The instructional materials reviewed for HMH Into AGA meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials address all aspects for most of the non-plus standards across the courses of the series. Examples of non-plus standards addressed by the series include:

• A-SSE.1a: In Algebra 1, Lesson 2.1 defines terms, factors and coefficients. In the three tasks, students explore what variables, terms, and expressions represent in situations. For example, in Task 2, students explain the reason 44 - a is an expression for the weight of grapes. In Algebra 2, Lesson 4.4, students write an equation for volume and identify the real-world meaning of each factor in an example using the volume of ice sculptures.
• A-REI.4b: In Algebra 1, Lessons 17.2 and 17.3, students solve quadratic equations by factoring; in Lesson 18.1, students solve quadratic equations by inspection with square roots; in Lesson 18.2, students complete the square; and in Lesson 18.3, students use the quadratic formula. In Algebra 2, Lesson 2.3, students solve quadratic equations using completing the square and the quadratic formula and find the value of the discriminant to tell the number and type of solutions. They also solve quadratic equations with complex roots in this same lesson.
• F-BF.4: In Algebra 2, Lesson 7.1, students find the inverse of a function. In Task 3, students explore the relationship between the distance an image is sent from an orbiting spacecraft and the number of seconds since the message has been sent. Students see the inverse of the function D(s) is s(D).
• F-IF.2: Students use function notation throughout the materials. In Algebra 1, Lesson 4.1, students use multiple versions of function notation to become accustomed to the variety of forms it can take. Students use h(t), g(x), R(b), A(t) and S(p) in various tasks to meaningfully represent the problems they are addressing.
• F-IF.4: In Algebra 1, Lesson 19.2, students determine the axis of symmetry, vertex, maximum or minimum, and intercepts. In Lesson 19.3, students use quadratic functions situated in various real-world contexts to identify and interpret x-intercepts and vertices. In Lesson 20.4, students use a graphing calculator to find x-intercepts, end behavior, turning points, and intervals over which cubic functions are positive and negative. In Algebra 2, Lesson 16.3, students use the key features of periodicity and maximum and minimum to write trigonometric functions. 
• G-CO.12: Compass and straightedge constructions are included throughout the Geometry materials. In Lesson 1.1, students construct a midpoint. In Lesson 1.2, students copy and bisect an angle. In Lesson 3.2, students construct parallel lines through a point. In Lesson 3.3, students construct a perpendicular bisector and a perpendicular line through a point.
• G-SRT.6: In Geometry, Lesson 13.1, students analyze the relationships of opposite and adjacent side lengths of two right triangles that have the same acute angle. They also use tracing paper to investigate the relationships of opposite and adjacent side lengths of two right triangles that have the same acute angle, defining the tangent ratio. In Lesson 13.2, students use drawing tools to create right triangles, examine side ratios, and discover the sine and cosine ratios.
• S-ID.8: In Algebra 1, Lesson 6.2, students use a spreadsheet to create a table of data and residuals. They also use a graphing calculator to find the equation of the line of best fit and correlation coefficient, which is interpreted for linear models. 

The non-plus standards that are not addressed, or partially addressed, include:

• G-GPE.2: This standard was completely omitted from the materials.
• S-IC.4: The materials do not develop a margin of error through the use of simulation models for random sampling.

The instructional materials reviewed for HMH Into AGA partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards.

Each module within the units has a Performance Task, and each unit has a project which is accessible in the digital materials. The Performance Tasks are designed as either Spies and Analysts or STEM Tasks. Spies and Analysts tasks are described by the publisher as Mathematical Modeling in the Planning and Pacing Guide on page 17. Spies figure out the information needed and the process of obtaining that information. Analysts decide what is important to use and create a mathematical model. The Planning and Pacing Guide further states that Spies and Analysts must determine whether the model they created is good. This idea of validating is not stated in the Teacher Edition pages where the Module Performance Task is located. In most Spies and Analysts tasks, the student materials contain one question for them to investigate. Through these specific Spies and Analysts prompts, students work through the entire modeling process; however, the modeling content standards that students use to work these problems are limited.

The instructional materials omit the full intent of the modeling process for more than a few modeling standards across the courses of the series. Examples where the materials include various aspects of the modeling process in combinations without using the full intent of the modeling process include:

• N-Q.3; F-IF.5,6: In Algebra 2, Unit 1, Project Infection Detection, Part 1, students identify a suitable domain for a function that represents the approximate number of new HIV infections (in millions) for the world per year. Students identify the range associated with the identified domain and explain their reasoning. In Part 4, students use two functions to determine when the number of people receiving treatment in South Africa will be equal to the number of new HIV infections in the world. Students are not given the opportunity in this project to interpret, validate, or report on their model. 
• N-Q.1; A-SSE.3c: In Algebra 1, Unit 5, Project Oh, How We Grow, an industrial engineer is hired to optimize the process of getting products out the door and onto the shipping trucks. Students calculate yearly and monthly rates of increase in production using a given model. Students do not make assumptions, choose their own variables, compute, validate and interpret the given model. 
• A-CED.4: In Algebra 1, Lesson 2.3, Spark Your Learning, students determine if a cheetah will catch a gazelle. Students formulate, make assumptions to model the situation, and compute to find the variable of interest. Students do not validate, interpret and report on their model.
• A-REI.11: In Algebra 2, Lesson 10.3, Spark Your Learning, students determine how much energy is released by an earthquake of a given magnitude. Students define the variables and unit of measurement. Students compute by using a table to understand solving logarithmic equations and determine an equation. Students do not determine their own model, validate, interpret and report on their model.
• F-IF.4: In Algebra 2, Unit 2, Project Network Functions, Part 1, students sketch a graph of U(t) and describe the characteristics of the graph. Students do not make assumptions, create their own model, interpret, validate, and report on the model. 
• F-IF.7c: In Algebra 2, Lesson 3.2, Spark Your Learning, What is the maximum volume the box can have?, students compute using a table or graph the equation to find the maximum volume. Students determine if there are any limits on the possible dimensions of the corner cut-outs. Students do not create their own model, interpret, and report on their model. 
• F-LE.1c: In Algebra 1, Lesson 13.2, Problem 18, students complete a table to determine if the total number of comic books sold for several years is represented better by a linear or exponential model. Students answer, “What information can you use to create a model?” and “Determine the model.” Students do not compute, interpret, validate, and report on their model.
• F-LE.3: In Algebra 1, Lesson 13.2, Spark Your Learning, How can you model each training schedule given?, students formulate a model. Students compute by writing the function equation or using a table. Students interpret the two models to see if one runner can ever surpass the number of miles the other runner runs a week. Students do not make assumptions, validate, or report on their model.
• F-LE.4: In Algebra 2, Lesson 10.2, Spark Your Learning, students determine the position of a key played on a piano if the frequency is known. Students formulate an exponential function and find limitations on the domain of the function. Using the exponential function, students compute the position by graphing and finding the point of intersection. Students see if their answer makes sense in the context of the situation. Students do not make their own assumptions, define their own variables, and report on their model.
• F-LE.5: In Algebra 2, Lesson 8.2, Problem 26, students are given a function to model Newton's law of cooling. Students write an equation for the temperature of tea as a function of time and determine the time the tea will take to cool. The students also graph the function to determine when the tea will cool to room temperature. Students do not make assumptions, determine their own model, compute and interpret. 

The instructional materials reviewed for HMH Into AGA meet expectations for, when used as designed, spending the majority of time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs and careers. Examples of the ways the materials allow students to spend the majority of their time on the WAPs include:

• N-RN.1: In Algebra 1, Lesson 1.2, Task 3, students discover that the properties of integer exponents can be extended to rational exponents. In Algebra 2, Lesson 6.1, Task 1, students extend the definition of integer exponents to define an exponent in terms of a root. In Step it Out of the same lesson, students translate between rational exponents and radical expressions.
• A-SSE.A: In Algebra 1, Lesson 2.1, students write, interpret, and simplify linear expressions in one variable.
• A-SSE.B: In Algebra 1, Lessons 19.1-19.3, students write quadratic equations in different forms to find x-intercepts, maxima, minima, and vertices. In Algebra 2, Lesson 2.3, students complete the square to find complex solutions.
• A-CED.A: In Algebra 1, Lessons 2.2 and 2.4, students solve linear equations with grouping symbols and variables on both sides. In Algebra 2, Lesson 2.1, students write and solve quadratic equations.
• A-REI.7: In Algebra 1, Lesson 20.3, Task 1, students use a graphing calculator to translate y = 2x and y = x$$^2$$ to determine linear-quadratic systems with 2, 1, and 0 solutions. In the Check for Understanding of the same lesson, students use a graphing calculator to solve a system that includes a linear equation and a parabola. In Algebra 2, Lesson 2.4, students solve linear and quadratic systems using substitution, elimination, and graphs.
• G-CO.1:  In Geometry, definitions for lines, angles, and circles are introduced in the Build Understanding of Lessons 1.1, 1.2, 2.1, 3.3, and 15.1.
• G-SRT.B: In Geometry, students prove theorems including the triangle proportionality theorem in Lesson 12.3 and the triangle congruence theorem in Lesson 12.2.

