## Alignment: Overall Summary

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Alignment to the CCSSM. The materials meet expectations for all indicators in Focus and Coherence (Gateway 1), and the materials meet expectations for all indicators in Rigor and Mathematical Practices (Gateway 2).

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## Gateway 1:

### Focus & Coherence

0
9
14
18
18
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

## Gateway 2:

### Rigor & Mathematical Practices

0
9
14
16
16
14-16
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

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## Gateway 3:

### Usability

0
17
24
27
27
24-27
Meets Expectations
18-23
Partially Meets Expectations
0-17
Does Not Meet Expectations

## The Report

- Collapsed Version + Full Length Version

## Focus & Coherence

### Criterion 1a - 1f

Materials are coherent and consistent with “the high school standards that specify the mathematics which all students should study in order to be college and career ready”.

18/18
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content for all students; attending to the full intent of the modeling process; spending the majority of time on content widely applicable as prerequisites; allowing students to fully learn each standard; engaging students in mathematics at a level of sophistication appropriate to high school; making meaningful connections in a single course and throughout the series; and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards.

### Indicator 1a

Materials focus on the high school standards.

### Indicator 1a.i

Materials attend to the full intent of the mathematical content in the high school standards for all students.

4/4
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series 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 of all non-plus standards across the courses of the series.

Some examples of non-plus standards that were addressed by the series include:

• N-RN.2: In Algebra 2, Unit 3, Lessons 3 and 4, students engage with rational exponents. During Lesson 3, students rewrite rational exponents as radical expressions. In Lesson 4, students use rational exponents and the properties of exponents learned from integer exponents. In Activity 4.2 Students explain why $$(5^{\frac{1}{3}})^2$$ is equivalent to $$(5^2)^\frac{1}{3}$$. Students then rewrite both of the terms as radicals.

• N-CN.1: In Algebra 2, Unit 3, Lesson 10, students begin to develop an understanding of the imaginary unit. In Lesson 11, students evaluate expressions that result in imaginary numbers and plot those imaginary numbers on a coordinate plane. In Activity 11.4, students develop a complex number using a coordinate plane with a real number axis and an imaginary axis.

• A-REI.4a: In Algebra 1, Unit 7, Lesson 12, students solve quadratic equations by completing the square. In the warm-up, students make an argument of why $$x^2+10x+20$$ is not a perfect square. The teacher notes explain the reason $$\sqrt20$$ is not an option. In the lesson, students complete a table that starts with factoring and work backward from factored to polynomial form, allowing students to use the work at the top of the table to inform their work at the bottom. Students use two examples to solve equations by completing the square. Students compare and contrast the methods and solve several equations themselves. Later, in Lesson 19, students complete and examine the steps of deriving the quadratic formula. Students discuss the steps to understand how the quadratic formula relates to completing the square.

• F-IF.7a: In Algebra 1, Unit 6, Lesson 6, students graph quadratic functions that represent physical phenomenon, interpret key features of the graph in the real-world context given in the problem, and write and interpret quadratic functions that represent physical phenomenon. The teacher notes say: “Students use a linear model to describe the height of an object that is launched directly upward at a constant speed. Because of the influence of gravity, however, the object will not continue to travel at a constant rate (eventually it will stop going higher and will start falling), so the model will have to be adjusted (MP4). They notice that this phenomenon can be represented with a quadratic function, and that adding a squared term to the linear term seems to “bend” the graph and change its direction.” After guided exploration, the students answer the question: “Why do you think the graph that represents $$d=10+46t$$ changes from a straight line to a curve when $$-16t^2$$ is added to the equation?” Through the guided exploration using technology, students interpret different parts of the function, the vertex of the graph of the function, and the zeros of the function.

