Holt McDougal Larson Traditional Series (2012)

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Report for High School

Alignment Summary

The instructional materials reviewed for the Larson 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 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.

High School

Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Alignment (Gateway 1 & 2)

Does Not Meet Expectations

Gateway 3

Usability

Usability (Gateway 3)

Not Rated

Overview of Gateway 1

Focus & Coherence

Gateway 1

v1.0

Gateway 1

Focus & Coherence

Criterion 1.1: Focus & Coherence

Does Not Meet 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 Larson Traditional Series do not meet the expectation 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 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, allowing students to fully learn each non-plus standard, and making connections within courses and across 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 Larson Traditional series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students.

The following are examples of standards that are fully addressed:

A-APR.3: In Algebra I, Chapter 8, Lessons 4 and 5 and Chapter 9, Lesson 2, Extension Activity, students identify the zeros of polynomials using the zero-product property and use the x-intercepts to write the quadratic function in factored form to graph the function. In Algebra II, Chapter 2, Lessons 5, 6, and 8, students work with quadratics, find the zeros through factoring and from a graph, and extend their knowledge to finding rational zeros through the use of The Rational Zero Theorem, The Remainder Theorem, and The Factor Theorem. Each of these work to scaffold student learning so that they are able to analyze graphs of polynomial functions using those zeros.
A-REI.4a: In Algebra I, Chapter 9, Lesson 5 and Algebra II, Chapter 1, Lesson 7, students solve quadratic equations by completing the square. In Algebra I, Derive the Quadratic Formula, page 619, students derive the quadratic formula.
F-BF.3: In Algebra I, Chapter 4, Lesson 1, Graphing Calculator Activity, students investigate linear equations and draw conclusions as to how various slopes and y-intercepts affect a linear function. Students explain how they found the answers and describe a process for finding an equation of a line that has a particular slope and passes through a specific point on page 231, Problems 9 and 10. In Algebra I, Chapter 5, Lesson 5, Extension Activity “Graph Absolute Value Functions,” students apply transformations to compare various graphs with the graph of the parent function, $f(x)=|x\vert$ . In Algebra I, Chapter 9, Lessons 1 and 5, students apply transformations to compare various graphs with the graph of the parent function $f(x)=x^2$ . In Algebra II, Chapter 3, Lesson 5, students apply transformations to compare various graphs with the graph of the parent function $f(x)=\sqrt{x}$ and $f(x)=\sqrt[3]{x}$ . In Algebra II, Chapter 4, Lessons 1 and 2, students apply transformations to compare various graphs with the graph of the parent function $f(x)=b^x$ . In each of these lessons, students are provided with a number of opportunities to engage in using graphing calculators to explore the mathematics.
G-SRT.8: In Trigonometric Ratios of Complementary Angles, Apply the Law of Sines, and Apply the Law of Cosines and Chapter 7, page 474, problem 41, students make a conjecture about relationships, make a table, compare values, look for patterns, and explore whether their conjecture is true for triangles that are not special right triangles.
S-ID.6a: In Algebra I, Chapter 4, Lessons 6 and 7, students use correlation coefficient values, data, and scatterplots to make a line of best fit using linear regression. In Algebra I, Chapter 4, page 284, Extension, students calculate and interpret residuals. In Algebra I, page 636, Graphing Calculator Activity, the students use the data sets to find linear, quadratic and exponential regressions. In Algebra II, Chapter 4, Lesson 7, students use exponential regression to find models. In Algebra I, page 282-283, in the Internet Activity and Extension, students model data and then plot and interpret the residuals to determine correlation. In Algebra I, Chapter 9, Lesson 8, students compare linear, exponential and quadratic models using ordered pairs and table of values and use that information to make predictions.

The following are standards where some aspect of the non-plus standard is not addressed in the instructional materials.

A-REI.5: In Algebra I, Chapter 6, Lesson 4, students solve linear systems by elimination and use linear systems to solve problems on topics such as investments, farm products, and music. On page 398, problem 42, students explain what the answers mean in the context of the problem; however the students are not given the opportunity 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.”
G-CO.7: In Geometry, Chapter 4, Lesson 3, Challenge Problems 18 and 19, students determine whether a rigid motion can move one triangle onto the other and justify their answer. The materials do not use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
S-ID.4: In Algebra II, Chapter 6, Lessons 3-5, students use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages through the problems and activities. Students do not "recognize there are data for which such a procedure is not appropriate," and students do not use spreadsheets to “estimate areas under the normal curve.”

