational numbers in fraction and decimal form. There are also operations with radicals and numerical manipulations where students leave “pi” in the answer.
In Discovering Algebra, each chapter begins with a “Refreshing Your Skills” section which often allows for practice with varying number types. Discovering Algebra Lesson 4.0, Exercise 2 includes repeating decimals and radicals. In Lesson 6.0, students convert decimals to percents, and students review scientific notation in Lesson 6.4, which is reviewed throughout the exercises thereafter. Negative exponents are presented in Lesson 6.6 and used thereafter.
In Discovering Advanced Algebra, students encounter the same variety of number types with the addition of complex number operations in Lesson 5.4. Complex numbers are then used in later work.

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 instructional materials reviewed for the Discovering series partially meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series, where appropriate and where required by the standards. Overall, mathematical connections are made within courses, but connections between courses are not made.

The following are examples of connections not being made between courses:

In Discovering Algebra, Lessons 8.6 and 8.7, the materials address completing the square and the quadratic formula (A-REI.4). The materials revisit completing the square and the quadratic formula in Discovering Advanced Algebra Lessons 5.2 and 5.3 without connection to Discovering Algebra Lessons 8.6 and 8.7.
In Discovering Geometry, Lesson 8.2, students calculate the area of different shapes in different contexts. There is no indication in the materials that writing and solving equations in one variable (A-CED.1) and using units to understand problems and guide the solutions (N-Q.3) could be used to solve the problems on area. A-CED.1 is addressed in Discovering Algebra Lesson 2.8, and N-Q.3 is addressed in Discovering Algebra Lesson 2.3.
In Discovering Geometry, Lesson 1.9, the teacher notes state, "This lesson introduces the three rigid transformations (isometries) of the plane: translations, rotations, and reflections." There is no connection to how transformations were addressed in Discovering Algebra Lessons 7.5 (Translating Graphs) and 7.6 (Reflecting Points and Graphs). The standards' correlation document indicates G-CO.2 is addressed in Discovering Algebra, but the connection is not made in the Discovering Geometry materials.
Discovering Advanced Algebra, Chapter 1, Linear Modeling addresses and extends many of the concepts addressed in Discovering Algebra, Chapter 3, Linear Equations. The teacher notes at the beginning of Discovering Advanced Algebra, Chapter 1 state, “Much of this chapter reviews basic algebra concepts but is presented from a fresh perspective. Rather than skipping a topic, you may be able to spend less time on some lessons than on others. Many of the Investigations will allow you to assess prior understanding of familiar topics. In the later lessons of the chapter, students are exposed to the analysis of models that they will need throughout the course.” While there are references to content taught previously and subsequently, there are no clear indications of the connections between concepts or standards.

The following are examples of connections made within courses:

In Discovering Advanced Algebra, Lesson 8.2, students calculate probabilities of independent events. In Discovering Advanced Algebra, Lesson 8.3, students use that knowledge to calculate the probability of mutually exclusive events. The opening paragraph of Lesson 8.3 describes to students which content they will be using from the previous lesson to apply to the new concept.
In the opening of Discovering Algebra, Chapter 3, the teacher notes discuss how the work in Lesson 3.1 with recursive sequences will connect to Chapter 6 when students develop exponential functions. The notes also discuss how the work in Lesson 3.7 connects to the interpretation of fitting a linear function to a set of data.

Indicator 1E

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.

The instructional materials reviewed for the Discovering series meet expectations for explicitly identifying and building on knowledge from grades 6-8 to the High School Standards.

In Discovering Algebra, standards from grades 6-8 are explicitly identified in the teacher assistance portal on the left side of the page in the online teacher manual. Standards from Grades 6-8 are listed and aligned to lessons in the Discovering Geometry Correlation Guide. There are no standards from Grades 6-8 listed or aligned to lessons in the Discovering Advanced Algebra Correlation Guide nor in the lessons.

Some examples where the materials explicitly identify content from Grades 6-8, make connections between Grades 6-8 and high school concepts, and allow students to extend their previous knowledge include:

Discovering Algebra, Chapter 0, Lesson 0.4 explicitly identifies 6.NS, 7.NS, 7.EE.1, 8.EE, and 8.F.1 and builds upon them to introduce F-BF.1a. Through the context of operations with signed numbers, students look for patterns in order to determine an explicit expression, a recursive process, or steps for calculation.
In Discovering Algebra, Lesson 4.1, 8.F.1 is explicitly identified and built upon to address F-IF.1. Students examine functions through the use of “secret codes” and determine that a function is a rule (8.F.1) in order to understand that one element in the domain of a function corresponds to exactly one element of the range (F-IF.1).
In Discovering Geometry, Lesson 7.3, students solve problems using scale drawings of geometric figures (7.G.1) while simultaneously solving problems using similar triangles (G-SRT.4).
In Discovering Geometry, Lesson 2.5, students use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve problems (7.G.5). This knowledge is extended as students prove that vertical angles are congruent (G-CO.9) in Investigation 1.

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.

The instructional materials reviewed for the Discovering series explicitly identify the plus standards when included and use the plus standards to coherently support the mathematics which all students should study in order to to be college and career ready.

The materials address the following plus standards: N-CN.3, A-APR.7, F-IF.7d, F-TF.3, F-TF.4, F-TF.7, F-TF.9, G-SRT.9, G-SRT.10, G-SRT.11, G-C.4, G-GMD.2, S-CP.8, and S-MD.6. In general, the materials include these standards as additional content that extends or enriches topics within a unit, and their inclusion does not interrupt the flow of the course. No plus standards are located in Discovering Algebra.