• G-CO.9: In Geometry, Unit 1, there are three lessons that address this standard. In Lesson 19, students critique a conjecture which states that an angle formed between angle bisectors is always a right angle. Students proceed to develop their own conjecture about vertical angles, which leads to the proof that vertical angles are congruent. In Lesson 20, students translate and rotate one of two intersecting lines to produce parallel lines cut by a transversal, and they prove theorems related to the angles formed by parallel lines cut by a transversal. In Lesson 21, the Triangle Sum Theorem is proven in two different ways using transformations.

• G-GPE.1: In Geometry, Unit 6, Lesson 4, the three activities support students in deriving the equation of a circle given the center and radius using the Pythagorean Theorem. Students use the Pythagorean Theorem to calculate segment length, test whether a point is on a circle, and apply that perspective to build the general equation of a circle. The three activities in Geometry, Unit 6, Lesson 6, also support students in completing the square to find the center and radius of a circle. Students complete perfect square trinomials, analyze a demonstration of completing the square, and complete the square to find the center and radius of a circle.

• S-ID.4: In Algebra 2, Unit 7, Lesson 6, students find the area under a curve and connect this concept to using the mean and standard deviation to describe the proportion of the data in an approximately normal distribution. In Algebra 2, Unit 7, Lesson 7, students use the area under a normal curve to find the proportion of values in certain intervals and have the option to extend their learning by finding an interval that fits a certain percentage of the data.

• S-ID.9: In Algebra 1, Unit 3, Lesson 9, Activity 9.2, students distinguish between correlation and causation given different situations. Students look for relationships between the scatterplots and the words and have a discussion about causation and correlation. In Lesson 9, Activity 9.3, students describe situations that exhibit varying degrees of causal relationships. Students determine if situations have a very weak (or no) relationship, a strong relationship that is not causal, or a causal relationship.

### Indicator 1a.ii

Materials attend to the full intent of the modeling process when applied to the modeling standards.

2/2
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials use the full intent of the modeling process to address nearly all of the modeling standards across the courses of the series.

Each course in the series provides modeling prompts in the teacher materials, which are separate from the units and lessons. The modeling prompts include multiple versions of a task, sample solutions, and instructions for the teacher around implementing the modeling task in the classroom. There is also guidance provided about the best time to use each prompt (e.g., “Use after Unit 4, Lesson 8”). The second task statement for each prompt typically provides some scaffolding and removes some obstacles that might prevent all students from accessing the material in the first task statement. Students engage in parts of the modeling process with the alternate prompts throughout the series.

Examples where the full intent of the modeling process is used to address modeling standards across the courses of the series include:

• In Algebra 1, Modeling Prompt, “Giving Bonuses” (N-Q), students determine how bonuses should be distributed among the five workers who worked on a project. Students create a proposal with at least two different methods to distribute the bonuses (formulate) to present to their boss along with a recommendation for a specific method with an argument in support of the recommended method. Students compute each employee’s bonus (compute) and provide advantages and disadvantages for each method proposed (interpret). Students also discuss which of the five employees would be most likely to complain about each method and how they would justify the method to the employee (validate).

• In Algebra 1, Modeling Prompt, “Planning a Concert” (N-Q, A-CED.2, A-REI.4), students propose a cost for concert tickets based on provided survey data (formulate/compute). Students consider the cost of the band, the cost of a venue, and revenue that might be generated through the sale of concessions and merchandise and how these aspects impact the cost of a ticket (interpret/compute). The students create and present a plan that includes information about the cost and the profit based on information about the venue, expenses, and ticket prices (validate/report).

• In Geometry, Modeling Prompt, “A New Container” (G-GMD.3, G-MG.1,3), students design a new container that must hold a volume of 16 fluid ounces. Students use any three-dimensional shape or a combination of shapes (formulate) to create a design that is appealing and meets the criteria for volume. They provide a model or prototype of their container and the calculations that prove that the container will hold 16 fluid ounces (compute/interpret/validate). The students present their designs and defend their mathematics to others (reporting). Each student/group is able to have different shapes/designs, but all presentations are considered valid if the criteria have been met.

