Alignment to College and Career Ready Standards: Overall Summary

The instructional materials reviewed for the AMSCO Traditional Series do not meet expectations for alignment to the CCSSM for high school. The instructional materials attend to the full intent of the high school standards and spend a majority of time on the widely applicable prerequisites from the CCSSM. However, the instructional materials partially attend to engaging students in mathematics at a level of sophistication appropriate to high school, making connections within courses and across the series, and explicitly identifying standards from Grades 6-8 and building on them to the High School Standards. Since the materials do not meet the expectations for focus and coherence, evidence for rigor and the mathematical practices in Gateway 2 was not collected.

See Rating Scale
Understanding Gateways

Alignment

|

Does Not Meet Expectations

Gateway 1:

Focus & Coherence

0
9
14
18
9
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

0
9
14
16
0
14-16
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Usability

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Not Rated

Not Rated

Gateway 3:

Usability

0
21
30
36
0
30-36
Meets Expectations
22-29
Partially Meets Expectations
0-21
Does Not Meet Expectations

Gateway One

Focus & Coherence

Does Not Meet Expectations

Criterion 1a - 1f

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).
9/18
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Criterion Rating Details

The instructional materials reviewed for the AMSCO Traditional Series do not meet expectations for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses that is consistent with a logical structure of mathematics. The instructional materials attend to the full intent of the high school standards and spend a majority of time on the widely applicable prerequisites from the CCSSM. The instructional materials partially attend to engaging students in mathematics at a level of sophistication appropriate to high school, making connections within courses and across the series, and explicitly identifying standards from Grades 6-8 and building on them to the High School Standards. The materials do not attend to the full intent of the modeling process when applied to the modeling standards and allowing students to fully learn each non-plus standard.

Indicator 1a

The materials focus on the high school standards.*
0/0

Indicator 1a.i

The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
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Indicator Rating Details

The instructional materials reviewed for the AMSCO Traditional series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. There are few instances where all the aspects of the standards are not addressed, and there is one standard that is not addressed by the materials. Overall, most non-plus standards are addressed to the full intent of the mathematical content by the instructional materials.

The following are standards that have been addressed fully in the instructional materials:

  • A-CED: The standards from A-CED are addressed throughout the Algebra 1 and Algebra 2 courses. In Algebra 1, students create linear equations and inequalities in one and two variables to represent relationships in problems. Students extend their understanding of creating inequalities to linear programming problems in Algebra 2 when they consider viable solutions within given constraints. Additionally, students create absolute value (Algebra 1, Algebra 2), quadratic (Algebra 1, Algebra 2), exponential (Algebra 1, Algebra 2), and rational (Algebra 2) equations to solve problems.
  • F-BF.3: Transformations of functions are emphasized throughout the Algebra 1 and Algebra 2 courses. In Algebra 1, students transform absolute value functions in Lesson 4.4 (pages 164-170), quadratic functions in Lesson 8.7 (pages 288-290), and exponential functions in Lesson 9.2 (pages 326-334). In Algebra 2, students transform quadratic and absolute value functions in Lesson 1.1 (pages 49-54), square root and cube root functions in Lesson 5.5 (pages 251-255), exponential functions in Lesson 6.1 (pages 264-265), and logarithmic functions in Lesson 7.2 (pages 302-305).
  • G-CO.9: Students prove theorems about lines and angles. Proofs include: vertical angles are congruent; alternate interior angles are congruent when a transversal intersects parallel lines; alternate exterior angles are congruent when a transversal intersects parallel lines; corresponding angles are congruent when a transversal intersects parallel lines; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints; segment addition postulate; if two angles are complementary to the same angle they are congruent to each other; if two angles are supplementary to the same angle they are congruent to each other; and two angles that form a linear pair have measures that sum to 180 degrees. Through these proofs, students enhance their understanding of the relationship between lines and angles in geometric figures.

The following standards are partially addressed in the instructional materials:

  • A-REI.11: In Algebra 1, Lesson 5.1, the materials state, “The coordinates of the point of intersection, or points of intersection, of the lines define the solution of the system” (page 185). However, an explanation as to why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equations f(x)=g(x) is not provided. The materials do provide examples where f(x) and g(x) are linear, absolute value, polynomials, rationals, exponential, and logarithmic functions.
  • G-CO.4: In Geometry, Lesson 1.6, materials define reflections in terms of line of symmetry, but not in terms of angles, circles, perpendicular lines, parallel lines, and/or line segments.
  • G-CO.13: Students construct an equilateral triangle, square, and equilateral hexagon; however, not all of these constructions are inscribed in a circle. Students construct a hexagon inscribed in a circle in Geometry, Lesson 8.3, pages 372-373.
  • S-IC.5: Students compare two treatments in an experiment in an online activity paired with Algebra 2, Lesson 10.7. However, students do not compare differences between two parameters or a statistic and a parameter to determine if the data is statistically significant. The concept of significance is not defined or discussed in materials.
  • S-CP.4: In Algebra 2, Lesson 10.4, pages 490-493, students interpret two-way tables to calculate probabilities and determine if two categorical variables are independent. However, students do not construct a two-way table to represent categorical data.

