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)

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.