The instructional materials reviewed for HMH Into AGA partially meet expectations for, when used as designed, allowing students to fully learn each non-plus standard. Examples of the non-plus standards that would not be fully learned by students include:

• A-SSE.4: In Algebra 2, Lesson 13.3, students use the formula for the sum of a finite geometric series to solve problems. The derivation is provided in this lesson, so students do not have the opportunity to derive the formula.
• A-REI.4a: In Algebra 1, Lesson 18.2, students solve equations by completing the square. In Algebra 1, Lesson 18.3, the derivation of the quadratic formula by completing the square is provided. In Algebra 2, Lesson 2.3, the materials state "You can use completing the square to derive the Quadratic Formula." The derivation is then given step-by-step without any student input, so students are not provided the opportunity to derive the quadratic formula.
• A-REI.5: In Algebra 1, Lesson 9.4, students solve systems of equations by elimination. However, students do not 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.
• A-REI.11: In Algebra 1, Lesson 8.3, Problems 18-21, students use the graph of an absolute value function and a constant function to solve each equation or inequality. Students explain how the x-coordinates of the intersection points relate to the solutions of the equation or inequality. In Algebra 1, Lesson 12.2, Problem 1, students explain by using a graph of two exponential functions at what values f(x) = g(x). There are a limited number of opportunities for students to fully learn this standard. Additionally, in Algebra 2, Module 10, Review, Question 16, an exponential expression is set equal to a constant expression. The materials state, "To estimate by graphing, what two equations should she graph together on her graphing calculator?" and "Estimate the solution to the nearest hundredth using a table."
• F-LE.1a: In Algebra 1, Lesson 13.2, Task 2, students complete a proof that linear functions change by equal differences over equal intervals by filling in three missing reasons. In Task 3 of the same lesson, students complete a proof about exponential functions by filling in two missing reasons. Students do not prove that linear functions grow by equal differences over equal intervals and exponential functions grow by equal factors over equal intervals.
• F-TF.8: In Algebra 2, Lesson 15.3, Task 1, students justify given reasons for the proof of the Pythagorean Identity, but do not complete the proof on their own. In Algebra 2, Lesson 15.3, students calculate values for sine and cosine. In the Review for Module 15, there are two instances where the value of tangent is calculated, Problems 13 and 14. Students have a limited number of opportunities to learn how to use the Pythagorean Identity to find tangents of angles.
• G-CO.9: Students do not prove several theorems about lines and angles due to the amount of scaffolding built into problems. For example, in Geometry, Lesson 3.1, students provide justifications in four steps that are given in a flow proof to prove the alternate interior angles theorem. In other instances, the theorem is provided, and students do not prove the theorem. For example, in Geometry, Lesson 2.4, a proof of the linear pairs theorem is provided in Task 4. Students fill in two blanks in the reasons side of the two-column proof. These reasons are scaffolded with, “What property justifies line segment BA and line segment BC being opposite rays?” and “What property justifies the measure of angle 1 added to the measure of angle 2 is the same as the measure of angle ABC?”
• G-CO.10: In Geometry, Lesson 9.3, students write a two-column proof of the converse of the isosceles triangle theorem. In Geometry, Lesson 9.1, Tasks 1-3, the triangle sum theorem, exterior angle theorem, and isosceles triangle theorem proofs are provided to students.
• G-CO.12: Students learn all aspects of the standard, except “with a variety of tools and methods”. Students are often instructed to use a compass and straightedge. For example, in Geometry, Lesson 3.3, Tasks 3 and 4, and in the Practice Workbook, Lessons 1.2, 1.3, and 3.3,  students are directed to use a compass and straightedge.
• G-GPE.1: In Geometry, Lesson 15.4, Task 1, the derivation of the equation of a circle is provided. Students do not derive the equation.
• S-ID.7: In Algebra 1, Lesson 6.1, Problem 21d, and Lesson 6.2, Problems 1d and 14b, students find the slope and y-intercept of linear models in the context of data. Students have a limited number of opportunities to interpret the slope and y-intercept of a linear model in the context of data.
• S-IC.2: In Algebra 2, Lesson 19.1, Problem 20, students use a data-generating process to model two number cubes being rolled 9 times. Students state the probability of success, create a table to show the theoretical probability distribution, and make a histogram with the probabilities within this same problem. Students have a limited number of opportunities to decide if a specified model is consistent with results from a given data-generating process.

The instructional materials reviewed for HMH Into AGA meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from grades 6-8.

Examples where the materials use age-appropriate contexts include:

• In Algebra 1, Lesson 1.3, Problem 38, students determine the average rate of speed when going down a zipline, given the vertical and horizontal distances.
• In Algebra 1, Lesson 18.3, Journal and Practice Workbook, Problem 9, students create and solve a quadratic function to determine how long after a football player kicks a football to a returner does the returner catch the ball. 
• In Geometry, Lesson 15.4, Problem 31, students find the equation of a ferris wheel given the diameter. Students also explain if a given point value can be possible for one of the cars of the ferris wheel to be attached to the wheel.
• In Geometry, Lesson 13.1, Journal and Practice Workbook, Problem 8, students use right triangle trigonometry to find the measure of an angle located at a shed formed by the ground and the line to the top of a cell phone tower. 
• In Algebra 2, Lesson 7.3, On Your Own, Problem 29, students describe the overall rate of change in the context given a square root function that models the number of likes received on a video after d days.
• In Algebra 2, Lesson 14.1, On Your Own, Problem 27, students write a recursive and explicit formula that gives the loan balance after n months and the amount of the monthly payment for a loan at a car dealership.

Examples where the instructional materials use various types of real numbers include:

• In Algebra 1, Module 18, students solve many types of quadratic equations that result in whole, rational, and irrational solutions. 
• In Algebra 1, Lesson 4.3, students identify the zeros and extreme values of functions. Problems 18, 19, 23, 24, and 29 have fractional values for these features.
• In Geometry, Module 14, students solve missing sides and angles using Law of Sines and Cosines that result in integer and rational numbers.
• In Geometry, Lesson 4.2, students write the equation of a line perpendicular to a line passing through a given point. Problems 8-10 have slopes and y-intercepts that are fractions. 
• In Algebra 2, Lesson 5.2, students solve polynomial equations with integer, rational, irrational, and imaginary roots.

Examples where students apply key takeaways from Grades 6-8 include:

• In Algebra 1, Lesson 4.3, students answer questions regarding key features of graphs from graphs of functions, which applies knowledge from Grade 8 Functions (8.F).
• In Algebra 1, Lessons 22.1-22.2, students create dot plots, box plots, and histograms. Students apply concepts and skills of basic statistics and probability from 6.SP.4.
• In Geometry, Lesson 6.2, students apply understanding of congruence and similarity through translations, rotations, reflections, and dilations (8.G) to learn about compositions with rigid motions.
• In Geometry, Lesson 12.3, students find the distance between two streets using proportional relationships, which applies key takeaways from 7.RP.2.
• In Algebra 2, Lesson 15.1, students find the relationship between arc length and radius by applying ratio and proportional relationships, applying learning from Ratios and Proportional Relationships (RP) in Grades 6 and 7.
• In Algebra 2, Unit 5, Project Epoxy Proxy, Problem 2, students write rational expressions for the total weight of various mixtures. Students apply Ratios and Proportional Relationships (RP) from Grades 6 and 7.

The instructional materials reviewed for HMH Into AGA meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials foster coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the standards. Each Module in the Teacher Edition has Teaching For Success that identifies Mathematical Progressions with bulleted lists of prior learning, current development, and future connections. Each lesson has Mathematical Progressions for teachers, which is more specific and targeted. In Prior Learning, grade levels and lessons are identified, without standards. In Future Connections, specific lessons within the course and/or series are identified.

