The instructional materials reviewed for the CORD Traditional series, when used as designed, partially meet expectations for spending the majority of time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs and careers. Overall, the materials spend a majority of the time on non-plus standards from CCSSM. However, the instructional materials do not spend a majority of time on the WAPs, and some of the remaining materials address prerequisite or additional topics that are distracting.

Examples where students are distracted from the WAPs by engaging in prerequisite materials from earlier grades include:

• In Algebra 1, Chapter 1, the majority of the lessons (according to the provided pacing guide) address the real number system, operations with real numbers, absolute value, expressions and equations, and unit analysis. These lessons align with standards from Grades 6-8.
• In Algebra 1, Chapter 2, the majority of the lessons address properties of addition, subtraction, multiplication and division, and solving proportions and percent equations, multi-step equations, and equations with variables on both sides. These lessons align with standards from Grades 6-8.
• Algebra 1, Lesson 3.1 addresses solving inequalities that align to 6.EE.8 and 7.EE.4b.
• Algebra 1, Lessons 7.2 and 7.3 address the properties of exponents for simplifying expressions and simplifying expressions containing zero and/or negative exponents which align to 8.EE.1.
• In Algebra 1, Chapter 11, the majority of the lessons address: finding mode, median, and mean of a set of data; determining which measure of central tendency best describes a data set; organizing data into a frequency distribution and histograms; creating and interpreting dot plots and boxplots; calculating measures of dispersion; and creating and interpreting two-way frequency tables. Measures of center and displaying numerical data in plots align with 6.SP.4,5.
• Geometry, Lesson 1.1 addresses drawing points, lines, line segments and rays which aligns with 4.G.1, and Lessons 1.2 and 1.3 address measuring line segments and angles using a protractor with all measures in whole-number degrees which align to 4.MD.6.
• Geometry, Lessons 9.1, 9.2, 9.3, and 9.5 address areas of squares, rectangles, irregular figures, parallelograms, triangles, rhombuses, trapezoids, and the circumference and area of circles which align to 6.G.1,3 and 7.G.4.

Examples where students are distracted from the WAPs by engaging in topics outside the CCSSM include:

• Geometry, Lessons 2.1 through 2.5 address patterns, inductive reasoning, deductive reasoning, converse, inverse, contrapositive, valid arguments, and types of proofs. These lessons are not aligned to any standards from CCSSM.
• Geometry, Lesson 3.2 addresses vectors on a coordinate plane which aligns to plus standards, N-VM.A.
• Geometry, Lesson 4.6 addresses tessellations which does not align to any standards from CCSSM.
• Geometry, Lesson 11.2 addresses perspective drawing which does not align to any standards from CCSSM.
• Algebra 2, Lesson 10.6 addresses the secant, cosecant, and cotangent reciprocal identities, and Lesson 8 addresses double-angle and half-angle identities. These identities are not aligned to any standards from CCSSM.
• Algebra 2, Lesson 12.1 addresses distance and midpoint formulas, and Lesson 2 addresses general conic sections. These topics do not align to any standards from CCSSM.

The instructional materials reviewed for the CORD Traditional series partially meet expectations, when used as designed, for letting students fully learn each non-plus standard. Overall, students are frequently not given the opportunity to develop their own definitions, and where the standards expect students to prove or develop a concept, the materials often provide students the information.

The following are some examples of how the series instructional materials, when used as designed, do not enable students to fully learn some of the non-plus standards. (Those standards that were not attended to by the materials, as noted in indicator 1ai, are not mentioned here):