The following standard was not addressed in the student or teacher materials.

G-SRT.1a: In Geometry, Chapter 6, Lessons 5 and 6, students work with dilations; however, the activity does not address a parallel line or a line passing through the center as indicated by the standard.

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 Larson Traditional series do not meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. Most aspects of the modeling process are present in isolation or combinations, but students do not have to revise their process and/or solution after interpreting their solution in the context of the problem. Opportunities for students to engage in the complete modeling process are absent for the modeling standards throughout the instructional materials of the series.

In the series, many of the real-world problems provide students with all of the needed information, including variables. Some questions ask students to determine the relationship between the variables while others ask the students to find a solution. The materials do not provide the opportunity to make assumptions about the real-world problem as part of the modeling process. Occasionally, students draw conclusions, make interpretations, or justify how they arrived at a solution. Examples of how students do not engage in the full modeling process include:

In Algebra I, Chapter 7, Lesson 4, problem 41, students create a model which is based on the data given about two trees. All of the information needed to write the equation is provided in the problem or on the labeled diagram.
In Algebra I, Chapter 9, Lesson 9, Challenge problem 12, students are provided with an approximate population, the growth rate, and the defined variables. The directions in the materials state that the data be represented graphically and interpreted. Students do not analyze, validate, or report the results.
In Geometry, Chapter 7, Lesson 5, the materials provide real-world scenarios but do not allow for the complete modeling process. The majority of these problems ask the students to find a height or distance, and all the variables are provided for the students. Page 466, problem 36 uses the context of an eye chart, but the eye chart scenario isn’t necessary to complete the problem. On page 466, problem 37, students are given the information about requirements for a wheelchair ramp.
In Geometry, Chapter 11, Lesson 7, problem 30, students find the volume of a small cone-shaped cup and a large cylindrical cup. Students also determine which container is a better buy based on their volume and the price. A diagram with the dimensions labeled is provided for the students, rather than allowing the students to create their own visual representation to support the modeling process. The volumes are computed based on given formulas without identifying the variables or drawing a model.
In Algebra II, Chapter 1, Lesson 5, problem 41, students create an equation to find the radius of a circular lot, solve for the radius, and generalize and justify the answers algebraically. Since each portion of the problem contains scaffolded questions for defining variables and determining which plan to follow or direction to take, students do not independently engage in the full modeling process.
In Algebra II, Chapter 4, Lesson 7, students write and apply exponential and power functions. On page 287, problem 34, students examine the relationship between the boiling point of water and atmospheric pressure by creating a scatter plot from given data, determining an equation from the given variables, and making a prediction. Students do not make assumptions about the given problem or determine the variables to be used, and results are not interpreted. On page 287, problem 35, students draw scatter plots, analyze the scatter plot to determine a function that would fit best, and create an equation based on the given variables. Variables are given, assumptions are not tested, and the interpretation is limited to the type of function that models the graph.

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

The materials allow students to spend the majority of their time on the WAPs except for those standards that were not adequately addressed as noted in indicator 1ai. There is time spent with standards and plus standards from Grades 6-8, but that time does not detract from the students spending the majority of their time on the WAPs.
In Algebra I, students spend the majority of their time working with WAPs from Number and Quantity, Algebra, and Functions. Some lessons from Chapters 1 and 2 address content that is aligned to 6.EE, 7.EE.1-3, and 8.EE.7, but the majority of the chapters in Algebra I do not include distracting or additional topics.
In Geometry, students spend the majority of instructional time on WAPs from the Geometry category. In Chapter 4, the materials address congruence (G-CO.A, B and G-CO.10) with a few references to rigid transformations in Lessons 4, 5, 6, and 9. In Chapter 5, Lessons 2, 4, 5, and 6, students prove theorems that address G-CO.9,10. In Chapter 6, the materials address some standards from G-SRT.A, and in Chapter 7, Lessons 3, 5, and 6 address standards from G-SRT.B, C.
In Geometry, Chapter 6, Lessons 1 and 2, students use the Pythagorean Theorem, and a proof of the Pythagorean Theorem is shown (8.G.B). In Geometry, Chapter 11, students solve volume problems involving 3-D figures (6.G.4, 7.G.3,6, and 8.G.9), but the inclusion of these topics from Grades 6-8 does not detract from the students spending the majority of their time on the WAPs.
In Algebra II, students spend the majority of their time working with WAPs from Number and Quantity, Algebra, and Functions. A-SSE is addressed multiple times in lessons from Chapters 1, 2, 4, and 7, and F-IF is addressed multiple times in lessons from Chapters 1, 2, 3, 4, 5, 7, 9, and 10. N-RN is addressed throughout Chapter 3.