Examples where the materials foster coherence within courses include:

• In Algebra 1, Lesson 2.2, students create and solve linear equations in real-world context (A-CED.1). In Lesson 3.1, students create and graph linear equations in two variables (A-CED.2). In Lessons 9.1-9.4, students solve systems of linear equations through graphing, substitution and elimination (A-REI.5).
• In Algebra 1, Lesson 4.2, students write and graph linear functions (F-IF.7a). In Lesson 9.1, students write and graphically solve systems of linear equations (A-REI.6).
• In Geometry, Unit 3, Lessons 5.1-5.3, students use descriptions of rigid motions to transform figures (translate, rotate, and reflect) on a coordinate plane. In Unit 4, Lessons 7.1-7.3, students continue to transform functions on a coordinate plane and use the definition of congruence in terms of rigid motions (G-CO.5,7).
• In Geometry, Module 12, students use the definition of similarity to decide if two figures are similar (G-SRT.2). In Lessons 13.1 and 13.2, students use the information they have learned about similar triangles to discover the tangent, sine, and cosine ratios in right triangles (G-SRT.6). Students also use trigonometric ratios in Lesson 19.2 to develop a formula for the volume of a cone (G-GMD.1).
• In Algebra 2, Lessons 7.3 and 7.4, students graph square root and cube root functions and analyze the graphs in real-world contexts (F-IF.7b). In Lesson 7.5, students solve radical equations and graph the equations using graphing calculators (A-REI.2).
• In Algebra 2, Lesson 2.3, students use completing the square and the quadratic formula to find complex solutions to quadratic equations (N-CN.7). The materials connect this to previous learning in Algebra 2, Lesson 1.3, by graphing and transforming quadratic functions (F-BF.3). Students further build on their knowledge of complex solutions by finding real and complex solutions to higher order polynomials in Algebra 2, Lesson 5.1 (A-APR.3).

Examples where the materials foster coherence across courses include:

• In Algebra 1, Lesson 5.2, students transform linear functions, including translations, reflections, and stretches or compressions (F-BF.3). In Geometry, Module 5, students define and apply translations, rotations, reflections, and symmetry. In Geometry, Lesson 6.1, students define and apply dilations, stretches, and compressions (G-CO.2, G-SRT.1). In Algebra 2, Lesson 1.3, students explore transformations of functions, including translations, stretches, compressions, reflections, and combined transformations (F-BF.3). The Build Understanding section of the lesson reminds students that in previous courses they performed transformations of figures and function graphs.
• In Algebra 1, Lesson 2.2, students solve multi-step equations. In Geometry, Lesson 3.2, students solve multi-step equations from diagrams to find the value of x that makes two lines parallel (A-REI.3).
• In Geometry, Lessons 13.1 and 13.2, students define trigonometric functions in terms of the ratio of sides in a right triangle (G-SRT.8). In Algebra 2, Lesson 15.1, students understand the radian measure of an angle as the length of the arc on the unit circle subtended by the angle (F-TF.1).

The instructional materials reviewed for HMH Into AGA meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials explicitly identify content from Grades 6-8 and support the progressions of the high school standards.

The materials explicitly identify content from Grades 6-8 by using language directly from prior, grade-level standards in Mathematics Progressions Prior Learning. The materials identify the grade level of the prior learning and the unit where the prior learning is found. The prior grade level and unit are noted in parentheses, and examples include:

• In Algebra 1, Lesson 4.2, Mathematical Progressions Prior Learning, “Students: Identified linear relationships graphically.” (Gr8, 6.3 and 6.4) 
• In Algebra 1, Lesson 6.1, Mathematical Progressions Prior Learning, “Students: Have created scatter plots with bivariate data and examined association between variables. (Gr8) Have identified linear association in bivariate data, informally fit a line to data, and informally assessed the fit of the line of the data. (Gr8) Have used a linear model of bivariate data to solve problems, interpreting the slope and intercepts of the linear equation.” (Gr8) 
• In Geometry, Lesson 4.1, Mathematical Progressions Prior Learning, “Students: Explained why the slope is the same between any two distinct points on a nonvertical line in the coordinate plane.” (Gr8, 5.1)
• In Geometry, Lesson 17.1, Mathematical Progressions Prior Learning, “Students: Knew the formulas for circumference and area of a circle and used them to solve problems.” (Gr7, 10.1 and 10.2) 
• In Algebra 2, Lesson 2.1, Mathematical Progressions Prior Learning, “Students: Used square root symbols to represent solutions of equations of the form x$$^2$$ = p. “ (Gr8, 3.1)
• In Algebra 2, Lesson 11.1, Mathematical Progressions Prior Learning, “Students: studied linear proportional relationships.” (Gr8, 5.1)

Examples where the materials make connections between Grades 6-8 and high school concepts and allow students to extend their previous knowledge include:

• In Algebra 1, Lesson 9.3, (A-REI.C) students extend 8.EE.8 as they analyze and solve systems of linear equations using the elimination method and estimate or solve pairs of simultaneous linear equations by inspection. 
• In Algebra 1, Lesson 10.22, students extend their prior learning from 6.SP.A, understanding a set of data can be described by its center, spread, and overall shape, to S.ID.3, interpreting differences in shape, center, and spread in the context of the data sets.
• In Geometry, Lesson 1.4, students use knowledge of the pythagorean theorem (8.G.8) to justify the distance formula. They use this to find the perimeter/area of polygons (G-GPE.7) on the coordinate plane.
• In Geometry, Lesson 3.1, students prove theorems about parallel lines cut by a transversal (G-CO.9). This extends from 8.G.5 where students use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal.
• In Algebra 2, Lesson 1.1, students discover there are constraints on domain and range in function models (F-IF.5). This is an extension of 8.F.1 where students understand that a function is a rule that assigns each input to exactly one output.
• In Algebra 2, Lesson 2.3, students find the value of the discriminant and solve quadratic equations using the quadratic formula (A-REI.4). This is an extension of 8.EE.2 where students learn to solve equations of the form x$$^2$$ = p.

The instructional materials reviewed for HMH Into AGA do not explicitly identify the plus standards. Of the plus standards that are present in the materials, there is coherence to support the mathematics which all students should study in order to be college and career ready. In the Teacher Edition: Planning and Pacing Guide, the plus standards are listed along with the non plus standards and are not denoted as plus standards. The plus standards are also not denoted as such in Lesson Focus and Coherence in the teacher pages prior to a lesson. For example, in Geometry, Lesson 19.3, Volumes of Spheres, Teacher Edition, G-GMD.2 is not listed in the bulleted list of Mathematics Standards, yet the standard is listed in the actual lesson and also in the Planning and Pacing Guide document (without explicit identification). 

Plus standards that are addressed in the materials include:

• N-CN.9: In Algebra 2, Lesson 5.2, students are given the Fundamental Theorem of Algebra, and the materials include an explanation that the pattern they observed in Task 1 reflects the Fundamental Theorem of Algebra. In Problem 36, students show that the Fundamental Theorem of Algebra and its corollary are true for quadratic polynomials.
• A-APR.7: In Algebra 2, Lesson 12.1, students multiply and divide rational expressions. In Lesson 12.2, students add and subtract rational expressions. In Lesson 12.2, students are provided a table that shows the closure properties of the set of whole numbers, integers, and rational numbers. In Task 5, students make predictions about the closure of the set of rational expressions for each of the operations listed in the table.
• F-IF.7d: In Algebra 2, Lesson 11.2, Problems 30-37, students graph rational functions and show asymptotes and reference points for the graph. In Algebra 2, Lesson 11.3, Tasks 4 and 5 show students how to sketch a graph finding vertical asymptotes, holes, and horizontal or slant asymptotes. Students also identify x-intercepts and create a table for end behavior. Students sketch the graph of each rational function, state any excluded values of the domain, identify the type of break in the graph, represent asymptotes, and represent any holes of the graph in Problems 22-29.
• F-BF.1c: In Algebra 2, Lesson 7.1, Problems 1 and 2, students compose two functions. Students write the equations for each composition function and express their domains and ranges in set notation in Problems 6-9.
• F-BF.4b: In Algebra 2, Lesson 7.1, Problems 10-15, students state the inverse of each given function and use composition of functions to verify the functions are inverse functions.
• F-BF.4c: In Algebra 2, Lesson 7.2, Problems 3-5 and 10-13, students use tables of values to answer questions about the function. In Lesson 9.3, Task 3, students create a logarithmic model by finding the inverse of the exponential model and using the table to interchange x and y values to graph in a graphing calculator.
• F-BF.5c: In Algebra 2, Lesson 9.1, Task 1, Problems 22-27, students evaluate logarithmic functions by converting them to exponential form. In Lesson 10.1, Problems 24, 26, 28-30, students use logarithmic and exponential models to solve real-world problems. In Lesson 10.3, Problems 6-9, students solve logarithmic equations graphically, and in Problems 10-15, students solve logarithmic equations algebraically.
• G-SRT.11: In Geometry, Lesson 14.1, Build Understanding, students are given five variations of when the Law of Sines can be used to solve a triangle. In Lesson 14.2, students are told the Law of Cosines needs to be used in cases of SSS and SAS. Students solve triangles, with and without context, using both laws.
• G-C.4: In Geometry, Lesson 15.3, Task 1, students use a geometric drawing tool to draw a circle and a line tangent to a point on the circle. Students use a compass and straightedge to create a diagram that demonstrates the tangent-radius theorem in Problem 6.
• G-GMD.2: In Geometry, Lesson 19.1, Tasks 1 and 2, students stack right prisms and cylinders to see that the volume is the total of all the volumes. Students extend this idea to oblique solids. Cavalieri's Principle is provided to students in Problem 8. In Lesson 19.3, Task 1E, students answer, "How can you use Cavalieri's Principle to find the volume of the hemisphere? Use this result to write a formula for the volume of a sphere with radius r.”
• S-CP.8: In Algebra 2, Lesson 18.2, Task 4, students use the multiplication rule in two ways to find the probability, determine if the events are independent, and explain how they know. Students use the multiplication rule to find the probability and interpret the answer in terms of the model in problem 14.
• S-MD.6: In Algebra 2, Lesson 17.1, Task 1, students use theoretical probability to determine the outcome of events. Students determine if the given process will lead to a fair decision by using probabilities and explain why in Problem 13.
• S-MD.7: In Algebra 2, Lesson 18.3, Task 1, students use data in a two-way relative frequency table to decide if positive test results are a reliable way to determine if someone carries a virus. Students calculate probabilities on given information to determine decisions in Problems 3-4.