• In Geometry, Modeling Prompt, “So Many Flags” (G-SRT.8, G-MG.1,3), students create a flag for Nepal, of any size, using provided instructions about its construction. Students decide the sizes of large and small flags for a parade of flags (formulate). Students compute the measure of each of the angles in the flag they have constructed, the amount of material needed to create the flags, and the amount of ribbon needed to sew ribbon along the border of each flag.

• In Algebra 2, Modeling Prompt, “How Big Is That?” (N-Q, A-CED.A, G-GMD.3, G-MG), students write part of a children’s book that compares the sizes of different animals by relating their size to other objects that might be familiar (formulate). Students determine how they will scale the objects to communicate the different sizes of the objects being compared and how they will scale the smaller object in order to make an appropriate comparison (compute/interpret/validate). Students present a portion of their book as well as the relevant mathematics to justify their work (validate/report). Each student/group has a choice in what they are going to compare. All results are considered valid if there is evidence that students completed the task based on the criteria.

• In Algebra 2, Modeling Prompt, “Swing Time” (N-Q.2,3, S-ID.6) students formulate conjectures about variables that might affect the period of a pendulum and validate their conjectures by collecting data and determining if the data makes sense in the problem. Students formulate models to demonstrate the relationship between the identified variable and period of the pendulum. Students complete computations to predict different “timed” periods that cannot be performed in class. Students revisit their work and make any necessary changes after they decide how many significant digits to use. Students report their findings throughout the task by explaining and justifying their decisions.

There are also examples of modeling scenarios within the lessons. For example, in Algebra 1, Unit 2, Lesson 1 (A-CED.2), students estimate how much a pizza party would cost. Students write expressions to show how they arrived at their estimate. Students consider many factors and identify quantities in their expressions that could change on the day of the party. Students formulate expressions to represent the parameters of the pizza party, make assumptions during this process, and set constraints. Students also compute the cost of the party based on their estimates and parameters. Groups consider how the estimated costs would change if parameters changed, allowing students to interpret their results and make adjustments to their initial estimates.

### Indicator 1b

Materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.

+
-
Indicator Rating Details

### Indicator 1b.i

Materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

2/2
+
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for, when used as designed, spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs and careers. Examples of how the materials allow students to spend the majority of their time on the WAPs include:

• N-RN.A: In Algebra 2, Unit 3, Lesson 3, students apply knowledge of exponential equations and rules of exponents to develop understandings of how rational exponents are related to equivalent radical expressions. In Lesson 4, students rewrite fractional exponents as a unit fraction times a whole number and rewrite the expressions using radicals, and connect roots, rational exponents, graphs of exponential functions, and decimal approximations. In Lesson 5, students further develop their understanding to include rational exponents, rules of exponents, and graphs to make sense of negative rational exponents (N-RN.1,2). In Algebra 2, Unit 4, Lessons 4, 6, and 7, students interpret fractional inputs for exponential functions in context. They also use properties of exponents to interpret and transform expressions that represent decay, and students use fractional exponents to answer questions about amounts of radioactive isotopes in old artifacts. (N-RN.1)

• A-SSE: In Algebra 1, Units 5, 6, and 7 address many of the standards in A-SSE. For example, in Algebra 1, Unit 5, Lesson 9, students recognize and discuss similarities and differences in $$x^2$$ and $$2^x$$ as the beginning of interpreting exponential functions. Students interpret different parts of the exponential function in a real-world scenario (A-SSE.1,2). In Algebra 1, Unit 6, Lesson 8, students “explain why the diagram shows that $$6(3 + 4) = 6\cdot3 + 6\cdot4$$”. Students “draw a diagram to show that 5(x + 2) = 5x + 10.” The lesson continues using structure to find equivalent quadratic expressions (A-SSE.2,3). In Algebra 2, Unit 2, Lesson 3, students are introduced to polynomial functions and use graphing technology to write polynomials given specific characteristics (A-SSE.1,2).

