The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, students are expected to use multiple representations to further develop conceptual understanding. 

Examples of the development of conceptual understanding include:

• N-RN.1: In Algebra 2, Unit 4, Lesson 6.1, Warm-Up, students explain how given expressions with various exponents are equivalent and provide an additional equivalent expression. This helps students develop conceptual understanding of the properties of exponents.

• A-REI.6: In Algebra 1, Unit 2, Lesson 17, students determine solutions to a system of equations through inspection and use those solutions to determine that there are infinitely many solutions to the given system. Students recognize equivalent equations and explain what equivalence means in terms of solutions to systems of linear equations. Students also interpret what the solution of a system of equations would be if the equations represent parallel lines. 

• F-IF.2: In Algebra 1, Unit 4, Lesson 4, students match words to the symbolic rule of given functions. Students use symbolic notation to define the perimeter of a rectangle with a set height and varying width. Students graph the results and find both input and output values from the graph. In Algebra 1, Unit 4, Lesson 5, students expand this understanding to write functions that describe data plans for their phones and compare competing plans both symbolically and graphically. Students describe in writing how the graphic visualization matches the symbolic rule.

• G-SRT.6: In Geometry, Unit 4, Lesson 4, students connect angle measurements with ratios of side lengths in right triangles. In Lesson 6, students extend their thinking about the ratios of side lengths for any values of the triangle’s angles. Students define cosine, sine, and tangent and compare the answers they get using these definitions to the table used in the previous lessons. 

• G-GMD.1: In Geometry, Unit 5, Lesson 13, students partition prisms in order to build the volume formula for a pyramid as opposed to using a given formula to calculate volume. Students connect the volume of a prism to the volume of a pyramid with a base area equivalent to that of the prism. 

• S-ID.7: In Algebra 1, Unit 3, Lesson 4, students develop their understanding of slope while studying the line of best fit related to a scatterplot. Students answer a series of questions to help develop their understanding of what happens to the slope of the line of best fit if one of the elements changed. An example is as follows: “How would the scatter plot and linear model change if grapefruits were used instead of oranges?” Additionally, students discuss the role of the y-intercept during this lesson by explaining what the y-intercept means in the particular context of the problem. Students repeatedly demonstrate understanding of the concept of slope and y-intercept in given data sets throughout the remainder of Algebra 1.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series 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 provide opportunities to independently demonstrate procedural skills throughout the series. The curriculum guide states, “We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.” Throughout the series, procedural skills are developed through the lessons and the problem sets for each of the lessons. Each problem set has cumulative practice problems to review previously addressed procedural skills. 

Examples that show opportunities for students to independently demonstrate procedural skills across the series include: 

• A-SSE.1: In Algebra 1, Unit 6, and Algebra 2, Unit 2, students develop procedural skill and fluency, as they make observations related to the structure of a factored quadratic expression and the zeros of that expression when graphed as a function. Students predict possible factors and their forms by evaluating the expression for varying input values, looking at graphs, reading tables, and exploring end behavior.

• A-APR.6: In Algebra 2, Unit 2, Lesson 17, students perform polynomial division by using long division, and polynomial factorization in order to write a higher-order polynomial as a product of its linear and/or non-linear factors.

• F-IF.1,2: In Algebra 1, Unit 4, Lessons 2–5, students use function notation and develop fluency with substitution and calculations. 

• F-IF.4: In Algebra 1, Unit 5, and Algebra 2, Unit 6, students develop procedural skill and fluency, as they explore key features of multiple types of graphs. In Algebra 1, Unit 5, over the course of nine lessons, students build an understanding of how exponential growth differs from linear growth. Students encounter different contexts and use expressions, graphs, and tables to distinguish between the two types of functions. Students gain fluency in how to compare two exponential functions, how the functions differ in their expressions, and what that will mean for growth in context. In Algebra 2, Unit 6, students use these skills with trigonometric functions, recognizing and discussing amplitude, frequency, and shifts in many and varied contexts over the course of four lessons.

• F-BF.3: In Algebra 1, Unit 4, Lesson 14, students analyze the type of transformation a constant value creates with an absolute value function. In the cumulative practice problem set, there are more problems for students to practice this skill.

• G-GPE.4: In Geometry, Unit 6, Lesson 10 practice, students write equations using parallel slopes and identify equations that are parallel to a specific line. In Geometry, Unit 6, Lesson 11, students have the same opportunity with perpendicular slopes.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for supporting 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. 

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

• A-REI.11: In Algebra 1, Unit 4, Lesson 9, students engage with two or more graphs simultaneously, interpreting their relative features and their average rates of change in context. Examples are as follows: population, trends of phone ownership, and the popularity of different television shows. In Activity 9.2, students compare two functions by studying graphs and statements in function notation. In Algebra 2, Unit 4, Lesson 15, Activity 15.3, students solve a system of exponential equations involving a cicada population using logarithms and graphing; specifically, students explain why the intersection of the two graphs can be used to estimate when the cicada population will reach 100,000.