Plus standards that are partially addressed in the materials include:

• F-BF.4d: In Algebra 2, Lesson 7.2, Task 4, students determine why the domain of quadratic functions must be restricted to find the inverse functions. In Problems 15-20, students determine the inverse of quadratic functions with a given restricted domain. Students do not produce an invertible function from a non-invertible function by restricting the domain.
• G-SRT.9: In Geometry, Lesson 14.2, Task 3, students are given instructions to derive a formula for the area of a triangle. An auxiliary line is given in the drawing so that students do not have the opportunity to draw the line.
• G-SRT.10: In Geometry, Lesson 14.1, Task 1, students are led through a derivation for Law of Sines, and in Lesson 14.2, Task 1, students are led through the derivation for Law of Cosines. Students use the laws to solve problems, both with and without context in these two lessons, but students do not derive either of the Laws.

Plus standards that were not addressed in the materials include:

• N-CN.3-6,8
• N-VM.1-12
• A-APR.5
• A-REI.8,9
• F-TF.3,4,6,7,9
• G-GPE.3
• S-CP.9
• S-ID.1-5

The instructional materials reviewed for HMH Into AGA meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The series has Build Understanding in each lesson, which often includes Turn and Talks, designed for students to demonstrate their conceptual understanding.

Examples of the materials developing conceptual understanding and students independently demonstrating it include:

• N-RN.1: In Algebra 2, Lesson 6.1, students understand how the Power of a Power Property is extended to a rational exponent by solving an equation where the variable is an exponent and justifying each step. Students translate between rational exponents and radical expressions. In Lesson 6.2, students investigate the properties of rational exponents given the rules for integer exponents in a table. Students predict and show that the rules for integer exponents extend to rational exponents.
• A-APR.B: In Algebra 2, Lesson 5.1, students analyze the graph of a polynomial function in factored form, find zeros, write the function in standard form, and determine how the zeros are related to the standard form of the function.
• A-REI.A: In Algebra 1, Lesson 2.2, Journal and Practice Workbook, students solve equations, justify their solution steps, and check the solution. In Problem 9, students critique Kevin's reasoning by explaining his error, correcting his error, and completing the correct work to solve the equation. 
• F-IF.A: In Algebra 1, Lesson 4.1, Turn and Talk, students explain if a relation in which every domain value corresponds to the same range value is a function. Students state the change that would need to be made in a table of values so that the relation becomes a function. Students determine if it is sensible to complete a horizontal line test for functions and explain their reasoning.
• G-CO.8: In Geometry, Lesson 8.2, students explain the reason triangles provided on a coordinate plane are congruent using a triangle congruence theorem and describe a sequence of transformations that maps one triangle to the other. In Lesson 8.3, students use the SSS triangle congruence theorem and distance formula to show that two triangles provided on a coordinate plane are congruent.
• G-SRT.2: In Geometry, Lesson 12.1, students compare coordinates in a dilation to determine if two figures are similar. Students answer questions to explain or justify their answers using transformations. Students also determine whether each pair of figures is similar using similarity transformations and explain their reasoning.

The instructional materials reviewed for HMH Into AGA meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters.The instructional materials develop procedural skills, and students independently demonstrate procedural skills throughout the series. 

Examples from On Your Own in the student materials where students independently demonstrate procedural skills include: 

• N-RN.2: In Algebra 2, Lesson 6.1, Problems 52-60, students rewrite expressions with rational exponents as radicals. In Problems 66-74, students rewrite radicals as expressions with rational exponents.
• A-SSE.2: In Algebra 1, Lesson 15.3, Problems 14-37, students find special products of binomials. In Algebra 2, Lesson 4.4, Problems 7-24, students factor polynomials, including factoring by grouping.
• A-APR.1: In Algebra 1, Lesson 15.1, Problems 3-8 and 23-32, Lesson 15.2, Problems 7-9 and 23-30, Lesson 15.3, Problems 14-37, and Lesson 16.1, Problems 21-44, students add, subtract and multiply polynomials. 
• A-APR.6: In Algebra 2, Lesson 11.2, Problems 34-37, students rewrite rational functions to find the quotient and the remainder.  
• G-SRT.5: In Geometry, Lesson 8.3, Problems 7-12, students determine if triangles are congruent using SSS, ASA, or SAS triangle congruence.
• G-GPE.4: In Geometry, Lesson 4.3, Problems 8-13, students determine whether two lines are congruent by using the distance formula.

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially when called for in specific content standards or clusters. Application problems occur toward the end of each lesson, and the Journal and Practice Workbook follows the same pattern of having at least one application problem at the end of each lesson. 

Examples of students utilizing mathematical concepts and skills in engaging applications include:

• N-Q.A: In Algebra 1, Lesson 1.3, students choose a level of accuracy appropriate for the finances of a family budget of expenses that include rent, food, transportation, clothing, and entertainment. Students explain their reasoning of the level of precision that will be acceptable to assemble the budget.
• A-SSE.3: In Algebra 1, Lesson 19.3, students write a function in intercept form from a given function denoting the monthly profit of a party planning business after time, t, in months. Students provide intercepts and their meaning in a situation, write a function in vertex form, and provide an interpretation for the vertex in a situation.
• F-IF.B: In Algebra 2, Unit 2, Project Network Functions, students sketch a graph of a polynomial function that models the number of users connected to the network t hours after 6:00 AM. Students describe the characteristics of the graph and decide if they would classify the graph as an even or odd function and explain their reasoning. 
• F-BF.1: In Algebra 2, Lesson 11.1, students solve inverse functions using various applications, such as vibrating violin strings. Another problem provides a table of carpentry projects, and students determine the associated equation. Students also compare pressure on snow depending on the type of shoes worn and find the height of a tree from a given distance. 
• G-SRT.8: In Geometry, Lesson 13.1, students apply the tangent function to find the distance a person is standing from the Washington Monument given the height of the monument and the angle between the ground and the top of the monument from the place where the person is standing.
• G-GMD.3: In Geometry, Lesson 19.2, students find the amount of plastic needed to make a cylindrical-shaped, plastic cup. Students also calculate the cost per cubic inch of plastic, and the amount Dan would pay for a cup with the same radius if the height were 6 inches.
• S-ID.2: In Algebra 1, Lesson 22.1, students find measures of center and spread for each data set given about swim times for two swimmers. Students compare the two data sets and identify if either data set has an outlier, choose the better swimmer, and justify the statistics used to determine the answer.

The instructional materials reviewed for HMH Into AGA meet expectations for not always treating the three aspects of rigor together and not always treating them separately. All three aspects of rigor are present independently throughout the materials, and multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Each module begins with Are You Ready that reviews prior concepts and skills needed for that module; these problems are procedural. The lessons are intentionally developed so that students have opportunities to practice each aspect of rigor throughout each lesson.