The following standard is not addressed in the instructional materials:

  • A-REI.5: In Algebra 1, Lesson 5.3, students solve systems of equations using elimination by replacing one equation with the sum of that equation and a multiple of another equation in the system, but a proof by comparison of methods or how this method works is not provided in the materials.

Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.
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Indicator Rating Details

The instructional materials reviewed for the AMSCO Traditional series do not meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials do not include all aspects of the modeling process, and students do not engage in parts of the modeling process.

Students do not have an opportunity to work through the entire modeling cycle independently due to extensive scaffolding. Students do not have opportunities to make assumptions about problems, develop their own solution strategies, validate their solutions, and either improve upon their model or report their conclusions. Students apply and analyze given problems and scenarios, but the variables, parameters, units, equations, or problem-solving methods have been identified.

While aspects of the modeling process are attended to, there are components of the modeling process that are altogether missing from the series. Examples from the instructional materials include:

  • In Algebra 1, Lesson 3.8, Problem 9, page 135, students compare prices for two car repair services. Students are provided the variables and equations, yet they explain the equations in words. Students are directed to graph the equations (as to which variable goes on the x-axis and which goes on the y-axis) in order to find the difference in cost for service that would take 2, 3, or 5 hours before making a conclusion about which service to choose. Students do not formulate the equations nor do they have the opportunity to validate their conclusion. (A-SSE.1, F-IF.7a)
  • In Algebra 1, Chapters 1-6, Cumulative Review, Problem 23, page 238, students use information about hourly wages earned working two different jobs as well as the maximum number of hours that can be worked and the minimum amount of money needed to be earned weekly. Students write a system of linear inequalities to represent the situation and graph the system in order to describe the range of possible combinations of hours worked at each job to make at least $240 per week. Students formulate the inequalities representing the information provided, but they do not decide on a pathway for solving the problem or validate or report their conclusion. (A-CED.3)
  • In Geometry, Chapters 1-10, Cumulative Review, Problem 30, page 525, students determine a new height for a platform on a zip-line course in order to reduce the speed when participants approach the landing platform. Students are given certain parameters, change the height of the landing platform (not the take-off platform) by making the angle of depression between the two platforms smaller, and this removes assumptions students can make as they identify variables in the problem. In addition, students justify their answer but are not given an opportunity to validate their solution. (G-SRT.8)
  • In Geometry, Lesson 9.6, Multi-Part Problem Practice, page 56, students use coordinates representing street intersections that bound a park. Students apply properties of quadrilaterals to find the area of the park, determine how much fencing would be needed to enclose the park, and determine the coordinate for the center of the park where a tree will be planted. Students do not develop a plan to approach this multi-part problem, rather they are offered a step-by-step outline. Students compute area, perimeter, and midpoint and interpret their calculations in the context of the problem, but students do not validate or report their solution. (G-GPE.7)
  • In Algebra 2, Lesson 9.5, Problem 31, page 431, students write an equation to model the oscillating motion of a puck attached to a spring. The instructional materials identify the variables d (displacement in inches from its equilibrium position) and t (time in seconds) and that the motion is modeled by the sine or cosine function. Students formulate the equation, but they do not make assumptions about the variables and parameters involved in this scenario in order to develop the equation. Students do not interpret their equation in terms of the original scenario nor do they validate that their equation accurately represents the scenario. (F-TF.5)
  • In Algebra 2, Lesson 6.2, Problem 29, students write an equation to model the dilution of a solution of alcohol with water. The instructional materials identify the variables x (number of dilutions) and y (portion of alcohol that remains), so students do not make assumptions about the variables and parameters in order to develop the equation. Students graph the equation and use the equation or graph for two computations. Students do not interpret their equation in terms of the original scenario nor do they validate their equation to ensure it accurately represents the scenario. (F-LE.1c)

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.
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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.
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Indicator Rating Details

The instructional materials reviewed for the AMSCO Traditional series meet expectations, when used as designed, for spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs and careers (WAPs). Some examples of how the materials spend the majority of time on the WAPs include:

  • N-RN.2: In Algebra 1, Lessons 1.7 and 9.1 and Algebra 2, Lessons 5.1 and 5.3, students rewrite expressions involving rational exponents and radicals to create equivalent expressions.
  • A-APR.3: In Algebra 1, Lesson 8.5, students use zeros of quadratics to sketch a graph. This knowledge is extended in Algebra 2, Lesson 3.5 when students find zeros given a graph of a polynomial and write the function of the graphed polynomial.
  • F-IF.6: In Algebra 1, Lesson 3.3, students calculate and interpret the average rate of change of a linear function presented symbolically and as a table. Students later calculate and interpret the rate of change for quadratic functions in Algebra 1, Lesson 8.5. In Algebra 2, the concept of rate of change is extended to exponential functions in Lesson 6.1 and logarithmic functions in Lesson 7.5.
  • G-SRT.B, C: In Geometry, Chapter 7, students build upon knowledge of similarity from Geometry, Chapter 2. In Chapter 7, the materials address proving triangle similarity in order to develop the Pythagorean Theorem. This extends to solving problems with special right triangles and trigonometric ratios.
  • S-ID.7: In Algebra 1, Lesson 10.4 and Algebra 2, Lesson 1.2, students find a linear regression equation to represent a data set and describe the meaning of the slope and y-intercept within the context of the problem for linear models.

There is little evidence of where plus standards or non-CCSSM standards distract from the WAPs. An example of distracting topics include:

  • Algebra 2, Lesson 9.6 addresses reciprocal trigonometric functions. This topic is not included in the CCSSM high school standards. This lesson is labeled “Optional” in the instructional materials.

Indicator 1b.ii

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

The instructional materials reviewed for the AMSCO Traditional series do not meet expectations, when used as designed, for letting students fully learn each non-plus standard. Overall, where the standards expect students to prove, derive, or develop understanding of a concept, the materials often provide students the information. For some standards, the materials provide limited opportunities for students to learn the non-plus standard fully.

The following are examples of how the materials, when used as designed, do not enable students to learn the non-plus standards fully:

  • N-RN.3: In Algebra 1, Lesson 9.1, the materials state, “The sum or product of two rational numbers is always rational;” however, students do not explain why these two properties are true (page 322). Furthermore, students identify whether the following two properties are always, sometimes, or never true: (1) Multiplying a rational number by an irrational number results in a rational number, and (2) Adding a rational number and an irrational number results in a rational number (page 363). Students do not explain their reasoning.
  • A-SSE.3: Students regularly produce equivalent forms of expressions, yet do not use these equivalent expressions to explain properties of the quantity represented by the expression. For example, in Algebra 1, Chapter 8, students utilize procedures to produce equivalent quadratic expressions but do not make connections between factored form and x-intercepts of the graphed quadratic or vertex form and whether the vertex is a maximum or minimum.
  • A-SSE.3c: In Algebra 1, Lesson 9.3, Practice Problem 8b, students determine an equivalent way to express $$N(t)=50(2^{2t})$$ (page 339). In Algebra 2, Lesson 6.2, Practice Problems 26-28, students transform expressions for exponential functions to change the time measurement for the independent variable (page 277). These practice problems provide limited opportunities for students to learn the standard fully.
  • A-SSE.4: In Algebra 2, Lesson 8.4, the materials derive the formula for the sum of a finite geometric series (page 361). Students do not derive the formula but do use the formula to solve problems.
  • A-APR.4: The materials prove several polynomial identities - square of a binomial sum (Algebra 2, Lesson 2.1, page 81), product of the sum and difference of two terms (Algebra 2, Lesson 2.1, page 81), identity to generate Pythagorean triples (Algebra 2, Lesson 2.3, page 93), but students do not prove these identities.
  • A-REI.1: In Algebra 1, Lesson 2.1, students explain each step in solving a linear equation as following from the equality of numbers asserted at the previous step. Students do not explain each step in solving a non-linear equation.
  • A-REI.4a: In Algebra 1, Lesson 8.8, the materials derive the quadratic formula by completing the square on a general quadratic equation in standard form (page 291), but students do not derive the quadratic formula.
  • F-IF.8b: In Algebra 1, Lesson 9.2, Practice Problem 11, students identify six functions as exponential growth or exponential decay (page 332); however, students do not interpret parts of the exponential equations. In Algebra 1, Lesson 9.3, Practice Problem 21, students identify the decay rate for a given exponential function (page 340), yet properties of exponents are not needed to answer the question. In Algebra 2, Lesson 6.1, Practice Problem 24, students use properties of exponents to rewrite the function $$g(x)=(\frac{5}{2})^{3-x}$$ into an exponential function of the form $$a^{x+b}$$ (page 268), however, students do not need to interpret this function. In Algebra 2, Chapter 6, Review Problems 12 and 13, students use properties of exponents to interpret expressions for exponential functions involving rates of inflation (page 291).
  • F-BF.4a: Students write equations for the inverses of linear functions in Algebra 1, Lesson 3.7 and Algebra 2, Lesson 6.4, Opportunities, when students write equations for the inverses of non-linear functions, are limited to Algebra 1, Chapter 3, Review Problem 44, page 139, when students determine if $$f(x)^{-1}=\sqrt{x}$$ is the inverse of $$f(x)=x^2$$, Algebra 2, Lesson 6.4, Practice Problem 2, page 285, when students identify the graph of the inverse for a quadratic function with a restricted domain in a multiple choice question and Algebra 2, Cumulative Review Chapters 1-9, Problem 1 (page 460), when students identify the inverse of a rational function in a multiple choice question.
  • F-LE.1: In Algebra 1 and Algebra 2, students model situations with linear functions and exponential functions. However, students have limited opportunities to distinguish between situations that could be modeled with a linear or an exponential function. Limited opportunities for students to distinguish between linear and exponential models are provided in Algebra 2, Lesson 9.7.
  • F-LE.1a: In Algebra 2, Lesson 6.1, the materials show how linear functions grow by equal differences over equal intervals, and exponential functions grow by equal factors over equal intervals in an example (page 265). Students do not prove this relationship.
  • F-LE.3: In Algebra 1, Lesson 9.2, students observe the relationship that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically in Model Problem 5 (page 329) when comparing quadratic and exponential functions using a table and graph and Model Problem 6 (page 330) when comparing linear, quadratic, and exponential functions using a graph. These examples are completed for the students.
  • F-TF.2: In Algebra 2, Lesson 9.4, students apply understandings of the unit circle; however, they do not explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers.
  • G-CO.3: In Geometry, Lesson 1.6, students find the smallest angle of rotation that matches an image to the original figure. There was no evidence found where students take a given rectangle, parallelogram, trapezoid, or regular polygon and rotate it onto itself.
  • G-CO.8: In Geometry, Lessons 5.3 and 5.4, students use SSS, SAS, and ASA criteria to show that two triangles are congruent. Students do not explain how these criteria stem from the definition of congruence in terms of rigid motions.
  • G-GPE.6: In Geometry, Lesson 1.2, students use ratios to find the length of a partitioned segment or find the coordinate of an endpoint of a line segment. In Practice Problem 5, students find point B that partitions line segment AC in a ratio of 1:1, so students find the midpoint of segment AC (page 50). There is no evidence of students finding the point on a directed line segment between two points that partition the segment in a ratio other than 1:1.
  • G-GMD.4: In Geometry, Lesson 10, students identify the shape of two-dimensional cross-sections of three-dimensional objects (pages 475, 481-482). However, students' identification of three-dimensional objects generated by rotations of two-dimensional objects is limited to Problem 17 in the Practice Problem portion of the lesson.
  • G-MG.1: Throughout the Geometry course, Model Problems use geometric shapes, their measures, and their properties to describe objects include representing the shape of a yard using trapezoids (Lesson 9.6, page 450-451), a soup can as a cylinder (Lesson 10.2, page 486), and the shape of a hand using cylinders and prisms. Since these examples are completed for the students, students do not learn the standard fully. In Lesson 10.3, Practice Problem 36, page 507, students select a complex object and sketch the object reducing it to four or more solids that have a known formula for volume.
  • G-MG.3: Throughout the Geometry course, Model Problems apply geometric methods to solve design problems include building a new basketball court using specified ratios and limited gym space (Lesson 2.3, page 104) and deciding on the shape of an ornament to minimize the surface area (Lesson 10.3, page 503). Students apply geometric methods to solve design problems in Multi-Part Problem practice (Lesson 9.8, page 464) when designing a sculpture out of aluminum and steel with budget constraints. These problems combined provide limited opportunities for students to learn the standard fully.
  • S-ID.6b: In Algebra 1, Lesson 10.4, Practice Problems 1e, 2e, 3e, 4e, and 9d, students create residual plots (pages 388-389); however, students do not analyze the residuals to assess the fit of a function.
  • S-ID.9: In Algebra 1, Lesson 10.4, the materials state, “Correlation does not always mean causation,” (page 387) and provide two examples and Practice Problems 5 and 6 for students to distinguish the difference between causation and correlation. These problems combined provide limited opportunities for students to fully learn the standard.
  • S-IC.2: In Algebra 2, Lesson 10.1, Model Problem 1, page 468 and Practice Problem 16, page 471, students analyze whether results from a spinner are fair. These problems provide limited opportunities for students to learn the standard fully.
  • S-IC.6: In Algebra 2, Lesson 10.6, Practice Problems 9 and 10, page 512, students determine the validity of a decision based on reported data. In Algebra 2, Lesson 10.7, Practice Problem 22, page 522, students determine whether a polling company was accurate in their report given that the actual results fell outside their reported margin of error. These problems provide limited opportunities to learn the standard fully.