• F-IF.6 and F-BF.1: In Algebra 1, Unit 4, Lesson 18, students use functions to model real-life applications. Students create and analyze functions to model cell phone battery power using given data. In the cumulative practice, students engage with relevant applications related to distance driven over time and the relationship between temperature and cricket chirps.

• G-SRT.8: In Geometry, Unit 4, Lesson 9, Activity 9.3, students use the safe ladder ratio to determine the safe ladder angle, and students use the calculated angle to decide if a ladder’s length is sufficient for a given scenario. Students also determine if it is possible to adjust the ladder to a safe angle and explain their reasoning. In Lesson 10, students solve application problems using trigonometry. Students find the perimeter of figures inscribed in a circle using trigonometric ratios, and students solve problems involving an airplane’s angle of descent and path length as it descends to its destination.

• G-MG.2: In Geometry, Unit 5, Lesson 17, Activity 17.3, students apply volume and density to determine the number of fish that could be housed in a tube-shaped aquarium with an open 4-foot cylinder in the middle for viewing.

• S-ID.6 and N-Q.3: In Algebra 1, Unit 3, Lesson 6, students apply residual value, line of best fit, and percent error to data related to the weight of oranges in a crate. Students also apply these concepts in practice problems 4 and 5 involving applications of car sales and temperatures.

• S-IC.1: In Algebra 2, Unit 7, Lesson 3, students evaluate the randomness in population samples. Students determine the best way to have random samples and the factors that could affect the randomness in several different scenarios. Students draw conclusions from a variety of non-routine application problems.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations that the three aspects of rigor are not always treated together and are not always treated 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 lesson begins with a Warm-Up, and this is often an opportunity for students to develop their number sense or procedural fluency. After the Warm-Up, there are activities that do one or more of the following: provide context, introduce, formalize or practice vocabulary, work toward mastery, introduce a new concept, or provide an opportunity to model. The embedded classroom routines also contribute to a balance of the three aspects of rigor. These routines include the following: Analyze It, Math Talks, Group Presentations, Notice and Wonder, Think and Share, and others. At the end of the lesson, there is a synthesis activity where the teacher leads a discussion to formalize the learning. The lesson ends with a Cool-Down for students to work independently on the lesson concepts. Each of the aspects of rigor are addressed with this lesson structure throughout the series. 

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

• In Algebra 1, Unit 6, Lesson 9, students demonstrate a balance of the three aspects of rigor while addressing A-SSE.2,3. In the Practice Problems, students determine if two representations of quadratic expressions are equivalent. Students perform calculations related to quadratic functions in the context of real-world applications, one context of which is a football player throwing a pass. In the Student Task Statements, multiple representations of factorable quadratics are presented. Students examine standard form, factored form, and a geometric representation of a factorable quadratic expression. Students explain their reasoning when determining if an expression is in factored form. A-SSE.2 is also addressed in Geometry, Unit 6, Lesson 5 when students apply the distributive property and squared forms of a binomial to derive the equations of circles. Students practice procedural skills using the distributive property and writing different forms of equivalent expressions. They also solve application problems related to equations of circles and distances. In Algebra 2, Unit 2, Lesson 23, A-SSE.2 is addressed by having students build conceptual understanding of polynomial identities. Through an application problem, students define an identity. Then, they multiply expressions to generalize patterns in polynomials, which develops procedural skill.

• In Algebra 1, Unit 1, Lessons 2 - 5, the materials provide activities that engage students in all aspects of rigor with respect to S-ID.2. Specifically, in Lesson 2, Activity 2.2, students represent and analyze histograms; in Lessons 3 and 4, students consider dot plots to inform a conversation about the shape of a distribution; and in Lesson 5, students calculate the measures of central tendency. In Lesson 9, students perform statistical calculations. In ensuing lessons, students further develop statistical reasoning. In Lesson 10, students consider what variables are needed to analyze a situation and describe data displays used to compare two sets of data. In Lesson 11, students explore and reason about symmetry in a data set. In Lesson 12, students investigate standard deviation and other measures of variability; and in Lesson 14, students investigate the effect of outliers. In Lessons 15 and 16, students compare measures of center and variability in context as well as determine the best measure of center and variability for several data sets. Students also design an experiment to answer a statistical question, collect data, analyze data using statistics, and communicate the answer to the statistical question.

• In Geometry, Unit 6, Lesson 7, students solve problems related to distance and parabolas. In the Student Task Statements, students answer questions related to the distance from the focus to the vertex of a parabola and the effect that distance might have on the shape of the parabola. Students use the definition of a parabola and the distance formula to determine if a point is on the parabola. Students defend their answers and try to generalize how one would know if a point is on the parabola given a graph. Students also demonstrate an understanding of what happens to the shape of a parabola if one was to move the directrix closer to the focus.