The following are examples of balancing the three aspects of rigor in the instructional materials:

• In Algebra 1, Lesson 2.1, Spark Your Learning, students build conceptual understanding through application by comparing two stores offering different deals on the same-priced phone and determine the strategy to use to answer the problem. Students use procedural skills to write an expression to find the store that has the better deal.
  • In Algebra 1, Lesson 7.1, On Your Own, students use procedural skills to extend various types of patterns for arithmetic sequences to determine the common difference and write a recursive rule in function notation. Students extend this knowledge to write an explicit rule for arithmetic sequences. Students complete an application problem by writing a recursive rule and using that rule to determine the amount a caregiver earns in a home health job.
• In Geometry, Lesson 5.1, students use procedural skills to draw figures with given vertices and the images after a translation by a given value. Students use conceptual understanding as they reason a classroom map where students are standing at different spots, and students complete an application problem about a cyclist to determine what vector describes her position from the starting position to her final destination.
• In Geometry, Lesson 13.2, students build conceptual understanding of side ratios of a right triangle. In Turn and Talk, students predict how their answer would change if angle A is increased in a given triangle. Students use procedural skills to calculate Sine and Cosine ratios in the On Your Own problems. 
• In Algebra 2, Lesson 3.2, On Your Own, students use procedural skills to find key features of polynomial functions. Later, students engage in application problems using polynomial functions.
• In Algebra 2, Lesson 4.5, Journal and Practice Workbook, students use procedural skills when using synthetic substitution to find p(-3) for each polynomial, and determine whether the given polynomial is a factor of the polynomial p(x). Students apply their understanding of polynomial functions in the context of profit for a game by determining a lesser number of games a company could produce and still make the same profit. Students use conceptual understanding to determine if the set of polynomials is closed under subtraction and either justify or give a counterexample.

The instructional materials reviewed for HMH Into AGA partially meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards.

In the series, each task in Learn Together is aligned with an MP that is identified in the teacher materials, and the student materials identify the MPs, except for MP1, with a phrase next to problems in Check Understanding and On Your Own. The Planning and Pacing Guide indicates that MP1 is in Spark Your Learning and Spies and Analysts tasks, which appear on either the front page of every lesson, or on the first page of the modules. In the teacher materials, Persevere, which is on the first page of each lesson, states, “If students need support, guide them by asking:” and lists guiding questions.

Throughout the materials, the MPs are identified in multiple places, but there are examples of misleading identifications for some of the MPs across the courses of the series. Examples of the misleading identifications are listed in the criterion report for Practice-Content Connections, and as a result of those, 1 point is deducted from the scoring of this indicator.

Examples where students make sense of problems and persevere in solving them include:

• In Algebra 1, Unit 6, Fit Linear Functions to Data, Spies and Analysts Task, students collect data to model a rubber band bungee. Students collect data where x is the number of rubber bands and y is the distance in cm from the ground to determine if the relationship is linear. Students create a table of data and use a guess-and-check strategy to find the number of rubber bands needed.
• In Geometry, Module 9, Performance Task, Spies and Analysts, students answer the question, “How can we water all the grass?” Students determine the information necessary by looking for entry points to the solution.
• In Algebra 2, Module 13, Performance Task, students analyze constraints such as amount borrowed, interest rate, and length of a loan. They plan a solution pathway by changing one variable at a time and make sense of those solutions in regard to the content they are given.

Examples where the materials identify MP6 as MP Attend to Precision and students attend to precision include:

• In Algebra 1, Lesson 3.1,Task 3, students determine the points where a graph crosses the x- and y-axes by identifying the x- and y-intercepts.
• In Geometry, Lesson 12.4, students determine which of two roof plans requires the least amount of lumber and explain the reason. 
• In Algebra 2, Task 2, students verbally describe graphed functions and apply their understanding of domain, range, and end behavior.

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards.

Examples where the materials identify MP2 as MP Reason and students reason abstractly and quantitatively include:

• In Algebra 1, Lesson 2.4, students write an inequality and explain their reasoning of the amount of pillows that must be sold during the third month to have a profit for the quarter in a company with a fixed monthly expense of $3,200. 
• In Algebra 1, Lesson 3.1, students interpret graphs of a real-world scenario of purchasing T-shirts and sweatshirts with $100 and determine the solutions that make sense in the context of the situations.
• In Geometry, Lesson 4.3, students explain why x$$_1$$$$\not =$$x$$_2$$ and y$$_1$$$$\not =$$y$$_2$$ when using the Pythagorean theorem to prove that the distance between two points on the coordinate plane is given by the distance formula.
• In Geometry, Lesson 5.2, Workbook, students decide what regular polygon Cindy has and explain their reasoning. Students also decide the smallest angle of rotation for an image is, based on given information, and explain their reasoning.
• In Algebra 2, Lesson 4.4, students use spatial and algebraic reasoning to determine the dimensions of prisms, from given expressions for the volumes of four small rectangular prisms that form one large rectangular prism.
• In Algebra 2, Lesson 15.1, Workbook, students calculate the angle a larger gear turns if the smaller gear turns through a central angle of$$\frac{3\pi}{16}$$ radians.

Examples where the materials identify MP3 as MP Critique Reasoning or MP Construct Arguments and students construct viable arguments and critique the reasoning of others include:

• In Algebra 1, Lesson 17.1, students construct an argument about viewing windows where graphs cross the x-axis. 
• In Algebra 1, Lesson 16.2, students critique John’s answer to a problem of how a given expression represents a sequence and explain why or why not his given sequence is correct. 
• In Geometry Lesson 8.2, students determine if Zach has made a mistake and correct his error through an explanation of a given proof showing two triangles are congruent. 
• In Geometry Lesson 8.4, students construct an argument by explaining if the given claim that two right triangles share a hypotenuse, then the triangles must be congruent is correct.
• In Algebra 2, Lesson 4.4, students construct an argument if it is possible to factor any polynomial with four terms using factoring by grouping. 
• In Algebra 2 Lesson 13.3, students critique if John’s simplified formula for the sum of the first n terms of the geometric series is correct and explain why or why not.

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. The materials support the intentional development of MP4 and MP5.

Examples where MP4 is identified as MP Model with Mathematics and students engage in parts of model with mathematics include:

• In Algebra 1, Lesson 5.4, Problem 23, students write an equation that gives the university’s expected enrollment and write an equation for the inverse of that function. Students also predict the number of years it will take for enrollment to reach 10,000 students.
• In Algebra 1, Lesson 20.2, students choose possible dimensions and areas for a chicken coop given some restraints. There is one correct answer for each restraint, but students do revise their solutions based on the changing constraints.
• In Geometry, Lesson 4.3, students use the distance formula to write and solve an equation about a car traveling down a straight road starting at the origin. Students also find the midpoint of the path of the car.
• In Geometry, Lesson 17.3, students write an equation to solve a construction site problem with sectors from two different circles outlined. Students are provided a hint as to how to solve the problem if it is needed.
• In Algebra 2, Lesson 3.1, students write a polynomial function to model the mass of a cylindrical concrete pier used in house construction as shown in an image. Students graph the relationship between the mass and the radius of the cylinder to determine if a specified mass is a reasonable estimate for the radius and explain their reasoning.
• In Algebra 2, Lesson 10.2, students write an exponential equation to represent the cost of schooling given the average cost and percent increase. Students determine the number of years it will take for the cost of schooling to exceed a given amount.

Examples where MP 5 is identified as MP Use Tools and students choose appropriate tools strategically include:

• In Algebra 1, Lesson 4.3, Spark Your Learning, students develop a mathematical question about a high-speed train traveling between stations. Students are specifically asked what strategy and tool they would use to solve their question (Part C). 
• In Algebra 1, Lesson 17.1, Spark Your Learning, students develop a mathematical question about a pallet of rolled sod. Students are specifically asked what strategy and tool they would use to solve their question (Part C). 
• In Geometry, Module 1, Performance Task, students choose an appropriate tool for approximating the perimeter of the irregular region enclosed by the fire edge as well as the length of the controlled fire edge. Students might use the given coordinate grid, a ruler, and/or a piece of string.
• In Geometry, Lesson 15.1, Spark Your Learning, students develop a mathematical question about an outdoor amphitheater. Students are specifically asked what strategy and tool they would use to solve their question (Part B). 
• In Algebra 2, Module 8, Performance Task, students  choose appropriate research tools to determine a reasonable average annual growth or interest rate for an investment as well as the inflation rate for the period 1960–present in order to estimate how much an investment of $1,000,000 in 1960 would be worth today. 
• In Algebra 2, Lesson 13.2, Spark Your Learning, students develop a mathematical question about the Louvre Pyramid. Students are specifically asked what strategy and tool they would use to solve their question (Part C).

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. The materials support the intentional development of MP7 and MP8.