The following are examples of the materials addressing the full intent of plus standards:

In Discovering Advanced Algebra, Lessons 6.5 and 6.6, students graph rational functions while identifying asymptotes and end behavior. (F-IF.7d)
In Discovering Advanced Algebra, Lesson 7.7, students prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. (F-TF.9)
In Discovering Geometry, Lesson 12.3, students derive the area formula for a triangle by drawing an auxiliary line perpendicular to the base of the triangle. (G-SRT.9)
In Discovering Geometry, Lessons 12.3, 12.4, and 12.5, students apply the Law of Sines and Law of Cosines to solve typical problems involving non-right triangles. (G-SRT.11)

The following are examples of the materials not addressing the full intent of plus standards:

N-CN.3: In Discovering Advanced Algebra, Lesson 5.4, students find the conjugate of complex numbers and use the conjugates in addition, subtraction, and multiplication. However, students do not find the quotient or modulus of complex numbers.
A-APR.7: In Discovering Advanced Algebra, Lessons 6.7 and 6.8, students add, subtract, multiply, and divide rational expressions. However, the materials do not address rational expressions forming a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
F-TF.4: In Discovering Advanced Algebra, Lesson 7.3, students use the unit circle to define the periodicity of trigonometric functions. However, the materials do not address how the unit circle relates to symmetry of the functions (even or odd).
F-TF.7: In Discovering Advanced Algebra, Lesson 7.1, students use inverse trigonometric functions to solve word problems, but not within a modeling context.
G-SRT.10: In Discovering Geometry, Lessons 12.3 and 12.4, students use the Law of Sines and Law of Cosines to solve problems. Students prove the Law of Sines in Lesson 12.3, but the proof of the Law of Cosines is provided for them by the materials.

Criterion 1.1: Focus & Coherence

Overview of Gateway 2

Rigor & Mathematical Practices

Gateway 2

v1.0

Gateway 2

Rigor & Mathematical Practices

Criterion 2.1: Rigor

Criterion 2.2: Math Practices

Meets Expectations

Criterion 2.1: Rigor

Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.

The instructional materials for the Discovering series meet the expectation for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to developing students’ conceptual understanding, developing students' procedural skill and fluency, and providing engaging applications. Within the materials, the three aspects of rigor are not always treated together and are not always treated separately, and the three aspects are balanced with respect to the standards being addressed.

Indicator 2A

Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The instructional materials for the Discovering series meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, the instructional materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding.

The instructional materials develop conceptual understanding throughout the series. For example:

N-RN.1: In Discovering Advanced Algebra, Lesson 4.3 Investigation, students “describe what it means to raise a number to a rational exponent.” In the Investigation, students create a table and a graph for y = x^(½) in order to state a conclusion about raising a number to the power of ½. Students explain how they would evaluate numerical expressions involving a rational exponent, and they conclude the Investigation by generalizing “a procedure for simplifying a^(m/n)."
F-IF.A: Across the series, students develop an understanding of functions. In Discovering Geometry, Lesson 6.2, functions are developed through the algebraic nature of geometric transformations. In Discovering Advanced Algebra, functions are further developed in Chapter 3. Students learn about function notation and evaluate functions. They use real world situations to sketch and interpret graphs of functions. Students talk about reasonable domains and evaluate functions that are representing different situations. Students continue to develop their understanding of functions of different types in Chapters 4, 5, 6, and 7.
A-APR.B: In Discovering Algebra, Lessons 8.4 and 8.6, the materials initially address the relationship between zeros and factors of polynomials as students find the zeros of quadratic equations by factoring and completing the square. In Discovering Advanced Algebra, Lessons 6.2, 6.3, and 6.4, students further develop their conceptual understanding of the relationship between zeros and factors of polynomials through polynomial equations of degree 3 and higher. In both courses, students determine factors of polynomial equations from graphs in addition to finding the zeros for given polynomial equations.

The instructional materials provide opportunities for students to independently demonstrate conceptual understanding throughout the series. For example:

G-SRT.2: In Discovering Geometry, Lesson 7.1, students determine why two figures are similar. They are expected to answer that the angles are congruent and that the sides are proportional. On Chapter 7, Quiz 1 Form A, students explain why or why not two triangles are similar in Problems 3 and 4. On Chapter 7, Constructive Assessment Options, Problem 3, students extend the sides of a trapezoid to create similar triangles. Students explain why the triangles constructed are similar and determine the ratio of the corresponding sides. In Chapter 11, Constructive Assessment Options, Problem 7, students find the ratio between surface area and volume of two similar triangular pyramids and explain why their cross sections are similar.
G-SRT.6: In Discovering Geometry, Lesson 12.1, students explore right triangles with acute angle measures of 20 and 70 degrees. As students draw similar triangles with these angle measures, students develop an understanding that side ratios in right triangles are properties of the angles in the triangle, which leads to definitions of trigonometric ratios for acute angles.
S-ID.7: In Discovering Algebra, Lesson 3.3, students interpret the slope of a graph (speed) and starting location (intercept) in order to provide walking directions to another student. In Chapter 3, Quiz 1 Form A, students complete similar problems.

Indicator 2B

Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The instructional materials for the Discovering series meet expectations that the materials provide intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The materials routinely address procedural skills and start each chapter with “refreshing your skills.” The materials include an Exercise section so students can independently practice skills and concepts addressed in the lesson.

The instructional materials develop procedural skills throughout the series. For example:

F-BF.3: In Discovering Algebra, Lesson 7.5, Problems 4 and 5, students describe the graph of an equation based on the graph of the parent function. Students also write an equation given the description of the transformation and the graph of the parent function. In Chapter 7, Quiz 3, students describe the transformation and write the equation given a parent function.
A-APR.6: In Discovering Advanced Algebra, Lesson 6.8, students multiply and divide rational numbers without context to develop procedural skill. Students solve these stand-alone problems during the lessons in Example A and B as well as during the Exercise portion. In Quiz 3, students solve four problems involving rational equations without context to further develop procedural skill regarding rational expressions.
G-GPE.7: In Discovering Geometry, Coordinate Geometry 9, students use the distance formula to determine the perimeters and areas of quadrilaterals and triangles.

The instructional materials provide opportunities to independently demonstrate procedural skills throughout the series in the following examples:

A-SSE.2: In Discovering Advanced Algebra, Lesson 6.2, students complete guided examples of how a cubic expression for volume can be converted from one form to another (i.e., standard to factored). The materials include methods for using each form to find information about the behavior of the function (the graphed path of the volume expression). Students solve similar problems individually during Exercise 6.2.
F-BF.3: In Discovering Algebra, Lesson 7.5, students individually practice transforming absolute value, quadratic, and exponential functions. In Discovering Advanced Algebra, Lesson 3.5, students individually practice transforming square root functions, in Lesson 3.7, circles, in Lesson 6.6, rational functions, and in Lesson 7.5, trigonometric functions.
G-GPE.4: In Discovering Geometry, Coordinate Geometry 11, students individually use coordinates to prove the definitions of polygons, such as specific quadrilaterals and triangles, by completing the distance formula or slope.