• In Algebra 2, Unit 5, Lessons 8 and 9, students consider the impact of scaling the input or the output values of a function. Students examine how graphs change based on the scaling of the input (horizontal) or output (vertical) values. In Lesson 9, Practice Problems, students determine if different statements that are made based on scaled inputs or outputs are correct given different representations of functions. Students also use data to determine an appropriate scale factor that would model the population of sloths given an initial function.

• In Algebra 2, Unit 2, Lesson 20, students write a simple rational equation about batting average to develop procedural skill, and they demonstrate conceptual understanding when working with a word problem and writing it algebraically. The extension and what-if questions about the rational equation address application of rational equations.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards.

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

• In many units throughout the series, students answer the questions “What do you notice?” and  “What do you wonder?” in lesson activities. The goal of these questions is to guide classroom conversation toward the mathematical material that the class is about to address. These questions increase accessibility for students by providing entry points to the context, which aids in making sense of the tasks. 

• In Algebra I, Unit 4, Lesson 1, students analyze the relationship between the number of bagels purchased and the cost of the bagels to determine how three different costs could all be true. 

• In Geometry, Unit 4, Lesson 11, students examine how inscribed polygons with increasing numbers of sides can lead to an approximation of $$\pi$$. Students make sense of the problem to determine appropriate methods for finding a formula to calculate the perimeter of the inscribed polygons, which leads to an approximation of $$\pi$$.

• In Algebra 2, Unit 2, Lesson 24, students create multiple right triangles from a given set of instructions. Students persevere in finding at least one example that does not create a right triangle. Towards the end of the lesson, students develop an identity that can be used to generate Pythagorean Triples.

Examples where students attend to precision include:

• In Algebra 1, Unit 4, Lesson 6, students analyze a graph containing two mappings related to two objects in time. One graph shows a linear piecewise function produced by a drone, and the other graph shows a quadratic function produced by a toy rocket. Students describe the graphs’ representations in terms of the real-world contexts.

• In Geometry, Unit 6, Lesson 7, students articulate what they notice and wonder by attending to precision using language to describe their observations. Students may initially propose less formal or imprecise language, then restate their observation with more precise language in order to communicate clearly. Relevant vocabulary includes the following: equidistant, congruent segments, and parabola.

• In Algebra 2, Unit 3, Lesson 6, students solve simple equations involving squares and square roots. The teacher notes state: “Students attend to precision when they reason about solutions to equations involving squares and square roots from the meaning of the √ symbol (MP6).” In this lesson, students explore the idea that every positive number has two square roots. The convention of giving only the positive root is also discussed in terms of its precise meaning. Students explore the use of the radical symbol as a tool of precision.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. Across the series, there is an intentional development of MP2 and MP3 that reaches the full intent of the MPs. There are many examples in the instructional materials of MPs 2 and 3 where students reason abstractly and quantitatively or construct viable arguments and critique the reasoning of others.

Examples of where and how the materials use MP2 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1, Unit 1, Lesson 14, students determine whether or not to exclude an outlier. In the Student Task Statements, Problem 14, students determine if there are outliers for a data set, explain why any outliers might exist, and determine if the outliers should be included in the analysis of the data.

• In Algebra 1, Unit 5, Lesson 10, students examine data from the number of coffee shops since 1987. Students must determine, from given intervals, the average rates of growth and support their choice.

• In Algebra 1, Unit 7, Lesson 1, students write a quadratic equation and are prompted not to solve it. In writing an equation and interpreting the solution in its context, students practice reasoning quantitatively and abstractly.

• In Geometry, Unit 3, Lesson 2, students create a scale model of the Solar System to verify the distance the Moon would be from the Earth when fully eclipsing the Sun.

• In Algebra 2, Unit 7, Lesson 16, students take two readings of their pulse. For one reading, students count the beats out loud while watching the clock, and for the second reading, they take a few deep breaths, close their eyes and have someone else watch the clock. Students then compare the two rates. Data from the whole class is collected and a discussion held as students reason quantitatively and abstractly together.

Examples of where and how the materials use MP3 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1, Unit 1, Lesson 11, students explain their reasoning and critique the reasoning of others as they determine if a data display matches a written statement or not. In this activity, students have small-group discussions and examine scenarios from a classmate’s point of view. Students also construct arguments they can defend for their own matches as well as arguments for why they might disagree with their partner.

• In Algebra 1, Unit 7, Lesson 23, students explain why a vertex is a maximum or a minimum. Students explain their reasoning concerning which performance gives the greater maximum revenue without creating a graph. Students construct an argument, and during the class discussion, they critique the reasoning of others in the class.

• In Geometry, Unit 2, Lesson 3, students construct arguments to describe that congruence through transformations requires a series of transformations where corresponding parts match each other. In the Cumulative Practice Problem Set, Problem 2, students argue for the congruence of two triangles based on a rotation and explain their reasoning, citing the argument for congruence based on the transformation.

• In Geometry, Unit 6, Lesson 9, students construct viable arguments during the lesson synthesis by answering a question about which form of an equation of a line they prefer. In the previous lesson, students used equations in multiple forms to find out what the slope of the line is and what point each line passes through. Students explain why they prefer a specific form over another.