Indicator 1B.ii

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

The instructional materials reviewed for the Larson Traditional series do not meet expectations, when used as designed, for allowing students to learn each non-plus standard fully. Standards which students do not get to learn thoroughly include:

A-APR.1: In Algebra 1, Additional Lessons 10 and 11 and Algebra II, Chapter 5, Lesson 5, Extension, students determine if a set is closed under an operation for integers and rational numbers. In Algebra I, Chapter 8, Lessons 1-3 and Algebra II, Chapter 2, Lesson 3, students practice operations on polynomials. However, within these lessons, students do not determine that polynomials are closed under the operations of addition, subtraction, and multiplication.
A-REI.6: In Algebra I, Chapter 6, Lessons 1-3, students solve systems of linear equations by graphing, substitution, and elimination. In each of the opportunities, students find exact solutions, but students do not have an opportunity to estimate a solution.
A-REI.10: In Algebra I, Chapter 3, Lesson 2, students graph linear equations. Students plot a few points and notice that the points “appear to lie on a line,” but students do not have the opportunity to show understanding that the graph of the equation is the set of all its solutions plotted in the coordinate plane.
F-IF.6: In Algebra I, Chapter 9, Lesson 9, Graphing Calculator Activity, page 645, students calculate the rate of change between two given points for a linear, quadratic, and exponential function, but there is no context to interpret this change. Students do not estimate the rate of change from a graph. In Algebra II, Chapter 5, Lesson 7, students find the average rate of change between several intervals for an exponential function and use that to determine whether the graph is increasing or decreasing, but students do not estimate from a graph.
F-LE.1: In Algebra I, Chapters 5 and 7 and Algebra II, Chapter 4, students address linear and exponential functions; however students do not compare linear and exponential functions throughout these chapters. Students do not have opportunities to distinguish between situations that can be modeled with linear functions from those that can be modeled with exponential functions.
F-LE.1a: In Algebra I, Chapter 4, Lessons 1 and 2, students write and use linear equations in slope-intercept form, but students do not prove that linear functions grow by equal differences over equal intervals. In Algebra I, Chapter 7, Lessons 4 and 5, students write and graph exponential functions, but students do not prove that exponential functions grow by equal factors over equal intervals.
G-CO.2: In Geometry, Chapter 4, Problems 14, 15, and 16, students decide whether the transformation to move triangle MNP onto triangle PQM is a translation, reflection, or rotation. On the Chapter 6 Test, Problems 10 and 11, students determine whether the given dilation is a reduction or an enlargement and find its scale factor. On the Chapter 9 Test, students compare transformations by completing problems that involve translations, reflections, and rotations. However, students do not describe transformations as functions that take points in the plane as inputs and give other points as outputs.
G-CO.4: In the Geometry materials, students do not get to develop their own definitions of translations as moving points of a figure along parallel lines, reflections as moving points along line segments that are perpendicular to the line of reflection, or rotations as moving points around circles by angles of given measures.
G-CO.8: In Geometry, Chapter 4, there is an Investigation after Lesson 6 that addresses constructing congruent triangles using the SSS and SAS criteria. Students are shown how ASA, SSS, and SAS can follow from the definition of congruence in terms of rigid motions, but students do not explain independently how the criteria follow.
G-SRT.1: In Geometry. Chapter 6 Investigation after Lesson 1, students explore the properties of dilations, but the students do not verify the properties of dilations given by a center and a scale factor.
G-SRT.2: In Geometry, Chapter 6, Lesson 2, students use dilations to show figures are similar in Problems 13-16; however, students do not explain using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles, and the proportionality of all corresponding pairs of sides.
G-SRT.7: In Geometry, Chapter 7, Lesson 6, problem 41, students make a conjecture about the relationship between sine and cosine values by creating a table and recognizing patterns. In Additional Lessons 2 and 3, the term cofunction is introduced, and students find the trigonometric values for complementary angles. Students do not use the relationship between the sine and cosine of complementary angles to solve problems.
G-GPE.7: In Geometry, Chapter 1, page 22, Problem 53 and page 50, Problem 8, students use coordinates to find the perimeter and area of a right triangle. There are no other opportunities for students to use coordinates to find perimeters of polygons and areas of triangles and rectangles.
S-ID.2: In Algebra I, Chapter 10, Lesson 2, the Extension introduces variance and standard deviation; however, the problems and questions include the calculation of standard deviation with one data set. On page 671, Problem 3, students calculate the mean, median, and mode of two data sets, but the problem does not expect students to use the shape of the data to compare the two data sets.
S-ID.5: In Algebra I, Chapter 10, Lesson 3, students create two-way frequency tables and interpret the data. Students do not analyze and recognize associations and trends within the data.
S-IC.6: In Algebra II, Chapter 6, Lesson 4, Problem 29, students examine a report from an election between Kosta and Murdock, determine it is reasonable to assume that Kosta will win the election, and explain their answer. In Lesson 5, students use a report to determine if the study described is a randomized comparative experiment. Students do not evaluate reports based on data as indicated by the standard.