Indicator 2C

Attention to Applications: The materials support the intentional development of students' ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

The instructional materials for the Discovering series meet expectations that the materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Overall, the instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematics and independently demonstrate the use of mathematics flexibly in a variety of contexts.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematics throughout the series. For example:

N-Q.2: In Discovering Algebra, Lesson 2.3, students use conversion rates within contextualized problems. In Exercise 11, students find the conversion factor from a table and use it to solve multiple problems. This problem is multi-step and non-routine.
S-IC.1: In Discovering Advanced Algebra, Lesson 9.1, students use statistics as a process for making inferences about population parameters based on a random sample from that population within contextualized problems.
Chapter 6 of Discovering Geometry addresses applications of transformations. Students explore transformations through activities such as "Finding A Minimal Path" and "Exploring Tessellations."

The instructional materials include opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. For example:

F-IF.4: In Discovering Algebra, Lesson 4.3, Graphs of Real World Situation, students are provided with four problem contexts and need to match them to their corresponding graphs (six are given).
A-SSE.3: In Discovering Advanced Algebra, Lesson 5.1, students construct a function from a table and answer questions using their function related to the problem context.
G-SRT.8: In Discovering Geometry, Lesson 12.2, students use trigonometric functions to solve single and multi-step contextualized problems.

Indicator 2D

Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

The instructional materials for the Discovering series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The materials represent each aspect of rigor both independently and together.

The materials frequently prompt students to explain their reasoning to demonstrate conceptual understanding, and students complete at least one investigation in each lesson to better understand the concept in the lesson. The lessons also provide opportunities for students to practice problems to increase procedural skills with certain topics. The materials frequently use application/contextualized problems to relate concepts to real-world scenarios. Problems oftentimes address conceptual understanding with procedural skill or conceptual understanding with application.

All three aspects of rigor are present independently throughout the program materials in the following examples:

Functions and transformations are addressed in Discovering Algebra and Discovering Geometry. The “rules” related to different transformations, presented in Discovering Algebra, represent the procedural skills for this topic. The use of transformations with functions and combinations of transformations in Discovering Geometry represent conceptual understanding of this topic.
In Discovering Algebra, Lesson 5.1, students solve systems of linear equations. Most of the problems in this lesson do not have contexts, so students develop procedural skills in relation to A-REI.6.
In Discovering Geometry, Lesson 4.6, students complete several proofs. Students develop conceptual understanding of the triangle congruence criteria in regards to G-CO.8 and G-SRT.5 through the completion of the proofs.
In Discovering Advanced Algebra, Lesson 7.6, Exercise 14, students apply trigonometric functions to model a person’s distance from the ground at different places on a double ferris wheel and at different points of the ferris wheel's rotation.

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a topic throughout the materials in the following examples:

In Discovering Advanced Algebra, Lesson 6.1, students write a function to model the height of a falling object due to gravity (application). Students graph the function and write a description of the graph in the context of the problem (conceptual understanding) for S-ID.6a.
In Discovering Algebra, Lesson 6.1, students compare adding the same amount to an account each year with earning interest on the amount each year (application). From this, students compare linear growth to exponential growth (conceptual understanding). In Example C, students divide consecutive terms to find the constant multiplier of a sequence of numbers (procedural skill).

Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials for the Discovering series partially meet expectations that the materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials develop each of the Mathematical Practices, except MP5. For MP5, students do not have opportunities to choose an appropriate tool to use to solve a problem because the materials include directions which specify which tool(s) to use. Throughout the Discovering series, connections to the Standards for Mathematical Practice are listed in the “Blackboard” on the left side of the screen in the teacher manual. The Mathematical Practices are identified numerous times, and there are multiple occurrences when the Mathematical Practices are misidentified. The multiple misidentifications of the MPs are reflected in the scoring of indicator 2e, and these misidentifications do not affect the scoring of Indicators 2f, 2g, or 2h. Examples of the misidentifications are included in the reports for 2e, 2f, and 2h.

Indicator 2E

The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for the Discovering series partially meet expectations that the materials support the intentional development of overarching mathematical practices (MP1 and MP6) in connection to the High School Content Standards. Overall, MP1 and MP6 are used to enrich the mathematical content, and there is intentional development of MP1 and MP6. However, for all of the MPs across the series, there are many examples of misleading identifications as evidenced in the EdReports.org Criterion Summary for the MPs.

Some examples where MP1 (Make Sense of Problems and Persevere) is used to enrich the mathematical content include:

In Discovering Algebra, Lesson 3.4, Investigation the Teacher Notes state: “In Steps 1-5, students are making sense of the problem and looking at correspondences between representations of the situation.” Students use MP1 as they make connections between recursive rules and linear equations.
In Discovering Advanced Algebra, Lesson 4.3, Teacher Notes, students persevere (MP1) in determining that any point on the graph can serve as a starting place for solving the problem.

Some examples where MP6 (Attend to Precision) is used to enrich the mathematical content include:

In Discovering Algebra, Lesson 5.2, the Teacher Notes state that students attend to precision (MP6) by “using complete sentences and appropriate mathematics as evidence in stating and supporting their conjectures” about the slopes of parallel and perpendicular lines.
In Discovering Geometry, Lesson 6.2, Teacher Notes, students use precise mathematical language (MP6) as they write conjectures about the results of composing two reflections.

Examples of the misidentifications for MPs 1 and 6 include:

In Discovering Algebra, Lesson 6.2, the teacher’s notes for the Investigation state: “Step 1. Students who did not do Chapter 0 may need help in seeing how to generate one stage from another. Have them write a rule.” MP1 is identified for this lesson, but students do not have to make sense of the problem or persevere in solving it as they can reference Lesson 0.3 to see further stages of the pattern.
In Discovering Advanced Algebra, Lesson 1.1, MP6 is identified, and the Teacher’s Edition prompts the teachers to ask: “What’s behind the pattern?” An explanation for the teacher states: “Mathematics isn’t only about seeing patterns but also about explaining them.” However, students do not need to provide precise explanations in this lesson.

Indicator 2F

The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for the Discovering series meet the expectation that the materials support the intentional development of reasoning and explaining (MP2 and MP3), in connection to the High School Content Standards. The majority of the time, MP2 and MP3 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others.