Indicator 1c

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

The instructional materials reviewed for the AMSCO Traditional series partially meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts and apply key takeaways from grades 6-8, yet do not vary the types of numbers being used.

The materials provide opportunities to solve problems in real-world contexts that are relevant to high school students. Examples include:

  • In Algebra 1, Lesson 2.4, students determine how many miles a car can drive on 24 gallons of gas (Practice Problem 5, page 71).
  • In Algebra 1, Lesson 9.3, students determine the value of an autographed baseball card given that it grew 10% in value every year (Practice Problem 7, page 339).
  • In Geometry, Lesson 7.2, students use similar triangles and the Law of Reflection from physics to determine how to tag a friend in a game of laser tag (Practice Problem 34, page 296).
  • In Geometry, Lesson 7.8, students use the Law of Cosines to determine the length of a slide at an amusement park (Practice Problem 30, page 344).
  • In Algebra 2, Lesson 2.4, students determine whether a field goal kicker kicked the ball high enough to clear the goalpost (Practice Problem 21, page 102).
  • In Algebra 2, Lesson 6.2, students compare how the amount of money they invest changes depending on whether the account compounds interest annually, quarterly, monthly, or daily (Multi-Part Problem Practice, page 277).

The instructional materials offer opportunities for students to apply/extend key takeaways from Grades 6-8. Examples include:

  • In Algebra 1, Lesson 10.4, students apply statistical concepts by graphing scatterplots, using a calculator to find the correlation coefficient, and plotting residuals.
  • In Algebra 1, Lesson 9.2 and Algebra 2, Lesson 6.1, students apply the rate of change for a linear function to the average rate of change for an exponential function.
  • In Geometry, Lesson 7.6, students extend their understanding of ratios as they define the trigonometric ratios of sine, cosine, and tangent.
  • In Geometry, Chapters 2 and 7, students extend their understanding of proportional relationships by dilating geometric figures, proving triangles similar, and using trigonometric functions to solve problems.

The types of real numbers being used are not varied throughout any single course or the entire series. Examples include:

  • In Algebra 1, Lesson 2.1, students solve two-sided equations in Practice Problems 5-20, page 55. While four of the equations consist of non-integer values, all but one of the equations results in a positive whole-number answer.
  • In Algebra 1, Lesson 8.4, students solve quadratic equations by graphing. Of the quadratic equations graphed in Practice Problems 1-20, Practice Problem 4 contains a decimal: $$x^2+7x+12.25=0$$.
  • In Geometry, Lesson 1.2, problems involving the distance formula and midpoint formula include integer coordinates.
  • In Geometry, Lessons 4.1 and 4.2, students learn about angle relationships resulting from a transversal intersecting a pair of parallel lines. Model Problems and Practice Problems include positive integer values for all angle measures.
  • In Geometry, Lesson 7.8, integer values are used for given side lengths and angle measures of triangles in all but three problems (Problems 7, 8, and 14 provide the measure of a side length that is 6.2, 9.1, and 4.5 units in length, respectively).
  • In Algebra 2, Chapter R, students review topics from Algebra 1. Most Model Problems and Practice Problems use integer values in the given problems and result in positive whole number solutions.

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.
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Indicator Rating Details

The instructional materials reviewed for the AMSCO Traditional series partially meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. Connections within a course or between courses are often not made explicit for teachers and students.