• In Geometry, Unit 8, Lesson 2, students, in groups, draw slips of paper with a name on them from a bag with an unknown number of slips. They record the name, replace the slip, pass the bag, and draw again. After 15 draws, each student in the group makes predictions of how many names and how many slips were in the bag. If the group has consensus, they draw another round. Each time, they construct new arguments and critique the thinking of others.

• In Algebra 2, Unit 2, Lesson 4, students answer questions about operations on polynomials. They experiment to develop reasons to support their answers. During the group discussion, students defend and critique the reasoning of their classmates as they describe events as subsets.

• In Algebra 2, Unit 4, Lesson 14, students construct logical arguments to justify solutions and explain why a certain value is a reasonable estimate for a given logarithm.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. There are multiple problems and activities throughout each unit in which students use and create mathematical models to enrich the mathematics. Students also choose from a variety of tools throughout the lessons, including digital tools provided in a drop down menu in the online materials (MP5). 

Examples where students model with mathematics include:

• In Algebra 1, Unit 2, Lesson 20, Activity 20.2, students solve a problem about gas in a lawn mower. The teacher materials state, “To reason about the problem, students need to interpret the descriptions carefully and consider their assumptions about the situation. To make sense of the situation, some students may define additional variables or use diagrams, tables, or other representations. Along the way, they engage in aspects of modeling (MP4)”. Students identify the important quantities in this scenario, identify the relationships, and write at least one inequality to represent their conclusions.

• In Algebra 1, Unit 4, Lesson 17, students write a linear function for data concerning the percent of cell phones in homes in the United States since 2004. Students answer questions leading to finding a model for the inverse of the function. 

• In Geometry, Unit 1, Lesson 9, students use perpendicular bisectors to decide which stores in the city will be responsible for orders based on the store’s location compared to where the order will be delivered. Students use real-world situations to apply their knowledge and make approximations on their calculations to simplify distributions for a local store.

• In Geometry, Unit 7, Lesson 14, students work with a pizza slice as a model of a sector of a circle. Students compute the cost per square inch of pizza slices from four vendors by computing the sector area. This engages students in the modeling process by reporting their findings and considering other variables.

• In Algebra 2, Unit 2, Lesson 16, students engage in aspects of the modeling process by making reasonable estimations and determining reasonable constraints in the context of real-world scenarios. In “Are You Ready for More?”, students consider different aspects of manufacturing, other than simply minimizing materials, in order to make sense of an open- ended problem. 

• In Algebra 2, Unit 7, Lesson 14, students speculate whether the differences of the means in small experimental groups can be reduced by randomly regrouping the data. Students approximate the distribution of simulated differences of means by using a normal distribution. 

Examples where students choose and use appropriate tools strategically include:

• In Algebra 1, Unit 7, Lesson 17, students use the form of a quadratic equation as a “tool” to solve problems. Students “write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward.”

• In Algebra 1, Unit 5, Lesson 19, students compare linear and exponential growth involving simple and compound interest. Students strategically use technology, whether they make a graph (for which they will need to think carefully about the domain and range) or continue to tabulate explicit values of the two functions (likely with the aid of a calculator for the exponential function). 

• In Geometry, Unit 7, Lesson 7, Activity 7.2, students create an arbitrary triangle, use angle bisectors and constructions to find the incenter, and construct the triangle’s inscribed circle. The narrative states: “Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).” The narrative also states: “Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.”

• In Geometry, Unit 8, Lesson 9 students use two-way tables as a sample space to decide if events are independent and to estimate conditional probabilities. Although technology is not required, it is recommended that technology be made available as there are opportunities for students to choose to use appropriate technology to solve problems. 

• In Algebra 2, Unit 4, Lesson 5, students create an exponential function given a table of values. During this lesson, teachers make sure that students have access to a spreadsheet tool to reason about the given questions. This helps students focus on the questions rather than the calculations, and students use tools to their advantage during the lesson.

• In Algebra 2, Unit 5, Lesson 11, students apply transformations on functions to determine the best model for a data set, specifically temperature data from heating objects. The Lesson Narrative states: “This can be done by hand via experimenting, but students may also choose to use graphing technology to help choose the appropriate translations, scalings, and reflections (MP5).”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. Each of these MPs is cited numerous times across the series, and the Algebra 1 Extra Support Materials cite each of these MPs. Additionally, across the series, the majority of the time MP7 and MP8 are used to enrich the mathematical content, and there is intentional development of MP7 and MP8 that reaches the full intent of the MPs.

Examples of where and how the materials use MP7 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1, Unit 4, Lesson 12, students use piecewise notation to look for structure when graphing those functions. 

• In Geometry, Unit 2, Lesson 13, students look for and make use of structure by working backward from the statement they are trying to prove about parallelograms.

• In Geometry, Unit 3, Lesson 5, students use Notice and Wonder to examine triangles whose midpoints connect to form other smaller triangles. Students notice these smaller triangles are dilations of the larger triangle.