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 Larson Traditional series partially meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials, at times, use age-appropriate contexts. However, some key takeaways from Grades 6-8 are not applied, and the materials do not vary the types of real numbers being used.

The materials provide a variety of problems within real-world contexts that are appropriate for high school students such as amusement parks, skateboard ramps, DVD players, sports, money, baking, video games, nutrition, and various job skills. Examples include the following:

In Algebra I, Chapter 6, Lesson 4, Problem 39, students use data from a table to create a system of equations and determine how many apple pies and batches of applesauce can be made if every apple is used.
In Algebra I, Chapter 9, Lesson 2, Problem 43, students use the equation of a parabolic arch in an aircraft hangar to determine how wide the hangar is at its base.
In Geometry, Chapter 1, Lesson 1, Problem 46, students identify points, lines, and planes in the context of different numbers of streets intersecting in a town to determine how many traffic lights would be needed.
In Geometry, Chapter 7, Lesson 1, Problem 34, students imagine there is a field in their town in the shape of a right triangle, find the perimeter of the field, and plant dogwood seedlings in the field at specified distances.
In Algebra II, Chapter 9, Lesson 4, Problem 39, the materials present a bike race where the bike passes an observer at 30 MPH. Students find the angle that the observer turns their head to see the cyclist t seconds later.
In Algebra II, Chapter 6, Lesson 4, Problems 30, 31, and 32, the contexts are a poll where 23% of the students surveyed say that math is their favorite subject in school, the number of voters who voted for candidate A or candidate B, and the number of people surveyed who prefer cola Y versus cola X. In these problems, students use statistical models, calculate the margin of error, and determine intervals that contain exact percentages.

Throughout the series, the majority of the problems utilize integers or simple rational numbers and do not vary the types of numbers being used. Students are provided few opportunities to practice with operations on a variety of rational and irrational numbers. Examples include the following:

In Algebra I, Chapter 2, Lessons 2-5 address solving equations, but the values within the equations are mostly whole numbers or simple rational numbers, such as $\frac{1}{2}$ or $\frac{1}{4}$ .
In Algebra I, Chapter 6 addresses solving systems of linear equations. The majority of coefficients are either whole numbers, simple fractions, or decimals to the hundredths place.
In Geometry, the majority of problems use whole numbers or decimals to the tenths place. In Chapter 4, Lesson 1, students use the Triangle Sum Property, and the majority of the angle values are whole numbers.
In Geometry, Chapter 6, Lesson 4, the triangle side lengths are predominantly whole numbers.
In Algebra II, Chapter 2, Lesson 8, students analyze graphs of polynomial functions that include whole number values when using a graphing calculator to find minimum and maximum values.
In Algebra II, Chapter 9, Lesson 6, Problems 43-46 include whole number values when using trigonometric ratios.