Some examples where MP2 (Reason Abstractly and Quantitatively) is used to enrich the mathematical content include:

In Discovering Algebra, Lesson 4.2, Investigation Step 1, students examine a series of tables, determine which relations are functions (MP2), and explain their reasons for their answers.
In Discovering Geometry, Lesson 8.1, Investigation 1, students transform a parallelogram and a triangle labeled with dimensions expressed as variables to derive the formulas for the area of a parallelogram and triangle, respectively. Students reason abstractly by transforming general figures and manipulating variable dimensions, and they can reason quantitatively by contextualizing the general figures and calculating numerical areas to verify the derived formulas are valid (MP2).
In Discovering Advanced Algebra, Lesson 3.1, students examine a graph of the speed and time two cars traveled. Students determine which car will be in the lead after 1 minute (MP2) and explain their reasoning.
In Discovering Advanced Algebra, Lesson 2.7, students reason abstractly and quantitatively (MP2) by adding and subtracting rational numbers. Students add fractions using fraction bars, and they add and subtract rational expressions in an abstract manner.

Some examples where MP3 (Construct Viable Arguments and Critique the Reasoning of Others) is used to enrich the mathematical content include:

In Discovering Algebra, Lesson 2.1, students examine the work of three different students and answer the following questions: “There are many ways to solve proportions. Here are three student papers, each answering the question ‘13 is 65% of what number?’ What steps did each student follow? What other methods can you use to solve proportions?” Students analyze the different solutions to determine what steps were taken.
In Discovering Algebra, Lesson 5.2, "The Slopes Investigation," students plot separate rectangles and find the slopes of the four sides to conclude that opposite sides have equal slopes and adjacent sides have slopes that are opposite reciprocals. They move from the concrete shape to the abstract slopes and construct an argument to support their findings (MP2 and MP3).
In Discovering Geometry, Chapter 4, Constructive Assessment Options, Problem 2, students agree or disagree with "Chloe" as to whether her triangle on her quiz has enough information to solve the problem. Students provide reasoning to support their answer.
Discovering Advanced Algebra, Lesson 2.2, Exercise 12 contains a system of equations. Three fictitious students in the problem recommend different ways to solve the system by substitution, elimination, or graphing. Students determine which method works the best for this particular problem and why.

An example of the misidentifications for MPs 2 and 3 is in Discovering Geometry, Lesson 7.2. The teacher’s notes for summarizing the lesson include: “Return to the list of all potential shortcuts: AA, SSS, SAS, SAA, ASA, and SSA. 'You’ve considered the first three as similarity shortcuts in this lesson; what about the last three?' Ask whether it would help to consider cases in which SSA failed as a congruence shortcut [SMP 1,3,6].” Students do not construct a viable argument or critique the reasoning of others (MP3).

Indicator 2G

The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for the Discovering series partially meet expectations that the materials support the intentional development of modeling and using tools (MP4 and MP5), in connection to the High School Content Standards.

For MP4, throughout the Discovering series, students routinely complete portions of model with mathematics.

In Discovering Algebra, Lesson 4.6, Exercise 10, students compute the number of calories burned while walking. Students use the data in the table to write an equation and determine the real-world meaning of the equation. Students are also asked if a certain equation can model the situation.
In Discovering Algebra, Lesson 1.4, "The Hand Spans Investigation," students collect hand span measurements for the classroom and model the data in a histogram and a stem and leaf plot. Then, they assess which representation would be most appropriate to use under certain circumstances.
In Discovering Advanced Algebra, Lesson 7.6, Example B, students build a mathematical model which will find the vertical height of a seat on a ferris wheel at any time during the rotation.
In Discovering Geometry, Lesson 6.2, "Finding a Minimal Path Exploration," students use a protractor and straightedge on patty paper to model shots on a pool table.

For MP5, throughout the Discovering series, students do not have opportunities to choose an appropriate tool to use to solve a problem because the materials include directions which specify which tool(s) to use.

In Discovering Algebra, Lesson 4.6, the teacher notes state: “Step 3 uses technology to allow students to focus on how the tables and graphs are the same.” However, in Step 3, students are directed to: “Enter both your point-slope equation and your intercept equation into your calculator.” Thus, students are not choosing their own tools.
In Discovering Geometry, Lesson 4.2, MP5 is referenced multiple times in the teacher notes, but students do not choose their own tools in the investigations. In Investigation 1, students are directed to use patty paper and a protractor to construct a triangle and measure the angles in it. In Investigation 2, students are directed to use a compass to copy an angle during the construction of a triangle.
In Discovering Advanced Algebra, Lesson 4.2, the teacher notes provide the following: “Ask students to check the answers on their calculators. [SMP 5]” There is no indication that students are choosing their own tools, but they are directed to use the calculator.

Indicator 2H

The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for the Discovering series meet expectations that the materials support the intentional development of seeing structure and generalizing (MP7 and MP8) in connection to the High School Content Standards. The majority of the time, MP7 and MP8 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, support is present for the intentional development of seeing structure and generalizing.

Some examples where MP7 (Look for and Make Use of Structure) is used to enrich the mathematical content include:

In Discovering Algebra, Lesson 3.7, students examine the graph of two lines and use the structure of the graphed lines to determine how the lines and their equations are similar.
In Discovering Algebra, Lesson 6.1, students look for and describe patterns in the data they have collected. They look for structure when they analyze the pattern to see if it is linear. By examining data and determining that linear data grows at equal amounts over equal intervals, students look for and make use of structure.
In Discovering Geometry, Lesson 6.4, Congruence Shortcuts, students complete a series of compositions to see if certain compositions of transformations can be combined into a single transformation.
In Discovering Advanced Algebra, Lesson 5.2, Investigation, students expand a binomial raised to the second power resulting in a perfect-square trinomial. Students use the structure of the perfect-square trinomial to rewrite other expressions as perfect-square trinomials (completing the square) which develops the vertex form of a quadratic equation. Students also use the structure of completing the square to determine how to find the x-coordinate of a vertex given the general form of a quadratic equation. MP7 is not identified in the teacher materials for this lesson, though students use structure to proceed through the investigation.

Some examples where MP8 (Look for and Express Regularity in Repeated Reasoning) is used to enrich the mathematical content include:

In Discovering Advanced Algebra, Lesson 6.7 Investigate Structure, students graph rational functions. Students determine the slant asymptote equation for a general rational function in terms of variables by recognizing patterns from the graphs of the provided functions.
In Discovering Algebra, Lesson 3.1 The Toothpick Patterns Investigation, students complete a series of repeated steps to determine a recursive formula for finding the number of toothpicks in subsequent terms.
In Discovering Geometry, Lesson 5.1, students examine different angle sums in polygons and look for a pattern to determine the polygon interior angle sum formula.