Examples of where the materials foster coherence through meaningful mathematical connections in a single course and between courses include:

  • In Algebra 1, Lesson 3.4, students graph linear equations by creating a table of values or identifying the slope and y-intercept from an equation in slope-intercept form or point-slope form. In Algebra 1, Lesson 4.1, students make a table or use the slope and y-intercept to graph linear inequalities.
  • Students identify the effect on the graph of f(x) when it is replaced by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k when transforming functions throughout the series (F-BF.3). In Algebra 1, students work with transformations of absolute value functions (Lesson 4.4), quadratic functions (Lesson 8.7), and exponential functions (Lesson 9.2). In Algebra 2, students connect to their work from Algebra 1 when they transform polynomial functions (Lesson 3.7), exponential functions (Lesson 6.1), logarithmic functions (Lesson 7.2), and rational functions (Lesson 4.4). In Geometry, students translate parabolas (Lesson 11.4) from the origin.
  • Students work with the slope criteria for parallel and perpendicular lines to solve problems in Algebra 1, Lesson 3.3 and Geometry, Lesson 4.4. In Lesson 4.4, the materials explicitly state, “We recall from Algebra how to determine if two lines are parallel” and then prove why perpendicular lines have negative reciprocal slopes using rotations to find parallel lines.
  • Students solve quadratic equations using a variety of methods in Algebra 1, Chapter 8. In Algebra 2, Chapter 2, students review those methods for solving quadratic equations in Lessons 2.1 and 2.4, and in Lessons 2.5 and 2.6, students connect those methods to solving quadratic equations with complex number solutions.

Examples of where the materials do not foster coherence through meaningful mathematical connections in a single course and between courses include:

  • In Algebra 1, Lesson 4.3, students graph a piecewise function in which a portion of the graph is an absolute value function. The materials state, “The absolute value function we will study later can also be defined algebraically as a piecewise function,” (page 156). However, when students graph absolute value functions in Lesson 4.4, no connection is made to defining an absolute value function as a piecewise function.
  • In Algebra 1, Chapter 8, students solve quadratic equations algebraically by factoring and taking the square root (Lesson 8.2), completing the square (Lesson 8.3) and the quadratic formula (Lesson 8.8). Students solve quadratic equations graphically in Lesson 8.4. In Lesson 8.4. There are connections made for students in the Model Problem between the factored form of a quadratic equation and the x-intercepts of the graphed equation. However, students do not make these connections in the Practice Problem portion of the lesson.
  • In Geometry, Chapter R, Algebra Review, students review concepts and skills from Algebra 1 including solving equations, inequalities, and systems, the slope intercept form of a line, multiplying and factoring polynomials, simplifying square roots, completing the square, graphing parabolas, area and perimeter fundamentals, and the Pythagorean Theorem. There are no specific connections made between these topics and specific Geometry concepts or skills to be taught later in the course. For example, Geometry, Lesson R.8, pages 24-25 and Algebra 2, Lesson 2.4, pages 94-96 address completing the square, which was addressed in Algebra 1, Lesson 8.3. These two lessons are duplicate lessons with the same Model Problems and Practice Problems.
  • In Algebra 2, Chapter R, Review, students review concepts and skills from Algebra 1 including solving equations, the rate of change for linear functions, graphing functions, solving systems of linear equations and inequalities, operations with polynomials, and translating parabolas in vertex form. There are no explicit connections made between these topics and specific concepts or skills to be taught later in the course. For example, in Algebra 2, Lesson R.4, students solve systems of linear equations and inequalities by graphing or using the algebraic methods of substitution and elimination. However, students do not solve problems related to systems of linear equations or inequalities later in the course.
  • Statistics and Probability standards are addressed in Algebra 1, Chapter 10 (interpreting quantitative and categorical data), Geometry, Chapter 12 (probability), and Algebra 2, Chapter 10 (probability). These chapters are taught in isolation and have limited connections to other content in the series.

Indicator 1e

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

The instructional materials reviewed for the AMSCO Traditional Series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The instructional materials do not explicitly identify content from Grades 6-8, but they do support the progressions of the high school standards. Connections between the non-plus standards and standards from Grades 6-8 are not clearly articulated in the student or teacher materials. Certain lessons or parts of lessons are labeled “Review,” but these review sections do not identify standards from Grades 6-8.

Although the materials do not explicitly identify Grade 6-8 standards when addressed in the materials, evidence that the materials build on knowledge from Grades 6-8 standards and connect to the high school standards include:

  • Students extend their knowledge of properties of operations to add, subtract, factor, and expand linear expressions (7.EE.1) when adding, subtracting, and multiplying polynomials in Algebra 1, Lessons 6.1 - 6.5 and Algebra 2, Lesson 3.1 (A-APR.1). Connections between the multiplication of polynomials and factoring are made in Algebra 1, Lesson 7.1-7.3 as students determine the factored form of a quadratic expression, with an emphasis on the structure of the quadratic expression (A-SSE.2 connects to 7.EE.2).
  • Students build upon their knowledge of the formulas for the circumference and area of a circle (7.G.4) and proportional reasoning (7.RP.3) as they calculate the length of an arc and the area of a sector in Geometry, Lesson 8.5 (G-C.5).
  • Students extend their knowledge of the relationships between angles that are created when parallel lines are cut by a transversal (8.G.5) by proving postulates and theorems about angle measures in Geometry, Chapter 4 (G-CO.A).
  • Students review the properties of integer exponents used to generate equivalent numerical expressions (8.EE.1) in Algebra 2, Lesson 5.1 and extend their knowledge to generate equivalent numerical expressions for numeric and algebraic expressions in Lessons 5.1 and 5.2 (N-RN.2).