• F-IF: In Algebra 1, Unit 4, students interpret and use function notation, analyze and create graphs of functions, find the domain and range of functions, and find, write, and interpret inverse functions. Students extend their work with F-IF standards in Algebra 2, Unit 2, Lesson 1 as they construct an open box and calculate the volume. Students find the largest volume, write an expression, and use graphing technology to create the graph. Students use their knowledge of the key features of graphs and domain/range to create their box.

• G-SRT: In Geometry, Unit 3, Lesson 13, Activity 13.3, students use similarity criteria to write statements indicating why the three triangles in the picture provided are similar. Additionally, in Lesson 15, Activity 15.2, students complete an activity involving task and data cards while engaging in discussion about what information is needed and why it is needed to solve the problems about triangle similarity (G-SRT.5). This is prerequisite work for G-SRT.8.

• S-ID.2: In Algebra 1, Unit 1, Lesson 5, students calculate interquartile range and discuss the importance of outliers. Students also find the Mean Absolute Deviation (MAD) and use their understanding of the MAD to interpret given scenarios. Students create two different sets of six data points which could be possible locations of pennies along a meter stick that would result in a given MAD. In Lesson 11, Cool Down, students use the interquartile range and/or the MAD to compare the spread of four data sets by examining menu prices to determine the best menu based on the greatest variability in menu options relative to price. In Lesson 12, students use the MAD to develop standard deviation by learning what happens to the standard deviation (derived through technology) when specific numbers are manipulated as in the following: the lowest value is removed; the greatest value is removed; the greatest value is doubled, etc.

### Indicator 1b.ii

Materials when used as designed allow students to fully learn each standard.

4/4
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-
Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for, when used as designed, letting students fully learn each non-plus standard. The instructional materials for the series, when used as designed, enable students to fully learn all of the non-plus standards. Examples of how the materials allow students to fully learn all of the non-plus standards include:

• N-CN.7: In Algebra 2, Unit 3, Lessons 17, 18, and 19, students solve quadratics using the quadratic formula and by completing the square when solutions are complex. This standard is addressed in all three lessons as well as in the practice assignments. Students continue to engage in solving equations with complex solutions in practice sets throughout Algebra 2, Unit 3.

• A-APR.6: In Algebra 2, Unit 2, Lesson 12, students divide polynomials by linear factors using area models traditionally used to support the multiplication of polynomials. The activities include division that results in a remainder. In Lesson 13, students use long division to divide polynomials, and in Lesson 15, students engage with The Remainder Theorem. In Lesson 19, students apply long division to rewrite rational expressions in order to reveal the end behavior of the function.

• A-REI.4a: In Algebra 1, Unit 7, students have multiple opportunities to complete the square and derive the quadratic formula. In Lesson 12, students recognize perfect square expressions and build perfect square trinomials, and students develop the rule for completing the square. In Lesson 14, students examine visual models representing the process of completing the square and use “u-substitution” to complete the square. In Lesson 15, students find irrational solutions by completing the square. In Lesson 19, completing the square is further developed as students derive the quadratic formula. In Algebra 2, Unit 3, Lesson 16, students compare different methods for solving quadratic equations, including completing the square, and determine when it might be best to use each method. Practice sets for each lesson listed provide additional problems for students to practice completing the square.

• F-IF.7b: In Algebra 1, Unit 4, Lesson 12, students graph piecewise functions in applications, such as cost for shipping related to weight and renting a bike for minutes used. Absolute value functions are addressed in Algebra 1, Unit 4, Lessons 13 and 14. In Lesson 13, students create a scatter plot of the absolute guessing error calculated from guesses for the number of objects in a jar. In Lesson 14, students graph absolute guessing error again for temperatures and work with the distance function (absolute value function). Students plot graphs by hand and by the use of technology. In Geometry, Unit 5, Lesson 5, students scale the area of different objects (floor area, painting area), graph the area with the scale factor, which results in a square root function, and explain the behavior of the graph. In Geometry, Unit 5, Lesson 7, students apply the same technique when scaling volume to produce the cube root function. In Geometry, Unit 5, Lesson 18, students complete a volume problem that relates the scaling of a balloon to its volume and surface area, which revisits square root and cube root functions.