• N-Q.1: In Algebra 1, Lessons 11.2, 11.4, and 11.5, students do not choose and interpret the scale and the origin in graphs and data displays; rather, students are provided with pre-labeled tables or graphs with pre-determined scales making the quantities that they represent obvious to the student.
• N-Q.3: In Algebra 1, Lessons 1.7, 1.8, and Math Labs, students have few opportunities to choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
• A-REI.5: In Algebra 1, Lessons 6.4 and 6.5, students solve systems of linear equations, but students do not have the opportunity, in this chapter or the rest of the materials, to prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
• F-IF.3: In Algebra 1, Lesson 5.6 and Lesson 9.7, students write explicit and recursive functions for sequences, but students do not recognize that the domain of a sequence is a subset of the integers.
• F-IF.7b: In Algebra 1, Lessons 5.1-5.5 and 8.2 and Algebra 2, Lessons 3.1-3.5, 5.1-5.2, 7.4, 10.2-10.4, students graph square root and piecewise-defined functions but do not have an opportunity to graph cube root functions in the materials.
• F-IF.8b: Students use exponential functions to solve problems, but students do not interpret the relationship between the growth and decay factor and the exponent in the function. Students do not use the properties of exponents to write equivalent exponential expressions with different growth/decay factors.
• F-LE.1a: In Algebra 1, Lesson 9.6, students read that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals, but there are no opportunities for students to prove these things on their own.
• G-CO.6: Geometry, Lesson 5.4 states two triangles are congruent if one triangle can be carried onto the other triangle by a rigid motion. There are limited opportunities for students to use geometric descriptions of rigid motions to transform figures and predict the effect of a given rigid motion on a given figure.
• G-CO.7: Geometry, Lesson 5.4, Activity 1, Identifying Congruent Triangles uses the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding sides and corresponding pairs of angles are congruent. However, there are no opportunities for students to use the definition of congruence in terms of rigid motions to show that two triangles are congruent.
• G-CO.8: In Geometry, Lesson 5.5, students are given the criteria for triangle congruence and to identify the triangle congruence; however, there is no opportunity for students to explain how the criteria follow from the definition of congruence in terms of rigid motions.
• G-CO.9,10: In Geometry, Lessons 2.7, 2.8, 5.1-5.8, 6.2, and 6.3, many of the proofs listed in these standards are completed in the examples. There are no opportunities for students to prove the theorems or demonstrate that they understand how to complete the proofs given.
• G-CO.12: Students are given opportunities to create the constructions listed within this standard; however, all constructions are created using a compass and straightedge. No other tools or methods, such as using string, reflective devices, paper folding, dynamic geometric software, etc., are used.
• G-SRT.1b: In Geometry, Lesson 4.7, students are provided one example and one activity to verify experimentally the dilation of a line segment being longer or shorter. There are no other opportunities in the materials for students to develop their knowledge of this standard.
• G-SRT.6: In Geometry, Lesson 7.4, students are given several opportunities to work with tangent ratios. Lesson 7.5 defines the sine and cosine ratios, but students do not have an opportunity to use similar triangles to understand that the sine and cosine ratios are properties of the acute angles in the right triangle.
• G-SRT.7: In Geometry, Lesson 7.5, students are given the relationship between sine and cosine. In Problem 24a, students prove sin A = cos(90 - m∠A). This standard is not revisited for practice or application in the remainder of the materials.
• G-C.1: In Geometry, Chapter 10, Math Lab 1, Activity 1, students use step-by-step procedures to determine that all circles are similar by showing that the ratios of the lengths of the circumferences to the lengths of the radii are proportional. Students do not formally prove that all circles are similar.
• G-C.5: In Geometry, Lesson 10.3, students are given a description of arc length and a procedure to calculate it, but similarity is not addressed. In Problem 4, students explain how to find the area of a sector of a circle, but students are not required to derive the formula. In Algebra 2, Lesson 10.2, students have no opportunities to discover or define radian.
• G-GPE.1: In Geometry, Lesson 10.1, students use the distance formula to derive the equation of a circle given center and radius. Students have no opportunities to find connections between the Pythagorean Theorem and the distance formula.
• G-GPE.5: Students make a conjecture about slopes for parallel and perpendicular lines, but they do not prove the criteria as addressed in the standard.
• G-GMD.4: In Geometry, Lesson 11.10, students identify the two-dimensional cross sections of various three dimensional objects. In Activity 2, students rotate a line at a 30 degree angle around a vertical line to create a three-dimensional object. There are no other opportunities for students to generate three-dimensional objects from rotations of two-dimensional objects.
• S-ID.4: In Algebra 2, Lesson 11.5, students are given limited opportunity to use tables and no opportunities to use spreadsheets to estimate areas under the normal curve as described in the standard.
• S-ID.6a: In Algebra 1, Lesson 4.8, students calculate least squares regression lines and use the lines to solve problems in the context of given data. Students are not given opportunities to fit quadratic or exponential functions to a set of data and use those functions to solve problems in the context of given data.
• S-ID.7: In Algebra 1, Chapter 4, students identify but do not interpret the slope of a linear model in the context of the data. In Math Applications, Problem 9a, students state the slope of the graph but do not interpret it. In Problem 11b and c, students state the slope but do not interpret it, and they do interpret the y-intercept.
• S-CP.6: In Geometry, Lesson 12.4, students find the conditional probability as a ratio of probabilities. Evidence was not found where students are required to interpret a conditional probability in terms of the given model.
• S-CP.7: In Geometry, Lessons 12.2 and 12.4, students apply the Addition Rule in multiple problems. However, evidence was not found where students are required to interpret the answer in terms of the given models.

The instructional materials reviewed for the CORD 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. Materials do not always vary the types of real numbers being used, and some of the key takeaways from Grades 6-8 are not applied.