Some key takeaways from Grades 6-8 are not applied, and examples of this include the following:

In Algebra I, Chapter 2, Lessons 6 and 7 address ratios and proportional relationships, but the problems use cross products rather than applying the connections between ratios, proportional relationships, and linear functions from Grades 6-8.
In Algebra I, Chapter 3, students write and graph linear equations, but the examples and problems do not apply ratios or proportional relationships.
In Geometry, Chapter 6, Lesson 1, the materials address similarity through proportional relationships, ratios, and scale factors (7.G.1); however, the problems and examples do not apply key takeaways from either 7.G.1 or 8.G.A.
In Algebra II, Chapter 9 addresses trigonometric functions and does not make connections to ratios or apply key takeaways from 8.EE.B or 8.G.A.

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 Larson Traditional series do not meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials do not foster coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the Standards.

The Additional Lessons within each course address specific standards, but they are placed at the beginning of each course and not connected to any chapters or lessons within the course. There is also no clear indication of when or how the Additional Lessons are to be used within the series, which disrupts the coherence of the materials. Examples regarding the Additional Lessons are included in the evidence below.

Examples where the materials do not foster coherence by omitting appropriate and required connections within courses include:

The Algebra I, Additional Lessons 2 and 3 (S-ID.7), Interpreting Linear Models are not connected to Algebra I, Chapter 3, Graphing Linear Equations and Functions.
The Algebra I, Additional Lessons 10 and 11 (A-APR.1), Investigate Polynomials and Closure are not connected to Algebra I Chapter 8, Polynomials and Factoring.
The Algebra I, Additional Lessons 14 and 15 (A-SSE.3), Write Quadratic Equations are not connected to Algebra I, Chapter 9, Quadratic Equations and Functions.
The Algebra I, Additional Lessons 16 and 17 address S-ID.3, but they are not connected to Chapter 10, Data Analysis.
In Geometry, the materials address transformations in Chapter 9, but there are no connections to the congruence of triangles in Chapter 4 or reasoning and proof throughout the course.
The Geometry, Additional Lessons 2 and 3 (G-SRT.7), Trigonometric Ratios of Complementary Angles are not connected to Chapter 7, Right Triangles and Trigonometry.
The Algebra II, Additional Lessons 2 and 3, which address N-CN.2 are not connected to Chapter 1, Lesson 6, Perform Operations with Complex Numbers.
The Algebra II, Additional Lessons 4 and 5 (A-APR.4), Use Polynomial Identities are not connected to Algebra II, Chapter 2, Polynomials and Polynomial Functions.
In Algebra II, Chapter 3, Lesson 5, students graph square root and cube root functions as transformations of parent functions, but there is no reference made to quadratic functions previously graphed as transformations of a parent function from Algebra II, Chapter 1, Graphing is addressed differently in both chapters, and connections are not made between the two chapters to increase coherence within the course.

Examples where the materials do not foster coherence by omitting appropriate and required connections between courses include:

In Algebra I, Chapter 8, Lesson 6, students solve a quadratic equation by factoring. In Algebra II, Chapter 1, Lesson 3, students also solve quadratic equations by factoring, but there is no connection to prior learning from Algebra I. In Algebra II, Chapter 1, Lessons 6 and 8, students solve quadratic equations that involve complex numbers and use the discriminant to determine the number and type of solutions for a quadratic equation, respectively. However, the different forms of quadratic equations are not coherently connected across Algebra I and II or within the chapters of Algebra II.
Algebra I, Chapter 11 and Geometry, Chapter 12, both titled “Probability,” are identical. There is no connection between the chapters to build coherence between the courses.
The Algebra I, Additional Lessons 6 and 7, Use Inverse Functions are not connected to Algebra II, Chapter 3, Rational Exponents and Rational Functions.
The Algebra I, Additional Lessons 12 and 13 (F-BF.2), Translate Between Recursive and Explicit Rules for Sequences are not connected to Algebra II Chapter 7, Sequences and Series.
In Algebra II, Chapter 1, students graph from standard form in Lesson 1 and from vertex form in Lesson 2. There is no connection to Algebra I, Chapter 9, Lesson 1 where students graph as transformations (with no b value) or Algebra I, Chapter 9, Lesson 2 where students graph from standard form.

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 Larson Traditional series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The Plan and Prepare sections are included at the beginning of each chapter in order to assess, practice, and build on standards from previous grades, but in these sections, standards from Grades 6-8 are not explicitly identified.