• In Geometry, Unit 8, Lesson 9, students notice and make use of structure through a Math Talk to recognize fraction bars as part of a fraction, and as representing division. 

• In Algebra 2, Unit 2, Lesson 8, students make a conjecture based on creating their own function and analyzing the end behavior to see if it matches their conjecture. The focus of the lesson is “using the structure of the expressions to understand how the term with the highest exponent dictates end behavior even when other terms may have larger values at inputs nearer to zero due to coefficients.” 

• In Algebra 2, Unit 7, Lesson 7, students find the area under the normal curve and interpret the proportion of values at different intervals. These problems utilize a real-world scenario, and students connect these applications to the theoretical study of the normal distribution. 

Examples of where and how the materials use MP8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1, Unit 1, Lesson 6, students experiment by changing numbers in a provided spreadsheet to discover what and how the number impacts the outputs. Students express regularity in repeated reasoning as they observe the outcome of different inputs to generalize the operations in the formula cell.

• In Algebra 1, Unit 2, Lesson 8, students repeatedly rewrite equations to isolate different variables. Students rely on previous knowledge of solving equations to generalize their reasoning as they work with literal equations. 

• In Algebra 1, Unit 6, Lesson 5, Activity 5.3, students look for repetition in their calculations for a falling object and how this relates to quadratic functions. The teacher materials state, “To find a new expression that describes the height of the object, students reason repeatedly about the height of the object at different times and look for regularity in their reasoning (MP8)” as they write new quadratic expressions. Students also use repetition in their calculations to complete provided tables to determine where an object would be at a certain time and use that information to write new expressions.

• In Geometry, Unit 1, Lesson 6, students construct parallel and perpendicular lines, and students engage in MP8 as they repeatedly construct these different types of lines. 

• In Geometry, Unit 7, Lesson 8, students perform multiple arc length and sector area calculations to generalize formulas.

• In Algebra 2, Unit 1, Lesson 1, students look for a pattern within the Tower of Hanoi puzzle, where students complete a puzzle by building a tower. Students play the game and make some conjectures about the smallest number of moves you can make to complete the tower.

• In Algebra 2, Unit 1, Lesson 8, students use a table of values to generalize formulas for finding the nth term of a sequence. By examining patterns and applying repeated reasoning, students generalize the definition of a sequence into an equation and/or function.

• In Algebra 2, Unit 6, Lesson 10, Activity 10.3, students determine specific trigonometric values for large angles. “Students make connections between angles greater than 2 and between 0 and 2 that correspond to the same point on the unit circle (MP 8)”.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.  

The materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. The Implementation Guide includes:

• the general overview of how to use the materials,

• detailed explanation of the Math language routines,

• how to use the math prompts and the research behind them, and

• the research behind supporting diverse learners. 

The teacher materials provide the specific details of how to implement the lesson, with guidance in implementing all the components of the lesson. Examples include:

• Algebra 1, Unit 1, Lesson 1 includes warm-up 1.1 “Types of Data”, activity 1.2 “Representing Data about You and Your Classmates,” and cool down 1.3 “Categorizing Questions”. Each of these is supported by a launch, the instructional routine, and the synthesis.

• Geometry, Unit 5, Lesson 1, warm-up 1.1 ”Which One Doesn’t Belong”, activity 1.2 “Axis of Rotation”, activity 1.3 “From Three Dimensions to Two”, and cool down 1.4 “Telescope.”

• Algebra 2, Unit 1, Lesson 1 includes warm-up 1.1 “What’s Next?”, activity 1.2, “The Tower of Hanoi” activity 1.3, “Checker Jumping Puzzle,”, and cool down 1.4.

The materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes the lesson pacing, required preparation, materials, student and teacher goals, and math language routines to support the teacher. Examples include:

• Algebra 1, Lesson 1.2 states, “as students discuss their ideas, monitor for those who …”  The synthesis in the lesson provides key takeaways for the lesson as well as discussion prompts and suggested answers.

• Geometry, Unit 5, Lesson 1 Launch states, “Arrange Students in groups of 2-4. Display images for all to see.” The Activity Synthesis states, “During the discussion, ask students to explain the meaning of any terminology they use, such as round, corners, circular, or symmetric. Also press students on unsubstantiated claims.” 

• Algebra 2, Unit 1, Lesson Narrative includes both the content and practice standards as well as common misconceptions to support the standards alignment.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for  containing adult-level explanations and examples of the more complex course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. There are supports provided for teachers to develop their own understanding of the current material being taught, and there are explanations and examples  for concepts beyond the current courses.

Lesson Narratives provide specific information about the mathematical content within the lesson and are presented in adult language. These narratives contextualize the mathematics of the lesson to build teacher understanding and give guidance on what to expect from students and important vocabulary. Examples include:

• In Algebra 1, Unit 5, Lesson 17, students investigate compounding intervals. The explanation in the 17.1 activity synthesis provides guidance to assist teachers in developing student knowledge.