Examples where the materials identify MP7 as MP Use Structure and students look for and make use of structure include:

• In Algebra 1, Lesson 2.2, students see algebraic expressions as single objects being composed of several objects as they notice that 3(4x - 7) + 2(4x - 7) = 5 has the form 3[ ] + 2 [ ] = 5 where [ ] = 4x - 7. Students explain how to use this observation to solve an equation, then use the same method to solve 2(3x + 4) - 5(3x + 4) = 33.
• In Algebra 1, Lesson 5.1, students look for, identify, and generalize relationships and patterns in transformations of quadratic functions by matching equations relating a function written in transformational function notation to its corresponding pair of graphs. 
• In Geometry, Lesson 6.1, students look for patterns and make generalizations by writing a coordinate rule to dilate a figure with center (a, b) not at the origin. 
• In Geometry, Lesson 18.3, students use structure to break apart solids and explore familiar two-dimensional faces by decomposing solids and developing surface area formulas for a regular pyramid and a right cone.
• In Algebra 2, Lesson 3.2, students look for overall structure and patterns as it relates to the behavior of graphs by matching factored polynomial functions to their graphs.
• In Algebra 2, Lesson 4.1, students look for mathematical structures by evaluating two functions, f(x) and g(x), given x = -1, 2, and 5. Students use the values of f(x) and g(x) to evaluate f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x)/g(x) for each of the values of x. Students then combine the function expressions to form four new functions, and evaluate each new function when x = -1, 2, and 5 to verify the results are the same. 

Examples where the materials identify MP8 as MP Use Repeated Reasoning and students look for and express regularity in repeated reasoning include:

• In Algebra 1, Lesson 2.3, students use repeated reasoning to rewrite the given perimeter formulas for specific types of regular polygons so that each formula gives the polygon’s side length s in terms of its perimeter P. In Part B, students then express regularity in their reasoning by writing the general formula.
• In Algebra 1, Lesson 7.1, students use repeated reasoning by observing that each figure in a pattern contains 3 more dots than the previous figure. Students then express regularity in their reasoning by concluding that the numbers of dots form an arithmetic sequence and write an algebraic rule for the number of dots in the nth figure.
• In Geometry, Module 2, Performance Task, students reason repeatedly about the data points in the graph by observing that the y-coordinate of each point is about 10 times the x-coordinate, meaning that an object’s weight in newtons is about 10 times its mass in kilograms. Students can then express regularity in their reasoning by conjecturing that the equation y = 10x approximates an object’s weight y if its mass is x.
• In Geometry, Lesson 5.4, Task 4, students reflect several points across the y-axis and then generalize their results by observing that when a point (x, y) is reflected across the y-axis, its image is (–x, y).
• In Algebra 2, Lesson 3.2, Task 1, , students use repeated reasoning to determine features of the graphs of specific polynomial functions. Later in the task, students express regularity in their reasoning by drawing general conclusions about the graphs of polynomial functions.
• In Algebra 2, Lesson 13.3, Spark Your Learning, students examine the first five stages of a tree fractal, look for a pattern in the number of new branches at each stage, and then express regularity in their reasoning by answering the question, “What is a general rule for the number of new branches at each stage of the tree fractal?”

The instructional materials reviewed for HMH Into AGA meet expectations that there is a clear distinction between problems and exercises in the materials.

Each module presents lessons with a consistent structure. During the instructional sections, which include Spark Your Learning, Build Understanding, and Step It Out, students have opportunities to learn new content through examples and problems for guided instruction, step-by step procedures, and problem solving.

At the end of the lesson, Check Understanding and On Your Own provides a variety of exercises which allow students to independently show their understanding of the material. Exercises are designed for students to demonstrate understanding and skills in application and non-application settings. Assessment Readiness and Spiral Review also include exercises.

The instructional materials reviewed for HMH Into AGA meet expectations that the design of assignments is intentional and not haphazard.

Overall, lessons are intentionally sequenced and scaffolded, so students develop understanding of mathematical concepts and skills. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skills, and engage with multiple activities that utilize concrete and abstract representations and increase in complexity. Examples include:

• Spark Your Learning serves to motivate and set the stage for students to learn new material and persevere through a related mathematical task. 
• Build Understanding and Step It Out provide opportunities for students to learn and practice new mathematics, as well as “connect important processes and procedures” according to the Planning and Pacing Guide. 
• Check Understanding provides a formative assessment opportunity after instruction. 
• On Your Own, Exit Ticket, Assessment Readiness, and Spiral Review sections in each lesson support students in developing independent mastery of the current lessons as well as reviewing material from previous lessons.

The instructional materials reviewed for HMH Into AGA meet expectations for having a variety in what students are asked to produce, and examples include:

• Performance Tasks: Algebra 1, Module 15, Packaging Science states, “A team of designers wants to produce a package that can hold the largest possible volume while keeping the cost of production low. As illustrated below, the team can program a machine to cut out a square from each corner of a large sheet of construction material to form the framework of the package.” Students use polynomial multiplication to write and solve an equation for the surface area and volume of the packaging. Geometry, Module 18, On the Spot states, “How can you describe the area of a sunspot?” The Overview on the side provides, “ This problem requires students to ask questions in order to make decisions about how to approach the problem. Examples of questions: How do you measure the dimensions of the irregularly shaped sunspot? Do you only include the larger section, or do you include the smaller dots next to the larger section?”
• STEM activities: Geometry, Unit 2 states, “A triaxial woven carbon fiber provides a lot of strength while remaining lightweight. The fiber has an area density of only75 kg/m2. The frame of a drone is made from the fiber. One section of the frame is 150 mm × 20 mm. What mass does the section add to the drone?” Algebra 2,  Unit 8, states, “An archaeologist allocates an effective sweep width while searching a site for artifacts.” The activity provides a diagram of the effective sweep area. Materials further state, “How successful was the sweep?”
• Show written calculations and solutions
• Verbally defend or critique the work of others to show understanding
• Build models for a problem by using diagrams and equations
• Compare multiple representations - table, graph, equation, situation - of data
• Use a digital platform to conduct and present their work
• Use manipulatives, especially in small groups, to represent mathematics 
• Construct written responses to explain their thinking

The instructional materials reviewed for HMH Into AGA meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

• The series does not involve extensive use of manipulatives, however, when they are included, they are consistently aligned to the expectations and concepts in the standards. 
• Most hands-on manipulatives are integrated in supplemental, small-group, differentiated instruction activities, and warm-up options.

The instructional materials reviewed for HMH Into AGA are not distracting or chaotic and support students in engaging thoughtfully with the subject. 

 The entire series, both print and digital, follows a consistent format, which promotes familiarity with the materials and makes finding specific sections more efficient. The page layout in the materials is user-friendly. Tasks within a lesson are numbered to match the module and lesson numbers. Though there is a lot of information given, pages are not overcrowded or hard to read. Graphics promote understanding of the mathematics being learned. Student practice problem pages include enough space for students to write their answers and provide explanations. The digital format is easy to navigate, but students have to scroll without being able to view much of the information at one time.

The instructional materials reviewed for HMH Into AGA meet expectations for providing quality questions to help guide students’ mathematical development. 

There are Guided Student Discussion questions and sample student answers throughout the Teacher Edition including on the Module Opener page, Spark Your Learning, Build Understanding, and Step It Out pages that correspond to the tasks or exercises on the page. Each module review also contains suggested questions intended to have students summarize concepts and skills developed within the module.

The Spark Your Learning planning page in the Teacher Edition includes examples of student work which show two different strategies and Common Errors. Each example has suggested questions for teachers to correct or advance student thinking. For example, in Algebra 1, Lesson 13.1, Common Error about linear relationships states, “Now find the ratios of consecutive values of the stamp collection. What do you observe? How can you use this observation to solve the problem?”

The instructional materials reviewed for HMH Into AGA meet expectations for containing ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials.

In the Module planning pages, there is a variety of information that can help teachers understand the materials in order to present the content. Each lesson identifies the relevant content standards and Mathematical Practices, an “I Can” Objective, Learning Objective, Language Objective, and Mathematical Progressions Across the grade that contain prior learning, current development, and future connections. Unpacking the Standards provides further explanations of the standards’ connections. This section gives an explanation of the content standard contained in the lesson and Professional Learning, which sometimes contains information about the practice standard contained in that lesson. Teaching for Depth provides teachers with information regarding the content and how this relates to student learning.There are additional suggestions about activating prior knowledge or identifying skills in Warm-up Options, activities to Sharpen Skills, Small-Group Options, and Math Centers for differentiation.

There are  prompts in each module related to Online Ed: “Assign the Digital Module Review for built-in student supports, Actionable Item Reports, Standards Analysis Reports” and “Assign the Digital Module Test to power actionable reports including proficiency by standards item analysis.” Within lessons, there are multiple prompts. Warm-Up Options and Step It Out both have an icon “Printable & projectible” which states “More print and digital resources for differentiation are available in the Math Activities Center”.