• F-BF.2: In Algebra 2, Unit 1, over multiple lessons, opportunities for students to work with arithmetic and geometric sequences are presented. In Lesson 5, students represent sequences graphically, numerically, and in a table, and determine if a sequence is arithmetic or geometric. In Lesson 6, students match sequences with the appropriate recursive pattern and represent a given sequence recursively. In Lesson 7, students write recursive patterns/sequences and determine the terms of the sequence. In Lesson 8, students transform recursive sequences into explicitly defined sequences. In Lesson 9, students define given sequences both recursively and non-recursively. In Lesson 10, students compare arithmetic and geometric sequences in the context of real-world applications.

• G-C.2: In Geometry, Unit 6, Lesson 14, students consider a triangle that is inscribed in a semi-circle and examine the slope of the chords, relating the hypotenuse to the diameter of the circle. In Geometry, Unit 7, Lesson 1, students define chords, central angles, and inscribed angles, considering each one’s relationship to the circle as a whole. Students examine angle measures and arc measures related to central angles and inscribed angles. Students also examine chords, similar triangles that are formed by chords, and diameters. In Geometry, Unit 7, Lesson 2, students work with inscribed angles, the relationships to the intercepted arcs, and how the “rules” for finding those angle measures are related to central angles. In the practice problems, students explain or describe the difference between central angles and inscribed angles. In Geometry, Unit 7, Lesson 3, students complete problems related to radii and segments through the points of tangency and use principles of perpendicular lines to solve problems. In Geometry, Unit 7, Lesson 14, students synthesize content from previous lessons by solving problems in the context of real-world scenarios, such as pizza (circles) and flashlight beams (arcs).

• S-ID.9: In Algebra I, Unit 3, Lesson 9, students explore the definitions of correlation and causation. Students examine real-world scenarios and make determinations as to how the data is correlated, as well as if there is a causal relationship between the variables. In Algebra I, Unit 3, Lesson 10, students have additional opportunities to consider both correlation and causation and explain their reasoning.

### Indicator 1c

Materials require students to engage in mathematics at a level of sophistication appropriate to high school.

2/2
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The 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 illustrate age-appropriate contexts for high school students include:

• In Algebra 1, Unit 3, Lesson 5, students compare pounds of ice cream sold to the outside temperature and rider service prices compared to distance traveled, as they use technology to write equations of lines. (S-ID.6, S-ID.7)

• In Algebra 1, Unit 6, Lesson 7, students work with quadratic functions to describe the number of downloads of a movie and how it impacts the revenue generated. (F-BF.1, F-IF.7)

• In Geometry, Unit 3, Lesson 16, students explore similar triangles to make a bank shot in a pool game. (G-SRT.5)

• In Geometry, Unit 8, Lesson 11, students play “Rock, Paper, Scissors” and discuss how different events influence the outcome. (S-CP.6)

• In Geometry, Modeling Prompt 4, students discuss and determine their water usage daily and weekly during different tasks performed in everyday life. Students research and “describe a container that would hold the amount of water you use in a week, a month, a year, and a lifetime”. (G-GMD.3, G-MG.1, N-Q.1-3)

• In Algebra 2, Unit 4, Lesson 18 includes applications of exploring acidity and the corresponding pH scale, measuring the intensity of earthquakes, and calculating the balance in a bank account.

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

• In Algebra 1, Unit 4, Lesson 1, Activity 1.1, students apply their knowledge of functions (8.F.1) as they engage with contextual relationships that do and do not represent functions (F-IF.1). In Lesson 1, Activity 1.2, students use their understanding related to 8.F.5 as they reason graphically about the relationship between time and the distance of a dog from a post (F-IF.4).