Each chapter has one section called MathLab which contains problems that engage learners in the mathematics through contexts where students take measurements and/or interpret data given a set of constraints. Contexts include a variety of different levels of interests:

• Algebra 1, Chapter 1, Math Lab 1: Comparing Pulse Rates
• Algebra 1, Chapter 5, Math Lab 1: Price and Size of Pizza
• Geometry, Chapter 3, Math Lab 1: Two-Dimensional Coordinates with Stairs
• Geometry, Chapter 6, Math Lab, Activity 1: Indirect Measurement of Height
• Algebra 2, Chapter 3, Math Lab 1: Calculating the Value of a Used Car
• Algebra 2, Chapter 8, Math Lab 2: Man Your Battle Stations

The instructional materials do not vary the types of real numbers being used. For example:

• Algebra 1, Lessons 6.3 and 6.4 address solving systems of equations, and the practice problems for solving by substitution use whole number coefficients. Within Lesson 6.3, all of the solutions are integers. Within Lesson 6.4, the coefficients of the practice and problem-solving questions are integers; there is one fractional solution, but there are some decimal solutions on the extra practice worksheet.
• In Algebra 1, Chapter 7, the coefficients in polynomials are mostly integers between -10 and 10.
• In Geometry, Chapters 9 and 11, the majority of the given dimensions that students use to find the area, surface area, and volume of two- and three-dimensional shapes are positive integers.
• In Algebra 2, Chapter 1, almost all of the solutions to the equations are integers.

Below are examples of how key takeaways from Grades 6-8 are not applied within the materials:

• In Algebra 1, Lesson 6.1, students are given step-by-step instructions to solve a system of equations (8.EE.8) by graphing and creating a table. The materials do not have students apply this key takeaway from Grades 6-8 within the materials.
• In Algebra 1, Chapter 11, examples show how to create histograms, box plots, and dot plots and to calculate measures of center (6.SP.B) without applying key takeaways from Statistics and Probability in Grades 6-8.
• In Geometry, Lesson 3.1, students find a point on a directed line segment which divides the segment by a given ratio. The materials give the formula for finding the midpoint of a segment, but students do not apply the key takeaway from ratios and proportional relationships in Grades 6 and 7 (6.RP.A and 7.RP.A).
• Geometry, Chapter 4 materials address reflections, translations, rotations, and dilations (8.G.A) without applying key Geometry takeaways from Grades 6 through 8.

The instructional materials reviewed for the CORD Traditional series partially meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials partially foster coherence through meaningful mathematical connections in a single course and throughout the series where appropriate and where required by the CCSSM.

In each course, chapters include lessons that present mathematical concepts or skills but few connections are made between them. Connections between clusters and domains of standards are not always made for the teacher or student, and these missing connections decrease the coherence of the materials across courses.

The instructional materials miss opportunities to make coherent connections between courses throughout the series. For example:

• Materials address Statistics and Probability at the end of each course in isolation from the rest of the topics within each of the courses. Algebra 1 contains calculation of descriptive statistics and linear regression. Geometry focuses on probability, in particular, probabilities of conditional and compound events. Algebra 2 addresses probabilities of compound events and data distributions. These chapters addressing Statistics and Probability do not make connections to each other or to other chapters within each course.
• The Geometry standards are predominantly addressed in the Geometry course with minimal connections to the Geometry standards occurring in the other courses. Conversely, the Geometry course is largely devoid of connections to Number and Quantity, Algebra, and Functions. For example, there are no connections to quadratic functions in Geometry. There is no connection between an area model and a quadratic function for finding a maximum area of a shape within the Geometry course.
• Algebra 2, Chapters 3, 5, 6 and 7 have no meaningful connections made between these chapters in order to increase coherence across the course and develop students’ understanding of functions.

Some examples of how the instructional materials display coherence between courses and provide opportunities for students to make meaningful connections are as follows:

• In Algebra 1, Chapter 8, students solve quadratic equations through a number of methods. In Algebra 2, Chapter 4, students use those methods to solve quadratic equations with no real solutions.
• In Algebra 1, Chapter 3, The Addition Property of Inequality Start-Up states: “Students already will know how to solve equations using addition and subtraction. All they need to know in order to succeed is to bring down the inequality symbol in each step of their solution.” In Algebra 1, Chapter 8, The Quadratic Formula Start-Up states: “Point out to students that, like completing the square, the quadratic formula can be used to find exact solutions to any quadratic equation.”
• Algebra 1 starts with linear equations, moves to systems of linear equations, extends to monomials with some factoring of trinomials, and then quadratics. This sequence of topics makes meaningful connections to provide coherence throughout the Algebra I course.
• The Geometry course progresses from triangle congruence in Chapter 5 to similarity in Chapter 6. In Chapter 7 right triangle trigonometry meaningfully connects to triangle similarity criteria to provide coherence across these three chapters.