The instructional materials reviewed for HMH Into AGA partially meet expectations for containing adult-level explanations so that teachers can improve their own knowledge of the subject. The materials include adult-level explanations of the grade-level content, but the materials do not include adult-level explanations of advanced mathematics concepts so that teachers can improve their own knowledge of the subject. Examples of the grade-level explanations include:

At the beginning of each module, the teacher’s edition often includes Teaching for Depth that provides a brief overview of the mathematics contained in the module. For example, Algebra 2, Module 12 states, “Meaning of extraneous solutions. An extraneous solution is a solution of a simplified version of an equation that does not satisfy the original equation. It seems counterintuitive to many students that a simplified version of an equation would yield an extra solution that was not part of the original equation, so it helps to show them how this happens.” 

In addition, Teacher to Teacher From the Classroom gives tips or anecdotes about the module content. For example, Geometry, Module 10 states, “One way I begin introducing the idea of inequality relationships in geometry is to remind my students about the three possibilities that exist between quantities. That is, two quantities are greater than, equal to, or less than each other. I ask students to consider what happens in a triangle if two of its sides have equal length or two of its angles have equal measure. They usually know. What about if the side lengths or the angle measures are unequal? It seems logical to assume that the same “unequalness” also applies. What happens if corresponding side lengths of two triangles are not equal? or corresponding angle measures?”

The instructional materials reviewed for HMH Into AGA meet expectations for explaining the role of the grade-level mathematics in the context of the overall mathematics curriculum.

Each module in the Teacher Edition includes Mathematical Progressions Across the Grades which lists prior learning, current development, and future connections. Similarly, the beginning of each lesson in the Teacher Edition includes Mathematical Progressions that show connections to prior and future grades’ standards, as well as other lessons within the program.

The instructional materials reviewed for HMH Into AGA provide a list of lessons in the teacher's edition, cross-­referencing the standards addressed, and a pacing guide. 

Each course in this series includes a Planning and Pacing Guide that includes the standards and pacing (number of days) for each lesson. There is another standards chart in the Planning and Pacing Guide that lists each standard and correlation to Student Edition Lessons. In the Teacher Edition, pacing is provided in the module planning pages, and the standards contained in each lesson are identified with written descriptions as well as listed under Current Development in the Mathematical Progressions chart.

The instructional materials reviewed for HMH Into AGA include strategies for parents to support their students progress. The Planning and Pacing Guide describes strategies to Connect with Families and Community:

• The student materials contain Math on the Spot problems that have videos connected to them. The materials state, “Math on the Spot video tutorials provide instruction of the math concepts covered and allow for family involvement in their child’s learning.” There are generally 1-3 problems per module.
• The materials state, “School Home letters inform families about the skills, strategies, and topics students are encountering at school.” Each module includes a school home letter, found online in four languages, providing vocabulary, a home activity, and discussion prompts.

The instructional materials reviewed for HMH Into AGA explain instructional approaches used and how they are research-based. 

The Planning and Pacing Guide contains Teacher Support Pages that include a section on Supporting Best Practices. “Into Math was designed around research-based, effective teaching practices such as those described in Principles to Actions (NCTM 2014).” These include:

• Establish mathematics goals to focus learning.
• Implement tasks that promote reasoning and problem solving.
• Use and connect mathematical representations.
• Facilitate meaningful mathematical discourse.
• Pose purposeful questions.
• Build procedural fluency from conceptual understanding.
• Support productive struggle in learning mathematics.
• Elicit and use evidence of student thinking.

The Planning and Pacing Guide describes four design principles from the Stanford Center for Assessment, Learning, and Equity (SCALE) that “promote the use and development of language as an integral part of instruction.” These principles are: Support sense-making, Optimize output, Cultivate conversation, and Maximize linguistic and cognitive meta-awareness. To address this, the instructional materials include language routines that “help teachers embrace these principles during instruction.” Each module contains a Language Development page in the Teacher Edition that states where the language routines should be used. On the lesson pages of the Teacher Edition, there are Support-Sense Making boxes that describe how the language routine can be used. Also, there are notes in the margin of the teacher’s edition providing connections from the strategy to the principle.

The instructional materials reviewed for HMH Into AGA meet expectations for providing strategies for gathering information about students’ prior knowledge within and across grade levels.

• At the beginning of the year, students’ prior knowledge is gathered through a Prerequisite Skills Inventory, which states, “This short-answer test assesses core precursor skills that are most associated with on-grade success.” (Assessment Guide)
• Each module begins with Are You Ready?, a diagnostic assessment of prior learning related to the current grade-level standards. Intervention materials are provided to assist students not able to demonstrate the necessary skills. Commentary for each standard explains how the prior learning is relevant to the current module’s content. 
• Prior learning is identified in the Mathematical Progressions section at the beginning of each module and lesson of the Teacher Edition.

The instructional materials reviewed for HMH Into AGA meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

• The module overview in the Teacher Edition contains “Common Errors” as students engage in an introductory task and provides questioning strategies intended to build student understanding.
• The Spark Your Learning planning page for each lesson in the Teacher Edition includes a Common Error section related to the content of the lesson that identifies where students may make a mistake or exhibit misunderstanding. There is a rationale that explains the likely misunderstanding and suggests instructional adjustments or steps to help address the misconceptions. 
• There are also “Watch For” boxes and question prompts that highlight areas of potential student misconceptions.

The instructional materials reviewed for HMH Into AGA partially meet expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

• Each lesson ends with two or three Spiral Review questions for ongoing practice in the More Practice/Homework section.
• Online interactive lessons and homework practice provide students with immediate notification that answers are correct or incorrect.
• The online lessons are the same as in the print textbook and provide immediate notification of correct or incorrect answers, but do not provide feedback for changing incorrect answers.
• Each Module Review has a scoring guide/checklist, so students know which questions they answer correctly. The scoring guide/checklist does not provide feedback for changing incorrect answers.
• Digital assessments are auto-scored and generate recommendations that can provide feedback to teachers but not directly to students.

The instructional materials reviewed for HMH Into AGA meet expectations that assessments clearly denote which standards are being emphasized. 

The standards alignment for each item on the Prerequisite Skills Inventory, Beginning-of-Year, Middle-of-Year, End-of-Year, and Module Tests are listed in the Assessment Guide on Individual Record Forms. Each Performance Task includes the standards in the teacher pages of the Assessment Guide, although the individual questions do not indicate which standards are being assessed.

The instructional materials reviewed for HMH Into AGA partially meet expectations that assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

• Each lesson has a diagnostic assessment, Are You Ready?, correlated to standards and a suggested intervention for struggling students. The materials state that when using Online Ed, teachers can assign the Are You Ready? digitally “for immediate access to data and grouping recommendations.” 
• The Planning and Pacing Guide states,  “Check Understanding is a quick formative assessment in every lesson used to determine which students need additional support and which students can continue on to independent practice or challenges.” Check Understanding presents a limited number of questions, usually one to three, which includes a digital option that can be “auto-scored online for immediate access to data and recommendations for differentiation.” 
• Each performance task includes a task-specific rubric indicating a level 0 response through a level 3 response. The structure of the rubrics is the same, but specific words are changed to reflect the mathematical content of the module. Level 3 indicates that the student made sense of the task, has complete and correct answers, and checked their work or provided full explanations. Level 2 indicates that the student made sense of the problem, made minor errors in computation or did not fully explain answers. Level 1 indicates that the students made sense of some components of the task but had significant errors in the process. Level 0 shows little evidence that the student has made sense of the task or addressed any expected components and has an inability to complete the processes. 
• The Individual Record Forms in the Assessment Guide suggest Reteach Lessons that teachers can use for follow-up based on the module assessments, but there are no other suggestions for follow-up with students or guidance to teachers.
• The Individual Record Forms for the Prerequisite Skills Inventory, Beginning-of-Year, Middle-of-Year Test, and End-of-Year Tests do not suggest Reteach Lessons or provide other guidance that teachers can use for follow-up with students.
• The Performance Task Rubrics for the Unit Performance Tasks do not suggest Reteach Lessons or provide other guidance that teachers can use for follow-up with students.

The instructional materials reviewed for HMH Into AGA include Scales to Track Learning Goals at the end of each lesson. The Teacher Edition introduction states, “The scales below can help you and your students understand their progress on a learning goal.” The scale progresses from 1 to 4. For example from Algebra 2, Lesson 9.1 states:

1. “I can identify an exponential equation and a logarithmic equation.
2. I can convert exponential and logarithmic equations to their inverse forms.
3. I can define and evaluate logarithms.
4. I can define and evaluate logarithms, and I can explain my steps to others.”

Each lesson includes “I’m in a Learning Mindset!” which gives students a prompt regarding the purpose of the lesson. For example, Algebra 1, Lesson 3.1, Perseverance states, “How does defining variables clearly give you a more confident mindset when solving a problem? When can you use intercepts to find solutions? How can you check to see if your answer makes sense?”