A 4-year scope and sequence is provided that identifies skills and concepts to be taught in a Pre-Algebra course, but these skills and concepts are also not identified as standards from Grades 6-8. Many of the skills and concepts from the Pre-Algebra course are designated as Reinforce and Maintain under the Algebra 1 column of the scope and sequence document. Examples of these skills include: evaluate expressions with integer exponents, solve problems with proportional relationships, order of operations, 1-step, 2-step, and multi-step equations, ordered pairs, origin, axes, and graphing in four quadrants. Multiplying and dividing decimals by whole numbers, decimals by decimals, fractions by whole numbers, and fractions by fractions are also included.

Examples, where standards from Grades 6-8 are not identified, include:

In Algebra I, Chapter 1, Lessons 1 and 3, students evaluate and write expressions that align to 5.OA.1 and 6.EE.1,2, 6. For example, in Lesson 1, students evaluate “15x when x = 4, w - 8 when w = 20, and 5 + m when m = 7”. In Lesson 3, students write algebraic expressions given the following information: “8 more than a number x, the product of 6 and a number y, and the difference of 7 and a number n”.
In Algebra I, Chapter 1, Lesson 2, students apply order of operations that align to 6.EE.1 and 7.EE.1,2. For example, students evaluate “13 - 8 + 3, 8 - 22 and $3\cdot6-4$ ”.
In Algebra I, Chapter 1, Lesson 4, students write equations and inequalities that align to 6.EE.9, and 7.EE.4. For example, “The sum of 42 and a number n is equal to 51; the difference of a number z and 11 is equal to 35; and the product of 4 and a number w is at most 51”.
In Algebra I, Chapter 1, Lessons 7 and 8, students represent functions as rules, graphs, and tables that align to 8.F.1, 2.
In Algebra I, Chapter 2, Lesson 1, students find square roots and compare real numbers that align to 8.EE.2. A few of the problems include: “ $\sqrt{4}$ , $-\sqrt{49}$ and multiple choice, If x = 36, the value of which expression is a perfect square? A. $\sqrt{x}+17$ B. $87-\sqrt{x}$ C. $5\cdot\sqrt{x}$ D. $5\cdot\sqrt{x}+2$ ."
In Algebra I, Chapter 2, Lessons 2 and 3, students solve one-step and two-step equations that align to 6.EE.7 and 7.EE.4.
In Algebra I, Chapter 2, Lessons 4 and 5, students solve multi-step equations that align to 8.EE.7 and 7.RP.3.
In Geometry, Chapter 4, Investigating Geometry Activity before Lesson 1, students draw several triangles, tear off the corners of the triangles, and place the three angles from each triangle next to each other to form a straight angle (8.G.5).
In Geometry, Chapter 7, Lesson 1, students use the Pythagorean Theorem, apply it in real-world situations, and use it to find the distance between two points, which aligns to 8.G.B.

In the student materials, prerequisite content for each lesson is identified with “Before,” and content from within the lesson is identified with “Now.” Examples of the materials building on standards from Grades 6-8 through the “Before” and “Now” sections, even though the standards are not explicitly identified, include:

In Algebra I, Chapter 1, Lesson 1, Before: “You used whole numbers, fractions, decimals.” Now: “You will evaluate algebraic expressions and use exponents.”
In Algebra I, Chapter 1, Lesson 4, Before: “You translated verbal phrases into expressions” Now: “You will translate verbal sentences into equations or inequalities.”
In Geometry, Chapter 4, Lesson 1, Before: “You classified angles and found their measures.” Now: “You will classify triangles and find measures of their angles.”
In Geometry, Chapter 7, Lesson 1 Before: “You learned about the relationships within triangles.” Now: “You will find side lengths in right triangles.”

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 Larson Traditional series explicitly identify the plus standards in the Correlation to Standards for Mathematical Content at the beginning of each course. In some instances, the plus standards are fully addressed and coherently support the mathematics which all students should study in order to to be college and career ready, but for others, the materials do not fully address the plus standards.

The plus standards that are addressed include.