• In Algebra 1, Unit 5, Lesson 1, students apply ratios and proportional relationships (7.RP.A) to compare examples of linear growth and exponential growth while working on Activity 1.2 about a genie in a bottle. This builds to an understanding of exponential growth (F-LE.5, F-IF.4).

• In Grade 8, students used similar triangles to explain why the slope, m, is the same between any two distinct points on a non-vertical line in the coordinate plane, and they derived the equation y = mx + b for a line intercepting the vertical axis at b (8.EE.6). In Geometry, Unit 6, Lesson 9, students develop the point-slope form of a linear equation: y - k = m(x - h). In ensuing lessons, students write equations of lines, and intercepts are not always readily available. (A-SSE.A, G-GPE.6)

• In Geometry, Unit 2, Lesson 1, Activity 1.2, students describe the sequence of transformations on figures in an open space instead of on a coordinate plane with exact numbers. Students apply their knowledge of rotations, reflections, and translations from Grade 8 to move figures and visualize the movements in “Are you ready for more?”. In this activity, students draw additional line segments in the given figures to make two congruent polygons (8.G.1,1b). In Lesson 1, Activity 1.4, students are offered “another opportunity to practice reasoning based on corresponding parts of figures they know to be congruent”. Using a reflection of triangles that share a side, students prove that the side shared is an angle bisector.

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

• In Algebra 1, Unit 5, Lesson 4, students study exponential decay, and the numbers used are $$\frac{27}{4}$$ and $$\frac{81}{8}$$. Throughout the lesson, students work with fractions, and in Lesson 5, students work with decimals as they explore how much medicine remains in a patient's bloodstream over a period of time.

• In Algebra 1, Unit 7, students work with radicals where some are rational and others are irrational. In Lesson 13, Activity 13.2, students complete the square with fractions and decimals as coefficients $$(x^2 +1.6x +0.63=0)$$. In Lesson 20, students add and subtract with integers, radicals, and fractions to decide if the sums will be rational or irrational.

• In Algebra 1, Unit 2, Lesson 5, students graph equations using large numbers (which need to be addressed through scale) and decimals. Practice Problem 5 also incorporates decimals to the hundredths place, and students consider limitations to domains as they solve application problems.

• In Geometry, Unit 5, Lesson 18, students determine the amount of helium, in cubic feet, needed to fill balloons of different sizes. These numbers get large (~21,000), so students would need to adjust their graphing calculator windows.

• Algebra 2, Unit 3 includes two assessments on complex numbers and rational exponents. The Check Your Readiness assessment includes integers, integer and rational bases, radical expressions, and cube roots. The End-of-Unit Assessment extends to complex numbers.

### Indicator 1d

Materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

2/2
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series.

Examples where the materials foster coherence within courses include:

• In Algebra 1, Unit 7, Lessons 20-21, students solve quadratic equations using a variety of methods (A-REI.4b) and defend whether or not the solutions are rational or irrational. Students investigate sums and products of rational and irrational numbers to develop general rules about the type of number the sums and products will be (N-RN.3). After completing both of these lessons, students determine if the solutions to quadratic equations are rational or irrational.

• In Geometry, Unit 6, Lesson 4, Activities 4.1 and 4.2, students subtract coordinates as part of a method for calculating the distance between two points, and they use the Pythagorean Theorem to test whether points are on a circle with a given center and radius (G-GPE.4). In Lesson 4, Activity 4.3, students apply their work to build the general equation of a circle (G-GPE.1). In Lesson 7, students apply their understanding of distances to parabolas based on the location of a focus and directrix. In the Lesson Synthesis, students compare and contrast the work completed in Lesson 7 with the work they did in Lesson 4. In Lesson 8, students build the equation for a parabola given a focus and directrix applying their understandings from previous lessons (G-GPE.2).