The instructional materials reviewed for HMH Into AGA meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• At the beginning of each module, Teaching for Depth provides information on strategies to use when teaching the concept, including Represent and Explain, which focuses on ways for students to describe and picture a concept, or Make Connections, which helps students understand a new idea by connecting it to previous knowledge.
• At the beginning of each module, Mathematical Progression Across the Grades makes connections to both prior and future skills and standards to scaffold instruction.
• At the beginning of each module, Diagnostic Assessment, Are You Ready? allows teachers to “diagnose prerequisite mastery, identify intervention needs, and modify or set up leveled groups.”
• Each lesson provides Warm-up Options to activate prior knowledge such as Problem of the Day, Quick Check for Homework, and Make Connections.
• Throughout the lessons, there are notes, strategies, sample guided discussion questions, and possible misconceptions that provide teachers structure in making content accessible to all learners.
• Student practice starts with up to four Check Understanding exercises to complete with guidance before moving to independent work in On My Own or More Practice/Homework.

The instructional materials reviewed for HMH Into AGA meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

• There are Reteach and Challenge activities for each lesson.
• Each module includes Plan for Differentiated Instruction that provides teachers with teacher-guided, Small-Group Options and self-directed Math Center Options based on student need: “On Track/Mixed Ability, Almost There (RtI), and Ready for More.”
• Each lesson provides Leveled Questions in the teacher’s edition identified as DOK 1, 2, and 3 with an explanation of the knowledge those questions uncover about student understanding. 
• There are three “Language Routines to Develop Understanding” used throughout the materials: 1) “Three Reads: Students read a problem three times with a specific focus each time.” 2) “Stronger and Clearer Each Time: Students write their reasoning to a problem, share, explain their reasoning, listen to and respond to feedback, and then write again to refine their reasoning.” and 3) “Compare and Connect: Students listen to a partner’s solution strategy and then identify, compare, and contrast this mathematical strategy.”

The instructional materials reviewed for HMH Into AGA meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

• Each unit includes a STEM Task and a Unit Project which include multiple entry-points and a variety of solution strategies. Teachers are provided with possible answers as well as What to Watch For tips, which include “Watch for students who are struggling with learning mathematics despite their best efforts and hard work. Help them become more responsive to feedback by asking: How can you assert your own needs and viewpoints in non aggressive or defensive ways?” and “Watch for students who resist receiving feedback about their own work or who are reluctant to provide feedback to others by asking them these questions.” (Geometry, Unit 4)
• Each lesson begins with Spark Your Learning, which is an open-ended problem that allows students to choose their entry-point for applying mathematics and can be solved in a variety of ways. There are suggestions in the teacher’s edition to help students access the context of the problem. For example, in the side margin of the teacher’s edition, Motivate provides prompts. Algebra 1, Lesson 8.2 states, “Select groups of students who used various strategies and tools to share with the class how they solved the problem. As they present their solutions, have each group discuss why they chose a specific strategy and tool.” Geometry, Lesson 11.1 states, “Have students look at the photo in their books and read the information contained in the photo. Then complete Part A as a whole-class discussion. Give the class the additional information they need to solve the problem. This information is available online as a printable and projectable page in the Teacher Resources.” Algebra 2, Lesson 15.2 states, “Turn and Talk Ask students to draw a Ferris wheel that meets these conditions to help them visualize the situation.”
• Support for Turn and Talk in the teacher’s edition provides suggestions to help students using a variety of strategies. Teachers are often prompted to “Select students who used various strategies and have them share how they solved the problem with the class.”

The instructional materials reviewed for HMH Into AGA meet expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

In addition to the strategies for meeting the needs of a range of learners described in Indicator 3s, there is further support in place for English Language Learners (ELLs) and other special populations. Examples include:

• For ELLs, there is Language Development in each module which includes linguistic notes that provide strategies intended to help students struggling with key academic vocabulary, such as “Speak with students about words that can have multiple meanings,” “Listen for students who do not distinguish between minus...and the negative sign,” and “Visual cues help students…”
• Language Objectives are included in every lesson.
• There are Reteach, RtI Tier 2, and RtI Tier 3 worksheets that can be assigned online or printed.
• There are Turn and Talk prompts designed to support students in other special populations, such as “go back and reread the problem and break it into pieces. For example: What do you know? What do you need to find?”

The instructional materials reviewed for HMH Into AGA meet expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

In addition to the strategies for meeting the needs of a range of learners described in Indicator 3s, there is further support in place for advanced students. Examples include:

• There are optional lessons provided online that teachers may choose to utilize with advanced students. 
• Each lesson has a corresponding Challenge page, provided in print or online, addressing the same concepts and standards where students further extend their understanding and often use more complex values in their calculations.

The instructional materials reviewed for HMH Into AGA provide a balanced portrayal of various demographic and personal characteristics. Examples include:

• Lessons contain a variety of tasks that interest students of various demographic and personal characteristics. 
• Names and wording are chosen with diversity in mind. The materials include various names throughout the problems (e.g., Jayson, Suyin, Malik, Tressa, Anton, Jasmine, Yu, Felice, Sonia, Roselyn, Tracy, Tran, Arie, Miguel, Maria) that are used in ways that do not stereotype characters by gender, race, or ethnicity.
• When multiple characters are involved in a scenario, they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias, and there is no pattern in one character using more/fewer sophisticated strategies.
• When people are shown, there is a balance of demographic and personal characteristics.

The instructional materials reviewed for HMH Into AGA provide opportunities for teachers to use a variety of grouping strategies. Examples include: 

• Each lesson provides teachers with a differentiated plan that includes small-group options. 
• The materials provide students with self-directed activities at math centers.
• Throughout the materials, there are ample opportunities for students to Turn and Talk with a partner. 
• Using the Check for Understanding, the teacher is directed to pull students into small groups.

The instructional materials reviewed for HMH Into AGA encourage teachers to draw upon home language and culture to facilitate learning. Examples include:

• The student glossary is in both English and Spanish.
• Each module includes School-Home Letters in multiple languages: Spanish, English, Portuguese, and Haitian Creole.

The instructional materials reviewed for HMH Into AGA integrate technology, including digital lessons and virtual tools. Students can complete tasks and activities from the Student Edition through an interactive format. Examples include:

• All lessons and tasks are built into interactive digital formats as well as print versions.
• Students can draw pictures, create shapes, and type to show their thinking on the interactive lessons using a virtual sketchpad. Students also drag and drop correct values into tables, graphs, or diagrams. 
• On the Spot videos of specific lesson problems are in the online student resources and provide the opportunity for students to review their work with their families by watching the video. These focus on content rather than MPs.

The instructional materials reviewed for HMH Into AGA are web-based and compatible with multiple internet browsers. Examples include:

• The materials are platform-neutral and compatible with Chrome, ChromeOS, Safari, and Mozilla Firefox.
• Materials are compatible with iPads, laptops, Chromebooks, and other devices that connect to the internet with an applicable browser.

The instructional materials reviewed for HMH Into AGA include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. Examples include:

• Lesson problems from the Student Edition, assessments, and unit performance tasks are provided to be completed and scored using technology, providing students with feedback on whether the answers are correct or incorrect.
• Assessments can be created using a question bank that repeats the questions presented throughout the interactive lessons. Teachers can also create their own assessment items. However, teachers cannot modify existing questions.
• The online system has dynamic reporting by assignment or standards. If teachers are using the online system, they can view student progress for interim growth, module readiness, and lesson practice and homework.

The instructional materials reviewed for HMH Into AGA include opportunities for teachers to personalize learning for all students. Examples include:

• Teachers can assign lesson components, problems, and assessments, as well as view assessment analytics. 
• Teachers can group students according to individual needs. The online component has Recommended Groups which “synthesizes data from assessments and places students into leveled groups” (Pacing Guide). Recommended lesson resources can be assigned to each group.
• Teachers can create assessments using a bank of items.

The instructional materials reviewed for HMH Into AGA provide opportunity to be adapted for local use. Examples include:

• Pieces of a lesson or problems can be assigned directly to students or groups of students. 
• There is a question bank for teachers to create assessments. The bank repeats the questions that are already included in each lesson, and these questions cannot be modified. However, teachers can create their own questions as well.

The instructional materials reviewed for HMH Into AGA do not incorporate technology that provides opportunities for multiple students to collaborate with the teacher or one another. However, materials do provide an opportunity for teachers to collaborate with one another in the Teacher’s Corner under the Professional Learning tab. Teachers can attend virtual live events to ask questions or engage in self care practices such as Teacher Talk with Yoga.