• In Geometry, Unit 6, the guidance provided for teachers pertains to what students will be doing in the unit and does not extend to concepts beyond the course. “The first few lessons examine transformations in the plane. Students excounter a new coordinate transformation notation which connects transformations to functions.” The section continues in this manner relating the concepts that will be addressed throughout the unit.  

• In Algebra 2, Unit 6, Lesson 2, sample student responses and sample discussion questions and answers are provided to support teachers in teaching the material, but there are no explanations and examples beyond the current course. Questions listed for discussion include “Which side is the hypotenuse of triangle ABC and what is the length,” and “What is the sine of angle A?”

In the Implementation Guide, there are articles that contain adult-level explanations and examples of where concepts lead beyond the indicated courses. These can be used for study to renew and fortify the knowledge of secondary mathematics teachers and other educators. There are some articles that pertain to the entire series, some that pertain to courses, and some that pertain to individual units within courses. Examples include:

• In Algebra 1, Inverses is an essay from the Noyce-Dana project that pertains to the entire course. In the essay, Farrand-Shultz unifies the different contexts in which “inverses” are used in mathematics, including the multiplicative inverse of a number, the inverse of a function, and the inverse relationship between differentiation and antidifferentiation. Many of these notions can be unified by noticing the operation relative to which the inverse is defined. 

• In Geometry, there is Proof in IM's High School Geometry and Rigor in Proofs, which pertain to the entire course. In these posts, the authors, Ray-Riek and Cardone and Cardone and Rosenberg respectively situate the treatment of learning how to write proofs in IM Geometry within a structure that appears in different forms throughout a math learner’s education, including before and beyond high school mathematics.

• In Algebra 2, The Secret Life of the ax + b Group is an essay from the Noyce-Dana project that pertains to the entire course. In the essay, Howe delves into ways in which the affine group of the line (the ax + b group) surfaces in high school mathematics, and some extensions of standard topics which are suggested by this point of view.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. Correlation information is present for the mathematics standards addressed throughout the series. Each lesson has a Standards Alignment provided for the teacher. This alignment lists the standards the lesson is building on, addressing, and building towards. There is also a Standards Alignment in the Implementation Guide for all courses in the series.

Explanations of the role of the specific course-level mathematics are present in the context of the series. The lesson narrative explains the role of the course-level mathematics in most but not all lessons. Examples include:

• In Algebra 1, Unit 2, Lesson 6 students learn equivalent equations. The lesson narrative states “In middle school, students learned that two expressions are equivalent if they have the same value for all values of the variables in the expressions. They wrote equivalent expressions by applying properties of operations, combining like terms, or rewriting parts of an expression. In this lesson, students learn that equivalent …”

• In Geometry, Unit 5 has a Unit Planner that lists the alignment between the lesson and the corresponding standards. Lesson 1 lists the standards that the lesson is building on, addressing and building toward.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The materials explain the instructional approaches of the program, and reference research-based strategies for each of the instructional routines. The implementation guide includes all instructional routines at both the Unit and the Lesson levels. Appendices in the teacher editions also have references to the research. Examples include:

• Algebra 1, Unit 2, Lesson 13 utilizes the instructional routines of Math Talk, Mathematical Language Routines of Compare and Connect and Discussion Support, and Think-Pair-Share.

• Geometry, Unit 8, Lesson 1 utilizes the instructional routines of Think-Pair-Share and Which One Doesn’t Belong, as well as the Mathematical Language Routines of Stronger and Clearer, and Collect and Display.

Along with each of these routines is the research that supports the strategy. Examples include:

• The Anticipate Monitor…….Connect instructional routine states the research from “5 Practices for Orchestrating Productive Mathematical Discussions (Smith and Stein, 2011).” 

• The Language Routine Discussion Supports states the research, “To support rich discussion about mathematical ideas, representations, context, and strategies (Chapin, O'Connor, & Anderson, 2009).”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The materials include a comprehensive list of supplies needed to support the instructional activities at both the Unit and the Lesson level. The implementation guide also has a master list of all the needed materials. Examples include:

• In Algebra 1, Unit 4, the required materials are listed in the Unit Planner. The required materials are blank paper, copies of blackline master, glue or glue sticks, graphing technology, pre-printed cards cut from copies of the blackline master, pre-printed slips cut from copies of the blackline master, scientific calculators.

• In Geometry, Unit 7, the required materials are listed as colored pencils, geometry tool kits, pre-printed slips cut from copies of the blackline masters, protractors, rulers, scientific calculators, scissors, spreadsheet technology, string. 

• In Algebra 2, Unit 1, Lesson 9, the only required material is graph paper.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the content standards assessed for formal assessments, and the materials provide guidance, including the identification of specific lessons, as to how the mathematical practices can be assessed across the series.

The formal assessments include cool downs, pre diagnostics, mid unit, and end of unit for every unit in every course. There is an additional online test bank that can be searched by concept, content standard and question type. Examples include:

• The cool downs are included for every lesson in every course, and assess a standard(s) that is the focus of that particular lesson.   