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.
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Indicator Rating Details

The instructional materials reviewed for the AMSCO Traditional series use the plus standards to coherently support the mathematics which all students should study in order to be college and career ready. Plus standards are integrated into the chapters in such a way that omitting the lessons or portions of lessons aligned to plus standards would not disrupt the coherence of the remainder of the mathematical content in the series.

Generally, plus standards are explicitly identified as “Optional” in both teacher and student materials. For example, Geometry, Lesson 11.5 is titled, “Optional: Ellipses at the Origin” and aligns to G-GPE.3. If a portion of a lesson addresses a plus standard, then the portion is identified as optional, for example, Algebra 2, Lesson 6.4, “Optional: Domain Restrictions of Composite Functions.” There are instances when plus standards are not explicitly identified, such as Algebra 2, Lesson 4.4, “Graphing Rational Functions” that aligns to F-IF.7d.

The following plus standards are addressed fully in the instructional materials:

  • N-CN.8: In Algebra 2, Lesson 2.5, the materials derive the identity for the sum of two squares, and students use the identity in Practice Problems 1-18, pages 107 and 108.
  • N-CN.9: In Algebra 2, Lesson 3.5, the materials state the Fundamental Theorem of Algebra and explain why it is true for quadratic polynomials.
  • A-APR.5: In Algebra 2, Lesson 8.5, the materials state the Binomial Theorem, and students apply the Binomial Theorem in the Practice Problems for the lesson.
  • A-APR.7: In Algebra 2, Lesson 4.1, the materials state that operations with rational expressions are part of a closed system. Students multiply and divide rational expressions in Algebra 2, Lesson 4.1 and add and subtract rational expressions in Algebra 2, Lesson 4.2.
  • F-BF.1c: In Algebra 2, Lesson 6.4, students compose functions in problems on pages 286-289.
  • F-BF.4b: In Algebra 2, Lesson 6.4, Multi-Part Problem Practice, students verify that a function is the inverse of a given function using composition.
  • F-BF.4d: In Algebra 2, Lesson 6.4, the materials restrict the domain of a non-invertible function to produce an invertible function.
  • G-SRT.9: In Geometry, Lesson 7.8, the materials derive the formula A=$$\frac{1}{2}$$ bc sin A on page 338.
  • G-SRT.10: In Geometry, Lesson 7.8, the materials prove the Law of Cosines and Law of Sines. Students use these laws in the Practice Problems for the lesson.
  • G-SRT.11: In Geometry, Lesson 7.8, students apply the Law of Cosines and Law of Sines to find unknown measurements in right and non-right triangles.
  • G-C.4: In Geometry, Lesson 8.1, the materials outline the steps for constructing a tangent line from a point outside a given circle to the circle and prove the construction on pages 357 and 358.
  • G-GPE.3: In Geometry, Lesson 11.5, the materials derive the equation for an ellipse given the foci, and in Geometry, Lesson 11.6, the materials derive the equation for a hyperbola given the foci.
  • G-GMD.2: In Geometry, Lesson 10.4, the materials state Cavalieri’s Principle, and students use it to find the volume of solids.
  • S-CP.9: In Geometry, Lesson 12.2, students use permutations and combinations to compute probabilities of compound events and solve problems.
  • S-MD.5a: In Geometry, Lesson 12.1, students have opportunities in the Practice Problem set and an online activity to find the expected payoff for a game of chance.
  • S-MD.5b: In Geometry, Lesson 12.1, Multi-Part Problem Practice 3, students evaluate whether to plead guilty or go to trial when given expected chances of going to prison for each outcome.
  • S-MD.6: In Geometry, Lesson 12.1, Model Problem 4, the probability to make a decision based on randomly flipping a coin versus using a deck of cards is considered when determining which high school in a town gets a new football field.
  • S-MD.7: In Practice problems in Geometry, Lesson 12.1 and Algebra 2, Lessons 10.6 and 10.7, students analyze decisions and strategies using probability concepts.