• In Algebra 2, Unit 5, throughout many lessons, students work with transformations of functions, both graphically and algebraically (F-BF.3). In Lesson 1, students examine a scatter plot of data for a cooling function and use the graph of the data and two given functions to determine which function best fits the data (S-ID.6a). The same data set is also presented in Lesson 7, where students describe how a given function can be translated to better fit the given data and write the function with the identified translations. The unit ends with students modeling given data by applying transformations to write functions that best fit the data.

Examples where the materials foster coherence across courses include:

• In Algebra 1, Unit 7, Lessons 12, 13, and 14, students complete the square to solve quadratic equations (A-REI.4a). In Algebra 1, Unit 7, Lessons 22 and 23, students produce equivalent forms of quadratic expressions by completing the square to reveal properties of quadratic functions (A-SSE.3). Students use the skill of completing the square again in Geometry, Unit 6, Lessons 5 and 6, to write given equations of circles in standard form and to identify the center and radius of the circle (G-GPE.1). Completing the square is used again in Algebra 2, Unit 3, Lesson 17 to solve quadratic equations that include complex solutions.

• In Geometry, Unit 5, Lesson 7, students graph cube root functions while working backwards from the volumes of original and scaled solids to calculate scale factors (F-IF.7b). Students use the graph to analyze rates of change in the scale factor for different volume inputs. Students also graph cube root functions in Algebra 2, Unit 3, Lesson 2, as they reconnect the ideas of a square root representing a side length of a square and a cube root representing an edge length of a cube.

• In Algebra 1, Unit 5, students write exponential equations and use context to compare linear and exponential models. Throughout the unit, students graph exponential functions and identify key components (F-IF.7e). In Algebra 2, Unit 4, students build on their understanding of exponential functions from Algebra 1 where students only worked with exponential functions with domains of integers. In Algebra 2, the domain is expanded to include all real numbers as students use exponential equations to model growth and decay (F-IF.4).

### Indicator 1e

Materials explicitly identify and build on knowledge from Grades 6-8 to the High School Standards.

2/2
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials explicitly identify the standards from Grades 6-8 in the teacher materials. The Design Principles of the teacher materials state that the initial lesson in a unit is designed to activate prior knowledge and provide an entry point to new concepts. The lessons are organized in such a manner that each activity has a foreword that indicates standards by category: Building On, Addressing, and Building Towards, where appropriate. This information appears routinely in the design of the teacher materials but not in the student and family materials.

Examples where the teacher materials explicitly identify content from Grades 6-8 and build on them include:

• In Algebra 1, Unit 1, Lesson 1, the Lesson Narrative of the preparation indicates that the work of the lesson builds on 6.SP.1, although there is no specific reference to grade 6 learning in the Lesson Narrative. The standard is indicated in the Building On portion of the CCSS Standard alignments. Students begin by identifying which of four given questions does not belong; the sample responses provided for the teacher indicate that students may respond that the questions are or are not statistical (6.SP.1). Students then develop survey questions based on three given statistical questions and survey the class to collect data.

• In Algebra 1, Unit 1, Lesson 2, a connection is made to 6.SP.4 and the display of numerical data in plots on a number line, dot plots, histograms, and box plots. The materials state that this serves as a brief review of these representations and the way they are created prior to engaging in the work of S-ID.1 and S-ID.2 in ensuing lessons.

• In Algebra 1, Unit 2, Lesson 17, students build on their previous understanding of 8.EE.8 by considering systems of equations that have no solutions or infinitely many solutions. Students identify, without graphing or using algebra, if a system of linear equations is equivalent or parallel.

• In Geometry, Unit 5, Lesson 2, students build on 7.G.3, where they describe two-dimensional figures that result from slicing three-dimensional figures. Students analyze cross-sections of three-dimensional figures (G-GMD.4), and build toward G-GMD.1, where students identify three-dimensional solids given parallel, cross-sectional slices.