• Each unit of each course has a pre diagnostic. The teacher materials list an item description for each question and when the first appearance of the skill or concept occurs in the unit, then gives advice to assist teachers when most students are struggling with the topic. The standards include below-course-level standards on which course-level standards can be built.

• Online materials list the standards in the question details for each question.

In the Implementation Guide, there is a chart for each course that identifies specific lessons within units that can be used to assess individual mathematical practices. The Implementation Guide also includes general guidance for teachers on assessing the mathematical practices, including “I can” statements for each mathematical practice that can be used to monitor students’ engagement with each practice. Examples include:

• MP1: I can show at least one attempt to investigate or solve the problem.

• MP3: I can listen to and read the work of others and offer feedback to help clarify or improve the work.

• MP4: I can refine or revise the model to more accurately describe the situation.

• MP5: I can recognize when a tool is producing an unexpected result.

• MP8: I can notice what changes and what stays the same when performing calculations, examining graphs, or interacting with geometric figures.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for including an assessment system that provides multiple opportunities throughout the courses and series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. All assessments provide a possible rationale that also includes misconceptions to support students who choose those answers. Examples include:

• Rubrics are provided for short answer, constructed response, and extended response prompts on the mid-unit assessments and end-of-unit assessments. The implementation guide explains a three tier classification for constructed response. Extended response items have a 4 tier classification. Answers are included showing how students could get a correct answer and/or what the student could have done wrong. Sample errors and acceptable errors are provided to specify exclusions.

  • For example, item 7 on the Algebra 2 Unit 4 end-of-unit assessment provides a sample student response and rubric describing the 4 tier classification. To earn a Tier 1 classification “work is complete and correct, with complete explanation or justification” and gives a sample “1. (4.5, 310)(or comparable values); the two molds cover the same area at this time. 2. Use the equation $$100 \cdot e^{0.05d} = 100$$ and solves correctly with the natural logarithm.”

  • The three tier classification for constructed response has a general rubric stating, “Tier 1: work is complete and correct, Tier 2: Work shows general conceptual understanding and mastery with some errors, Tier 3: Significant errors in work demonstrate lack of conception understanding and mastery.”

• Rubrics are provided for the modeling prompts and include sample student responses.  All modeling tasks are assessed on two skills: “use your model to reach a conclusion” and “refine and share your model”. Suggestions are provided for how students can improve their score for each skill. 

• All Multiple Choice/Multiple Response options include an item analysis noting any misconceptions and why students may be selecting the wrong answers. 

• The pre-unit diagnostic for Algebra 1, Unit 4, Question 1 states, “If most students struggle with this item: Plan to spend additional time in Lesson 1, Activity 2 calculating how to scale the passport photo by different percentages and if additional practice is needed after Lesson 1 to connect growth factor and percent change, revisit Unit 1, Lesson 19, Activity 1, or use the optional Lesson 2 for more support.”

• The pre-unit diagnostic for Geometry, Unit 4, Question 2 states, “First Appearance of the skill: Lesson 1 Item Description: In a previous unit, students proved the Pythagorean Theorem and used it to find lengths of sides of right triangles. This item will assess whether they may compute an irrational length that needs to be rounded. If most students struggle with this item plan to invite students to find the length of the ramp in Activity 3 in addition to stating the dimensions of the legs.”

• An online customizable question bank allows the teacher to target specific objectives and standards.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices across the series. Examples include:

• In Algebra 1, Unit 7, End-of-Unit Assessment, Item 2, students engage in MP7 when they identify all expressions equivalent to $$x^2 + 6x = 16$$. 

• In Geometry, Unit 2, End-of-Unit Assessment, students have a Multiple Response item to select all statements that are true about a parallelogram. There are 2 Multiple Choice items about similar triangles, and 4 items that are short answer or extended response items about proofs of congruent segments in a parallelogram, measurements of angles of a parallelogram, measurements of an isosceles triangle and rigid motions in proving figures are congruent.

• In Geometry, Unit 3, End-of-Unit Assessment, Item 6, students engage in MP’s 5 and 6 when they reason through “Tyler’s” proof and find his mistakes. 

• In Algebra 2, Unit 2, Mid-Unit Assessment, Item 5, students engage in MP2 when they find a coefficient of a polynomial, a factor of the polynomial, and explain how they know this is the correct coefficient.

• In Algebra 2, Unit 6, End-of-Unit Assessment, students write an equation and sketch a graph for another item on the assessment, and students explain what the different parameters of the equation mean in context for another assessment item.

• All lessons across the series include cool downs at the end of the lesson, to allow students to show they understood the work of that day’s lesson.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning course-level mathematics. Across the series, every lesson has supports and resources provided to teachers to help engage and support students with disabilities in course-level mathematics, as well as students in special populations. Examples include:

• In Algebra 1, Unit 4, Lesson 1, "Three Reads" is provided to support students reading comprehension. The teachers notes state to, “use the first read to orient students to the situation. After a shared reading, ask students ‘what is this situation about?’” and goes on to provide support for the teacher in two additional reads with targeted student outcomes.