An example of the misidentifications for MPs 7 and 8 is in Discovering Geometry, Lesson 2.3. Students draw a model of handshakes using points and line segments and, after completing a table, are told, “Notice that the pattern does not have a constant difference. That is, the rule is not a linear function. So we need to look for a different kind of rule.” The teacher’s note states: “Step 3. Students may be drawing points rather randomly. “How could you arrange the points to be sure that every pair is connected by a line segment?” [Using vertices of a convex polygon is a good arrangement.] [SMP 1,2,4,7,8]" Duplicating a representation given in the investigation does not engage students in MP4. Furthermore, students do not use MP7 or MP8 as students are told that the pattern does not have a constant difference. Students do not engage in MP7 or MP8 on their own due to the steps that are provided for the investigation.

Criterion 2.2: Math Practices

Overview of Gateway 3

Usability

Gateway 3

v1.0

Gateway 3

Usability

Criterion 3.1: Use & Design

Criterion 3.2: Teacher Planning

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

Criterion 3.5: Technology Use

Meets Expectations

Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials reviewed for the Discovering series meet expectations for being well designed and taking into account effective lesson structure and pacing. The instructional materials have an underlying design that distinguishes between problems and exercises, exercises given in intentional sequences, variety in how students are asked to present the mathematics, and manipulatives that are faithful representations of the mathematical objects they represent.

Indicator 3A

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials reviewed for the Discovering series meet expectations that the underlying design of the materials distinguishes between problems and exercises. Each lesson presents problems designed to introduce new content along with investigations and explorations. After new content is introduced, students complete Exercises designed to engage them in content of the lesson. Many lessons include a performance task in the Exercises. The lesson resources also provide Launch problems, One-Step problems, Extra Examples, Closing Questions, and Developing Proof.

In Discovering Algebra, there are Dynamic Explorations which are designed to help students develop knowledge of the content in the chapter, and these explorations are aligned to specific lessons. In Discovering Geometry, there are Dynamic Math Investigations, where students use the sketch program to investigate concepts in each chapter. These dynamic explorations/investigations enable students to develop knowledge from problems as opposed to practicing and applying knowledge in exercises. In Discovering Advanced Algebra, there are Explorations which are similar to the Investigations found in the Student and Teacher Editions. These Explorations allow students to further develop knowledge of concepts from problems rather than exercises. The Explorations found in Discovering Advanced Algebra are not linked to any particular chapter or lesson.

In all three products, there are More Practice Your Skills (in Discovering Geometry, these are titled Practice Your Skills), which are more exercises for students to practice the content they have learned.

Indicator 3B

Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials reviewed for the Discovering series meet expectations that the design of the assignments is not haphazard and given in intentional sequences. The materials present problems for students to explore content initially and then exercises for students to practice the content. Students complete a launch problem to have productive struggle with the content and determine their prior knowledge. In One-Step problems, students further grapple with the content. Students develop understanding of the content through problems, investigations, and explorations within the lesson. Students then complete exercises to practice the new content. The teacher materials provide ample additional resources that include content-rich tasks that could be provided to students.

Concepts build upon each other within the courses. For example, in Discovering Algebra, Lesson 5.3 includes linear equations written in different forms, and it builds upon work in Lesson 5.1, where linear equation are expressed as y in terms of x. In Discovering Geometry, students encounter a variety of Dynamic Explorations placed within lesson presentations and exercises to allow students to explore concepts and develop understanding. For example, in Lesson 4.3, students use the Dynamic Exploration as a visual representation of triangle inequalities and the exterior angle theorem.

The One-Step problems offer ways to modify the investigations (present problems from the lesson in different ways), so there is not as much scaffolding. For example, in Discovering Advanced Algebra, Lesson 4.3, the One-Step problem presents Example B in only one question where Example B in the lesson uses three parts to scaffold the students’ answer to the ultimate question “How loud was the bell when it was struck?”

Indicator 3C

There is variety in how students are asked to present the mathematics. For example, students are asked to produce answers and solutions, but also, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials reviewed for the Discovering series meet expectations that there is variety in how students are asked to present the mathematics. In Discovering Geometry, students solve problems by creating sketches, using dynamic geometric software, completing flowmaps, constructing proofs, and writing descriptions and explanations of their work. In Discovering Algebra and Discovering Advanced Algebra, methods for presenting mathematics include creating graphs, tables, equations, descriptions, and reports.

The variety of solutions required for the problems extends to the Assessment Resources provided with each course. The Modeling Tasks also add some variety to the types of responses required throughout the series. In the Cohesive Assessment System, teachers may select exercises that require constructed responses (called Essay questions in the system), Subjective Short Answer questions requiring short responses, and multiple-choice questions with one correct answer.

Indicator 3D

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials reviewed for the Discovering series meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. Students work with virtual manipulatives and through explorations using physical objects as directed in the lessons. In Discovering Algebra, students roll a pencil down a book at a given angle and collect data about the the distance the pencil travels. These manipulatives assist students in understanding the concepts as students create a model from the exploration. Students use coordinate grids and graph paper to construct polygons in Discovering Geometry, as well as interactive graphing software for explorations.

In Discovering Advanced Algebra, students make algebraic connections primarily using equations, tables, and graphs. Also in Discovering Advanced Algebra, an Exploration on Geometric Probability is provided in the digital product resources. In this exploration, students use a ruler, penny, nickel, dime, quarter, and grid paper to investigate a problem. The coins are used to represent the different sizes of coins. The grid paper is used as a representation of congruent tiles mentioned in the problem.

Indicator 3E

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials reviewed for the Discovering series have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The student materials presented online are organized in a consistent manner with each lesson followed by student exercises. Resources, provided in a tab, are consistently organized with Launch, One-Step, Extra Examples, and Closing Questions. Students can bookmark pages to return to content as well as take notes on the sidebar. All materials are clearly labeled and consistently numbered. In each lesson across the series, the Investigations are condensed so that students may reveal one step at a time.

Criterion 3.1: Use & Design

Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials reviewed for the Discovering series meet expectations for supporting teacher learning and understanding of the Standards. The instructional materials provide quality questions to help guide students' mathematical development, contain ample and useful annotations and suggestions, contain full, adult-level explanations and examples of the more advanced mathematics concepts and the mathematical practices, and partially explain the role of the specific mathematics standards in the context of the overall series.