N-CN.8: In Algebra II, Chapter 2, Lesson 7, students are introduced to the Complex Conjugates Theorem and use it to write a polynomial function. Through error analysis problems and the assessment, students apply this standard in different situations.
N-VM.6: In Geometry, Chapter 9, Lesson 2, students add, subtract, and multiply matrices. Students also use matrices to represent and manipulate data in multiple contexts such as softball, computers, swimming, agriculture, and art.
N-VM.7: In Geometry, Chapter 9, Lesson 7, students use scalar multiplication to simplify a product. Students also use scale factors of 2, ½, 3, 6 to represent dilations.
N-VM.8: In Geometry, Chapter 9, Lesson 2, students add, subtract, and multiply matrices of appropriate dimensions.
N-VM.9: In Geometry, Chapter 9, Lesson 2, students determine when matrices cannot be multiplied based on the dimensions of the matrices. Students explore the Commutative Property of Multiplication, the Associative Property of Multiplication, and the Distributive Property for matrix multiplication. The three properties are each addressed in one problem.
A-APR.5: In Algebra II, Chapter 6, Lesson 1, students apply the Binomial Theorem and make connections to Pascal’s Triangle.
A-APR.7: In Algebra II, Chapter 5, Lessons 4 and 5, students add, subtract, multiply, and divide rational expressions.
F-BF.5: In Algebra II, Chapter 4, students understand the inverse relationship between exponents and logarithms and solve a variety of problems involving logarithms and exponents.
F-TF.7: In Algebra II, Chapter 10, Lesson 4, students evaluate and interpret trigonometric functions using technology in “Using Alternative Methods” on pages 642-643. Students also write and use the trigonometric functions in the context of a buoy’s displacement.
G-SRT.9: In Algebra II, Chapter 9, Lesson 5, students use the formula for the area of a triangle that includes the sine function to solve problems. On page 591 challenge problem 42, students derive the formula for the area of a triangle that includes the sine function.
G-SRT.11: In Algebra II, Chapter 9, Lessons 5 and 6, students apply the Law of Sines and Cosines to find unknown measures in both right and non-right triangles that are a part of various real-world situations.
G-C.4: In Geometry, Chapter 10, Lesson 4, students construct tangent lines from a point outside a given circle to the circle.
G-GPE.3: In Algebra II, Chapter 8, Lesson 4, students write the equation of an ellipse using the foci in challenge problem 47. In Chapter 8, Lesson 5, students write the equation of a hyperbola in standard form using the distance formula, the foci, and the difference in the distance from a point on the hyperbola to the foci in challenge problem 37.
S-CP.8, 9: In Algebra I, Chapter 11, Lesson 2, students use factorials and permutations to determine the number of ways letters can be arranged and how many different ways six friends can sit together in a row of six empty seats at a movie theater. In Lesson 3, students examine various scenarios to determine whether to use a permutation or a combination. In problem 26, a teacher is going to choose two students to represent a class, and students calculate the probabilities of you and your best friend being chosen and you being chosen first and your best friend being chosen second. In Lesson 5, students find probabilities of independent and dependent events and use conditional probability in a variety of ways. Students are provided two pieces of information and asked to find the missing probability if the events are independent. Students then complete a similar problem for dependent events.

The plus standards that are partially addressed include:

N-CN.3: In Algebra II, Chapter 1, Lesson 6, students are introduced to complex numbers and complex conjugates, and in Chapter 2, Lesson 7, students find the conjugate of a complex number. Students do not use conjugates to find moduli and quotients of complex numbers.
N-CN.4: In Algebra II, Chapter 1, Lesson 6, students represent complex numbers on the complex plane in rectangular form. Students do not represent complex numbers on the complex plane in polar form, and students do not explain why the rectangular and polar forms of a given complex number represent the same number.
N-CN.5: In Algebra II, Chapter 1, Lesson 6, problems 70-73, students verify that the given properties extend to complex numbers, the commutative property of multiplication, the distributive property, the associative property of multiplication, the commutative property of addition and the associative property of addition. Students do not connect to the modulus or degree arguments.
N-CN.6: In Algebra II, Chapter 1, Lesson 6, students read Absolute Value of a Complex Number in a Key Concept box and find the absolute value of various complex numbers and sums of two complex numbers. Students do not find the midpoint of the segment at any time.
N-CN.9: In Algebra II, Chapter 2, Lesson 7, students are introduced to the Fundamental Theorem of Algebra and use it to write the number of solutions, zeros, and equation for polynomial functions. Students do not make these connections for quadratic functions.
N-VM.12: In Geometry, Chapter 9, Lesson 3, students use 2x2 matrices for transformations in the plane, but students do not interpret the absolute value of the determinant in terms of area.
F-TF.3: In Algebra II, Chapter 9, Lesson 3, students use special triangles to determine values of sine, cosine, and tangent. On page 571, the materials demonstrate using the unit circle to evaluate trigonometric functions when $\Theta=270\degree$ . Students practice this skill in four problems, but students do not use the unit circle to express the values of sine, cosine, and tangent for $\pi-x$ , $\pi+x$ , and $2\pi-x$ , in terms of their values for $x$ .
F-TF.4: In Algebra II, Chapter 10, the period of each trigonometric function is explained using the graph of the functions, but there are no examples or problems that use the unit circle to explain the periodicity or symmetry of trigonometric functions.
F-TF.6: In Algebra II, Chapter 9, Lesson 4, students evaluate inverse trigonometric functions, but students do not restrict the domain of a trigonometric function so that its inverse can be constructed. Students are shown the inverse functions graphically, and the materials state that “domain restrictions allow the inverse sine, inverse cosine, and inverse tangent functions to be defined.”
F-TF.9: In Algebra II, Chapter 10, Lesson 6, students are given the sum and difference formulas and then evaluate, rewrite in equivalent forms, and solve trigonometric equations. There is not a proof of the formulas or an opportunity for students to prove the formulas.
G-SRT.10: In Algebra II, Chapter 9, Lesson 6, students derive the Law of Cosines on page 597 in challenge Problem 42, but the proof of the Law of Sines is not found.
G-GMD.2: In Geometry, Chapter 11, Lesson 6, students use Cavalieri’s principle to give an informal argument for the volume of cylinders but not for other solids. The volume of a sphere is addressed in Lesson 8, but Cavalieri’s principle is not used in that lesson.
S-MD.1: In Algebra II, Chapter 6, Lesson 2, random variables are defined for a quantity of interest and graphed using histograms, but other types of graphical displays for probability distributions are not used.
S-MD.3, 4: In Algebra II, Chapter 6, Lesson 2, students construct and interpret binomial distributions and classify the distributions as symmetric or skewed. Students do not find the expected values of the distributions.
S-MD.6, 7: These standards are present in multiple locations throughout the series (Algebra II, Additional Lessons 10 and 11, Algebra I, Extension, pages 743-744, and Geometry, Extension, pages 847-848). However, since the materials include the same lesson in Algebra I, Algebra II, and Geometry, students use probabilities to make fair decisions once. Furthermore, Algebra I page 711, Problem 20 is repeated in Geometry, page 815, Problem 20.

The following plus standards are not addressed in the materials: N-VM.1, N-VM.2, N-VM.3, N-VM.4, N-VM.5, N-VM.10, S-MD.2, S-MD.5, A-REI.8, and A-REI.9.

Criterion 1.1: Focus & Coherence

Overview of Gateway 2

Rigor & Mathematical Practices

Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One

Gateway 2

v1.0

Gateway 2

Rigor & Mathematical Practices

Criterion 2.1: Rigor

Criterion 2.2: Math Practices

Not Rated

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.

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.

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.

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.

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.

Criterion 2.2: Math Practices

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

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.

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.

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.

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.

Criterion 2.2: Math Practices

Overview of Gateway 3

Usability

This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two

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

Not Rated

Criterion 3.1: Use & Design

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

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.

Indicator 3B

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

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.

Indicator 3D

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

Indicator 3E

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

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.

Indicator 3F

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

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.

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.

Indicator 3I

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

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).

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.

Indicator 3L

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

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.

Indicator 3M

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

Indicator 3N

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

Indicator 3O

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

Indicator 3P

Materials offer ongoing assessments:

Indicator 3P.i

Assessments clearly denote which standards are being emphasized.

Indicator 3P.ii

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

Indicator 3Q

Materials encourage students to monitor their own progress.

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

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

Indicator 3R

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

Indicator 3S

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

Indicator 3T

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

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).

Indicator 3V

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

Indicator 3W

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

Indicator 3X

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

Indicator 3Y

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

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.

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.

Indicator 3AB

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

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.

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.).

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.

Criterion 3.5: Technology Use

Downloadable Resources

2012