• In Geometry, Unit 1, Lesson 6, Math Talk, there are discussion supports that provide sentence frames to support students in explaining their strategies. Teachers are encouraged to have students share their answers with a partner, rehearsing what they will say in large group discussions. The teacher notes clarify that rehearsing provides “opportunities to clarify their thinking.”

• In Geometry, Unit 1, Lesson 8, “Digital Compass and Straightedge Constructions,” encourages teachers to keep the construction on display for students to reference as they work to support an accessibility for “Memory: Conceptual Processing.”

• In Algebra 2, Unit 5, Lesson 8, teachers are encouraged to “represent the same information through different modalities by drawing a diagram. Encourage students who are unsure where to begin to sketch a diagram of a slice of break on graph paper and to share the area that is covered in mold after 1 day, after 2 days, …”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing extensions and opportunities for advanced students to engage with course-level mathematics at higher levels of complexity. The materials provide multiple opportunities for all students to investigate course-level content at higher levels of complexity. Most lessons throughout the series have Are You Ready For More?. The Implementation Guide indicates that Are You Ready For More? problems are differentiated for students who are ready for more of a challenge. There are also optional lessons and optional activities in some lessons that may support learning at a higher level of complexity. If individual students would complete these optional activities, then they might be doing more assignments than their classmates. Examples of these activities include:

• In Are You Ready For More?, Algebra 1, Unit 4,  Lesson 11, students describe and graph unique functions given specific limitations on the domain and/or range.

• In Are You Ready For More?, Geometry, Unit 7, Lesson 9, students analyze the relationships between a sector and the full circle. 

• In the Implementation Guide, “The reason an activity is optional is that it addresses a skill or concept below grade level or addresses a concept or skill that goes beyond the requirements of the standard, or provides an opportunity for additional practice on a concept or skill.”  For example, in Geometry, Unit 7, Lesson 14, there are optional activities 14.3, “A Fair Split”, and 14.4, “Let Your Light Shine.” Students use what they know about trigonometry and circles to decide how to divide a pizza slice between two people equally so that one of them doesn’t have to eat the crust.

There is no clear guidance for the teacher on ways to specifically engage advanced students in investigating the mathematics content at higher levels of complexity.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning course-level mathematics. The implementation guide states, “The framework for supporting English Language Learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” Examples include:

• In Algebra 1, Unit 6, Lesson 5, teacher guidance states “Speaking, Reading: MLR5 Co-Craft Questions. Begin the launch by displaying only the context and the diagram of the building. Give students 1–2 minutes to write their own mathematical questions about the situation before inviting 3–4 students to share their questions with the whole-class. Listen for and amplify any questions involving the relationship between elapsed time and the distance that a falling object travels. This routine meets the Design Principle(s): maximize meta-awareness and cultivate conversation.”

• In Geometry, Unit 6, Lesson 14, there is a Collect and Display language routine that directs teachers to note, “As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments BD and CD is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, ‘The lines make 90 degrees because they have opposite slopes’ can be improved by restating it as ‘Segments BD and CD are perpendicular because their slopes are opposite reciprocals.’ This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments BD and CD is 90 degrees.”

• In Algebra 2, Unit 4, Lesson 14, there is a Compare and Connect language routine that states, “Use this routine to prepare students for the whole-class discussion about the strategies for finding the year when the population of Country C reached 30 million. After students find the year when the population reached 30 million, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the strategies for finding the value of t so that f(t) = 30. This will help students understand and find connections between multiple approaches to solving this problem. Design Principle(s): Cultivate conversation.”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Guidance is provided for both online and physical manipulatives. The online materials have access to any digital manipulatives needed, without using any physical manipulatives.

When students use the interactive student edition, they are provided with various digital tools including a drawing tool, a compass and protractor, a ruler, a statistics tool, and a generic graphing calculator. Examples include:

• In Algebra 1, Unit 1, Lesson 9, interactive student edition, students create a digital dot plot, a box plot, and a histogram via an embedded statistics tool to create models and graphical representations.

• In Geometry, Unit 3, Lesson 1, students use the drawing tool to dilate lines. 

• In Geometry Unit 4, Lesson 4, students use a web applet and a slider to change a right triangle and observe how the side lengths and angle measures change, then compare the ratio of side lengths of different triangles.

Using the physical materials, students use graph paper, graphing/scientific/four-function calculators, bouncing balls, rulers, cubes, colored pencils, blackline masters, measuring tapes, markers, clay, dental floss, pasta, isometric dot paper, mirrors, etc. Examples include:

• In Algebra 1, Unit 2, Lesson 12, students use graphing technology to model various situations in developing the meaning of the solution to a system of equations.

• In Geometry, Unit 2, Lesson 4, students use linguini pasta to support proving that two triangles are congruent. 

• In Geometry, Unit 7, Lesson 2, students use manipulatives both online and physically to measure angles and describe the relationship between an inscribed angle and the central angle that define the same arc.