Indicator 3F

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials reviewed for the Discovering series meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. In the Teacher Edition under Teaching the Lesson, teachers are provided questions to guide students’ mathematical understanding and detailed support for how to teach each lesson. Teachers are also provided Critical Questions to ask students about the Big Idea of the activity. For example, in Discovering Geometry, Lesson 6.1, the Critical Question is “Is a rotation through 0° a rotation?”

The Critical Questions and Big Ideas are placed throughout the lesson in the teacher’s notes to help the teacher check for student understanding at different points in the lesson. In addition to the Critical Questions and Big Ideas, the materials provide questions labeled Ask in the teacher notes. For example, in Discovering Advanced Algebra, Lesson 4.1, the teacher notes for Example B include “Why are these functions equivalent?” along with an explanation on how to answer the question. In Summarize in the same lesson, teachers are presented with three additional questions for students to consider on increasing exponential functions and how they relate to the work with half-life completed in the lesson.

Indicator 3G

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials reviewed for the Discovering series meet expectations for containing a Teacher Edition with ample and useful annotations and suggestions on how to present the content in the student edition and ancillary materials. Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. The materials include useful annotations and suggestions on how to teach the content in Teaching the Lesson in the Teacher Edition for each lesson. The materials include step-by-step instructions about how to Launch the lesson and conduct the investigations and explorations. Teachers are also provided formative assessment instructions as well as ways to summarize each lesson. The materials provide teacher guides for lessons in the ancillary materials. Teachers also have detailed notes on how to use the TI-84 and Ti-Nspire during lessons, and support is provided around the interactive investigations in the Discovering Geometry ancillary materials.

In the Teachers Notes for each section, there are Objectives, Vocabulary, Materials needed for the lesson, and connections to math practices. There is also a brief explanation that gives an overview of the content to be addressed in the lesson, and in some cases, there are connections to previous lessons. Additionally, teachers are provided extra examples in Apply of the Teacher Notes along with Closing Questions and suggested exercises for students in Assigning Exercises. In the Teacher Notes, teachers are given explanations for what to watch for as students complete the problems. This may include questions to ask for various exercises that can help to deepen the understanding of the exercise, information about what is included in the exercise (in Discovering Advanced Algebra, Lesson 4.1 Exercise 8, “...the data have decimal values in the exponent.”), and alerts for possible misconceptions or errors.

Indicator 3H

Materials contain a teacher's edition that contains full, adult--level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials reviewed for the Discovering series meet expectations for containing a Teacher Edition that contains full, adult-level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary. The teacher ebook for each course contains guidance within Teaching the Lesson and Answer for each lesson. One tab of Teaching the Lesson and Answer provides The Mathematics at the beginning of the unit, and The Mathematics provides an overview of the terms within the unit and their connection to the content within the course and beyond.

Discovering Algebra and Discovering Advanced Algebra supplemental materials also include Problem Strings. Problem Strings provides guidance for both the student and teachers related to concepts for each unit. For example, the problem string for Discovering Advanced Algebra, Lesson 6.1, provides information that “the strategy we are calling ‘adding ordinates’ comes from the (abscissa, ordinate) language of ordered pairs. It simply means adding the y-values from the parts of a combination function.”

Indicator 3I

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.

The instructional materials reviewed for the Discovering series partially meet expectations for containing a Teacher Edition that explains the role of the specific mathematics standards in the context of the overall series. In Teaching the Lesson in the CCSS section of the teachers’ materials, the materials provide the relationship of the mathematics of the chapter to prior content. However, the mathematics of the chapter is not connected to future courses in the overall series. For example, in Discovering Geometry, Chapter 12, “The definitions of sine, cosine, and tangent ratios for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations. Students in Grade 8 applied the Pythagorean Theorem to determine unknown side lengths in right triangles (8.G.7). In high school, students apply their earlier experience with dilations and proportional reasoning to apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem to solve problems (G.SRT.6−8). The Pythagorean Theorem is then generalized to non-right triangles by the Laws of Sines and Cosines. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on their work with quadratic equations in algebra (G.SRT. 9−11).”

Also, in Discovering Algebra, the CCSS section of Teaching the Lesson for Chapter 6 (Exponents and Exponential Models) connects the content to standards from Grades 5-8. The remainder of the CCSS discussion describes standards addressed in Discovering Algebra, but it does not make a connection to Discovering Advanced Algebra, Chapter 4 (Exponential, Power, and Logarithmic Functions). In the notes for Discovering Advanced Algebra, Chapter 4, there are three sentences that describe the work completed in Discovering Algebra and how the work is extended, but there is no connection to future work in the course or future courses.

Indicator 3J

Materials provide a list of lessons in the teacher's edition, cross-- referencing the standards addressed and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials reviewed for the Discovering series do not provide a list of lessons in the Teacher Edition, cross-referencing the standards addressed and providing an estimated instructional time for each lesson, chapter, and unit. The materials do not include a pacing guide in the student ebook, teacher ebook, or supplementary materials. A list of standards is provided in a CCSS Correlation document in the supplementary materials for each course that provides the lesson(s) where each standard is located.

Indicator 3K

Materials contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials reviewed for the Discovering series contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support progress and achievement in Discovering Algebra and Discovering Geometry. In these two courses, there is Guide for Parents for each chapter in the supplementary materials, and Guide for Parents contains a content summary for the chapter along with review problems and associated answers. The Guide for Parents also includes a summary problem and questions the parents can ask their students as they complete the problem. Discovering Advanced Algebra does not include Guide for Parents in the supplementary materials.

Indicator 3L

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.

The instructional materials reviewed for the Discovering series contain explanations of the instructional approaches of the program and the identification of the research-based strategies. Guide for Teachers is provided in the supplementary materials and is the same for each course. This guide provides various instructional strategies and ways in which teachers can implement them. For example, teachers are provided guidance on how to use cooperative learning, investigation-based classrooms, and various student groupings among other instructional strategies. There are also questions about cooperative learning answered by the authors to help with implementing the strategy in classrooms. The authors also provide questions teachers should ask when planning a lesson; following those questions, there are three sample lessons with one from each course.

Criterion 3.2: Teacher Planning

Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials reviewed for the Discovering series partially meet expectations for offering teachers resources and tools to collect ongoing data about student progress on the Standards. The instructional materials support teachers in gathering information about student’s prior knowledge, identifying and addressing common student errors and misconceptions, and ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials partially support teachers with assessments clearly denoting which standards are being emphasized and providing sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Indicator 3M

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.