The following plus standards are partially addressed in the instructional materials:

  • N-CN.3: In Algebra 2, Lesson 2.5, students find the conjugate of a complex number. However, students do not use conjugates to find moduli and quotients of complex numbers.
  • F-IF.7d: In Algebra 2, Lesson 4.4, students graph rational functions as they transform reciprocal functions. While the materials have students identify asymptotes of a rational function, students do not factor in order to identify zeros and asymptotes of a given function. Furthermore, the end behavior for the parent function $$f(x)=\frac{1}{x}$$ is provided (page 205), and students do not determine end behavior for any other rational functions.
  • F-BF.5: In Algebra 2, Lesson 7.4, students solve exponential equations with logarithms on pages 316 and 317. However, neither the materials nor students explain the inverse relationship between exponential and logarithmic functions.
  • F-TF.3: In Algebra 2, Lesson 9.4, the materials use the unit circle to express the values of sine, cosine, and tangent for $$z$$, $$\pi-z$$, and $$\pi+z$$ in terms of given values for $$z$$, where $$z$$ is any real number. However, the materials do not express the values of sine, cosine, and tangent for $$2\pi-z$$.

The following plus standards are not addressed in the instructional materials:

  • N-CN.4
  • N-CN.5
  • N-CN.6
  • N-VM.1
  • N-VM.2
  • N-VM.3
  • N-VM.4a
  • N-VM.4b
  • N-VM.4c
  • N-VM.5a
  • N-VM.5b
  • N-VM.6
  • N-VM.7
  • N-VM.8
  • N-VM.9
  • N-VM.10
  • N-VM.11
  • N-VM.12
  • A-REI.8
  • A-REI.9
  • F-BF.4c
  • F-TF.4
  • F-TF.6
  • F-TF.7
  • F-TF.9
  • S-CP.8
  • S-MD.1
  • S-MD.2
  • S-MD.3
  • S-MD.4

Gateway Two

Rigor & Mathematical Practices

Not Rated

Criterion 2a - 2d

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.
0/8

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.
0/2

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.
0/2

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.
0/2

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.
0/2

Criterion 2e - 2h

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
0/8

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.
0/2

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.
0/2

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.
0/2

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.
0/2

Gateway Three

Usability

Not Rated

Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
0/8

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.
0/2

Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
0/2

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.
0/2

Indicator 3d

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
0/2

Indicator 3e

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
0/0

Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
0/8

Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
0/2

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.
0/2

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.
0/2

Indicator 3i

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.
0/2

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).
0/0

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.
0/0

Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.
0/0

Criterion 3m - 3q

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
0/10

Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.
0/2

Indicator 3n

Materials provide support for teachers to identify and address common student errors and misconceptions.
0/2

Indicator 3o

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.
0/2

Indicator 3p

Materials offer ongoing assessments:
0/0

Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
0/2

Indicator 3p.ii

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
0/2

Indicator 3q

Materials encourage students to monitor their own progress.
0/0

Criterion 3aa - 3z3ad

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
0/10

Indicator 3aa

0/

Indicator 3ab

0/

Indicator 3ac

0/

Indicator 3ac.i

0/

Indicator 3ac.ii

0/

Indicator 3ad

0/

Indicator 3r

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.
0/2

Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
0/2

Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
0/2

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).
0/2

Indicator 3v

Materials provide support for advanced students to investigate mathematics content at greater depth.
0/2

Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
0/0

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.
0/0

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
0/0

Indicator 3z

0/

Indicator 3z3ad

0/

Criterion 3aa - 3z

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
0/0

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.
0/0

Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
0/0

Indicator 3ac

Materials can be easily customized for individual learners.
0/0

Indicator 3ac.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.
0/0

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.
0/0

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.).
0/0

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.
0/0

Additional Publication Details

Report Published Date: Tue Jul 31 00:00:00 UTC 2018

Report Edition: 2016

Title ISBN Edition Publisher Year
AMSCO Algebra 1 Teacher Manual 978-0-7891-8916-5 Perfection Learning 2016
AMSCO Geometry 978-0-7891-8929-5 Perfection Learning 2016
AMSCO Geometry Teacher Manual 978-0-7891-8930-1 Perfection Learning 2016
AMSCO Algebra 1 978-1-62974-529-9 Perfection Learning 2015
AMSCO Algebra 2 Teacher Manual 978-1-63419-887-5 Perfection Learning 2016
AMSCO Algebra 2 978-1-68064-478-4 Perfection Learning 2016

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Math HS Rubric and Evidence Guides

The High School review rubric identifies the criteria and indicators for high quality instructional materials. The rubric supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our rubrics evaluate materials based on:

  • Focus and Coherence

  • Rigor and Mathematical Practices

  • Instructional Supports and Usability

The High School Evidence Guides complement the rubric by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

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