The instructional materials reviewed for the Discovering series meet expectations for gathering information about students' prior knowledge within and across grade levels/courses. The materials provide opportunities for teachers to access students' prior knowledge in the Launch at the beginning of each lesson. For example, in Discovering Geometry, Lesson 8.1, “A rectangle has dimensions 6 ft by 8 ft. What is the area and perimeter of the rectangle? Describe what the perimeter and area each represent.” In Discovering Algebra, each chapter includes Refreshing Your Skills, which includes exercises related to topics from previous courses that will be used in the chapter.

Indicator 3N

Materials provide support for teachers to identify and address common student errors and misconceptions.

The instructional materials reviewed for the Discovering series meet expectations for providing support for teachers to identify and address common student errors and misconceptions. In Teaching the Lesson in the teacher ebook, the materials provide common misconceptions to problems in the exercises and advice on how to further student understanding if these misconceptions occur. For example, in Discovering Geometry, Lesson 4.7, “Students may claim that they don’t have enough information to prove the given triangles congruent. Ask “Can you make any triangles that you could prove congruent from the given information?” Teachers also look for misconceptions during group activities and are provided support on how to address these misconceptions. For example, in Discovering Algebra, Lesson 8.2, “Note any groups that plot (width, length) instead of (width, area). As students describe their graphs and whether the points should be connected, ask whether others in the class agree or disagree and why.” In Discovering Advanced Algebra, Lesson 6.2, students investigate maximizing volume in boxes, and in Step 7, teachers are given information about how to address students’ understanding of maximum based on the real-world domain for the function they are using. This clarifies misunderstandings regarding maximum and minimum values for polynomial functions.

Indicator 3O

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials reviewed for the Discovering series meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills. In Teaching the Lesson provided in the teacher ebook, the materials provide feedback to give to students throughout the lesson and during the exercises on both concepts and skills. For example, in Discovering Advanced Algebra, Lesson 6.3, the materials give the following guidance, “To expand these expressions, students can either draw rectangle diagrams or distribute systematically. Support: To multiply three or more factors, suggest that students use the problem-solving technique of breaking down the problem into smaller problems.” The materials also provide suggested student feedback in the supplementary materials. The Discovering series provides ongoing review and practice with a review at the end of each chapter that includes additional exercises and has the same type of feedback provided for teachers that was provided throughout the lessons. In the ancillary materials, there are More Practice Your Skills that offer additional practice exercises for each lesson.

Indicator 3P

Materials offer ongoing assessments:

Indicator 3P.i

Assessments clearly denote which standards are being emphasized.

The instructional materials reviewed for the Discovering series partially meet expectations for assessments clearly denoting which standards are being emphasized. Standards are included with the items in the Kendall Hunt Cohesive Assessment System Test Generator. However, assessments given within the materials do not indicate the standards addressed.

Indicator 3P.ii

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials reviewed for the Discovering series partially meet expectations for assessments providing sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. Follow-up suggestions based on student performance on formative assessments are provided in Teaching the Lesson of the teacher ebook. However, teachers are not provided guidance for interpreting student performance and suggestions for summative quizzes or tests.

Indicator 3Q

Materials encourage students to monitor their own progress.

The instructional materials reviewed for the Discovering series frequently ask students to check their work using a variety of methods. For example, in Discovering Advanced Algebra, Lesson 5.1, the materials state “Graph each equation to check your answers.” In Discovering Algebra and Discovering Advanced Algebra, each chapter includes Assessing What You’ve Learned. This section offers possible ways students can demonstrate their understanding using a variety of methods including, but not limited to, performance assessments, portfolios, journal entries, and presentations. Each of these assessment methods is described in the Teacher Guide (which is the same for each course).

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The instructional materials reviewed for the Discovering series partially meet expectations for supporting teachers in differentiating instruction for diverse learners within and across courses. The instructional materials provide teachers with strategies to help sequence or scaffold lessons, strategies for meeting the needs of a range of learners, and embedded tasks with multiple entry-points. The instructional materials partially suggest support, accommodations, and modifications for English Language Learners and other special populations and partially provide support for advanced students to investigate mathematics content at greater depth.

Indicator 3R

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials reviewed for the Discovering series meet expectations for providing teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners. The materials provide scaffolding for the majority of problems, and the materials are differentiated for groups of students. In the Teacher Edition, there is a tab that includes Modifying Investigation, Conceptual Procedural, Differentiated Instruction, and CCSS. Modifying Investigation includes ways the teacher could modify the investigation for time or levels of support. Conceptual Procedural describes how the lesson addresses conceptual and/or procedural teaching/learning. Differentiated Instruction includes strategies/statements for addressing content for ELL students, how to offer additional support for the students who need it, and how to modify the assignment/lesson for advanced learners. The suggested adjustments to instruction are not described in depth, and the materials do not provide justifications for the recommended adjustments to instruction.

Indicator 3S

Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials reviewed for the Discovering series meet expectations for providing teachers with strategies for meeting the needs of a range of learners. General strategies to meet the needs of all learners are included in Teaching the Lesson and in the Guide for Teachers in the supplemental materials. The general strategies suggested by the materials include ways to incorporate student grouping and exploration activities. In Teaching the Lesson, there are strategies labeled as Support, ELL, Language, and Alert that help the teacher make the materials accessible to all students. For example, in Discovering Advanced Algebra, Lesson 3.5, Teaching the Lesson gives Support, “If students are having difficulty reflecting the entire graph at once, encourage them to reflect each of the four marked points separately before reconnecting the segments. Students could use tracing paper to trace the function y = f(x) and perform the reflection in one step.”

Indicator 3T

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials reviewed for the Discovering series meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. There are several problems with multiple entry-points or problems with multiple solutions or representations. These problems are found within each lesson during explorations and within the modeling tasks included in the supplementary materials.

In the Guide for Teachers, the materials state that there is flexibility by allowing teachers to choose which investigations to use and which investigations they might omit. Teachers also have the flexibility to use the One-Step problem in lieu of the full scaffolded investigation for most lessons, which can offer more opportunities for multiple entry-points and/or correct responses. There are also Projects included for each course. For Discovering Algebra, these Projects are embedded in the lessons in the student and teacher ebooks. For Discovering Geometry and Discovering Advanced Algebra, these Projects are included in the ancillary, digital materials and referenced in Teaching the Lesson for each chapter. These Projects offer additional opportunities for problems with multiple entry-points and multiple correct responses.

Indicator 3U

Materials provide support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials reviewed for the Discovering series partially meet expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that support their regular and active participation in learning mathematics. For each lesson in the tab on the right side of the lesson, there is Differentiated Instruction. When selecting this, three boxes appear at the bottom of the screen; one of the boxes is ELL. There are general statements given for the entire lesson about adjustments to the lesson or things to watch out for as it relates to ELL students. However, justifications are not provided for the suggested changes to instruction for ELL students or other populations.

Also in Teaching the Lesson, some lessons have comments that are tagged with ELL to offer suggestions on helping students with understanding the concepts/skills being addressed. For example:

In Discovering Algebra, Lesson 1.1, “Make sure that students understand the word category. You may want to do this example with students lining up as a human pictograph or as an activity in which students write their birthdays on sticky notes and build a pictograph of their birthday data.”
In Discovering Geometry, Lesson 5.3, Guiding the Investigation 1, “For the next several lessons, create a vocabulary wall that shows each quadrilateral with its definition and related terms. After students complete an investigation, call on students to illustrate the related properties on the wall. Make sure to include drawings of quadrilaterals in “nonstandard position,” such as a trapezoid with bases that are not horizontal.
In Discovering Advanced Algebra, Lesson 1.2, Exercises 7 and 8, “Having students describe the real-world meanings for these exercises will give them the chance to practice their vocabulary and will also serve as a checkpoint for comprehension.”

Indicator 3V

Materials provide support for advanced students to investigate mathematics content at greater depth.

The instructional materials reviewed for the Discovering series partially meet expectations for providing support for advanced students to investigate mathematics content at greater depth. For each lesson, there is a button for Differentiated Instruction. When selecting this, three boxes appear at the bottom of the screen, and one of the boxes is Advanced. There are general statements given for the entire lesson about possible assignments or questions to pose for extending the lessons, but there is not support specifically for advanced students. Some lessons have comments that are tagged with Extend or a section labeled Extensions, and examples include (these items are not identified for only advanced students but are extensions):

In Discovering Algebra, Lesson 2.2, there is an Extend which states “You could design a study to count a second or third kind of fish.”
In Discovering Geometry, Lesson 4.6 Exercises and Lesson 5.1 Exercises, there is a section labeled Extension which offers additional problems/questions to extend the thinking.
In Discovering Advanced Algebra, Lesson 1.3, there is a statement labeled Advanced which states “‘Will the value ever stop changing? We solve for it, but will the input value ever equal 300?’ [In theory, the input value would grow to 300 only in an infinite length of time.]”

Indicator 3W

Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials reviewed for the Discovering series provide a balanced portrayal of various demographic and personal characteristics. The photos and illustrations of people display a variety of demographics, and the names and situations portrayed in the series are diverse.

Indicator 3X

Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials reviewed for the Discovering series provides opportunities for teachers to use a variety of grouping strategies in Teaching the Lesson and within the Guide for Teachers in the supplementary materials. For example, in Discovering Geometry, Lesson 4.2, teachers are provided the following guidance, “If you used the pair-share cooperative group strategy for the first investigation, have students switch partners or roles for this investigation.” In Discovering Advanced Algebra, Lesson 1.5, Example A, the materials suggest having students work in pairs, and in Example B, the materials suggest having students work in small groups.

Indicator 3Y

Materials encourage teachers to draw upon home language and culture to facilitate learning.

The instructional materials reviewed for the Discovering series do not encourage teachers to draw upon home language and culture to facilitate learning. While the materials provide vocabulary for students within the lesson and in Teaching the Lesson, students are not provided materials that draw upon home language or a variety of cultures.

Criterion 3.4: Differentiation

Criterion 3.5: Technology Use

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

The instructional materials reviewed for the Discovering series typically support effective use of technology to enhance student learning. Digital materials integrate technology such as interactive tools, virtual manipulatives, and dynamic mathematics software in ways that engage students in the Mathematical Practices. The instructional materials are web-based and compatible with multiple internet browsers, and the materials formatively assess student understandings and skills using technology. The instructional materials include few opportunities for teachers to personalize learning for all students and are not easily customized for local use. The instructional materials also do not provide opportunities for teachers and/or students to collaborate with each other.

Indicator 3AA

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Mac and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The instructional materials reviewed for the Discovering series are web-based and compatible with multiple internet browsers. In addition, materials are “platform neutral” and can be used on tablets and mobile devices.

Indicator 3AB

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials reviewed for the Discovering series formatively assess student understandings and skills using technology with Dynamic Explorations in Discovering Algebra and interactive tools within the ebook for each course. The instructional materials do not summatively assess student understanding through the use of technology.

Indicator 3AC

Materials can be easily customized for individual learners.

Indicator 3AC.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.

The instructional materials reviewed for the Discovering series include few opportunities for teachers to personalize learning for all students, using adaptive or technological innovations. Assessments or assignments created through the Cohesive Assessment System (CAS) can be modified to personalize learning for all students, but the remainder of the materials are static and cannot be modified.

Indicator 3AC.ii

Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials reviewed for the Discovering series are not easily customized for local use. The materials do not include a range of lessons to draw from on a topic. The materials contain multiple resources for working with each lesson in the supplementary materials; however, these materials are not digital.

Indicator 3AD

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The instructional materials reviewed for the Discovering series do not provide opportunities for teachers and/or students to collaborate with each other. Through the Cohesive Assessment System, there is no opportunity for teachers to contact students. There is no reference to websites or forums where topics may be discussed that would be provided by the publisher. The Guide for Teachers references teacher collaboration, but no digital resources are provided.

Indicator 3Z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The instructional materials reviewed for the Discovering series integrate technology such as interactive tools, virtual manipulatives, and dynamic mathematics software in ways that engage students in the Mathematical Practices. Discovering Algebra contains a series of Dynamic Explorations that enable students to explore mathematical content, reason abstractly and explain their reasoning. For example, in Discovering Algebra, Lesson 4.8 Dynamic Exploration, students move data points around a graph to change drawn box plots while answering questions. Interactive tools can be found within the lessons of the ebook. For example, in Discovering Geometry, Lesson 10.1, students explore the connection between sine, cosine, and tangent within special right triangles using geometry software. There are also videos used in some lessons to help students in investigations. For example, in Discovering Advanced Algebra, Lesson 1.6, students watch a video and use the data in the video to model the rocket’s distance as a function of time.

Criterion 3.5: Technology Use

Downloadable Resources

2014, 2015, 2017