• In Geometry, Unit 3, Lesson 3, Measuring Dilations (G-SRT.1) builds on 8.G.3. During this lesson, students dilate a quadrilateral using different scale factors. The purpose of this activity is to understand that the different ratios of the dilations are equal.

• In Geometry, Unit 5, Lesson 16, students build on their previous understanding of 7.G.6 and 8.G.9 by solving surface area and volume problems with a real-world context. In the problems in the student materials, students maximize and minimize these geometric attributes which is an extension of previously learned skills in grades 7 and 8.

• In Algebra 2, Unit 3, Lessons 1-3 and 5, students build on previous knowledge of 8.EE. Student work extends beyond the rules of exponents that were learned in previous grades in the following ways: solving simple equations to find the missing exponents in an equivalent relationship; considering numbers expressed as square roots and determining which integers it falls between; considering unit fractions as exponents and how the rules of exponents extend to all rational numbers.

### Indicator 1f

The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.

Narrative Evidence Only
+
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Indicator Rating Details

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series do not explicitly identify the plus standards when plus standards are included. There are some plus standards that are explicitly taught and support the mathematics in the course. In other instances, there are plus standards for which the standard is not fully addressed. In cases where plus standards are partially addressed, the inclusion provides more detail and context for the non-plus standards, which supports the mathematics all students should study in order to be college and career ready.

The following plus standards were fully addressed:

• F-IF.7d: In Algebra 2, Unit 2, Lessons 17–19 students encounter zeros, asymptotes (including oblique), and end behavior of rational functions.

• G-GMD.2: In Geometry, Unit 5, the Lesson Narrative states “In this unit, students practice spatial visualization in three dimensions, study the effect of dilation on area and volume, and derive volume formulas using dissection arguments and Cavalieri’s Principle.” In Geometry, Unit 5, Lesson 10 students conclude that an oblique prism and a right prism that have the same height and whose bases are of equal area have the same volume. This is because their cross sections at all heights have equal area (Cavalieri’s Principle) which is then used to develop volume formulas.

• S-CP.8: In Geometry, Unit 8, the Lesson Narrative states “Conditional probability is discussed and applied using several games and connections to everyday situations. In particular, the Multiplication Rule, P(A and B) = P(A | B)⋅P(B), is used to determine conditional probabilities. Conditional probability leads to the definition of independence of events. Students describe independence using everyday language and use the equation P(A | B) = P(A) when events A and B are independent.” In Lesson 8, students engage with the Multiplication Rule to find conditional probabilities, and in Lesson 9, students estimate conditional probabilities and compare work done with the Multiplication Rule.

The following plus standards were partially addressed:

• N-CN.B: In Algebra 2, Unit 3, Lesson 11, the Lesson Narrative states “While a deep, geometric interpretation of complex numbers in the complex plane is beyond the scope of this course, some activities in this unit use the complex plane to support student understanding. The complex plane helps students conceptualize numbers that are not on the real number line and make sense of complex addition. This is similar to how the real number line can be used to understand signed numbers and signed number addition but is not a topic itself. There are purposefully no assessment items related to the complex plane in this course.” The complex plane is then used in Lessons 11 and 12 to support students’ understanding of imaginary numbers and arithmetic with complex numbers.

• N-CN.8: In Algebra 2, Unit 3, Lesson 17, Activity 17.1, students match $$x^2+25$$ with $$(x-5i)(x+5i)$$. Other than this problem presenting the factored form of a quadratic polynomial with imaginary roots, complex numbers are used in solving quadratic equations and not in polynomial identities.

• N-VM: In Geometry, Unit 1, Lesson 12, the Lesson Narrative states “The concept of a directed line segment is introduced to give students language for efficiently describing the direction and length of a translation. Students know the term line segment, and so the phrase directed line segment builds on a concept they already know and connects it to the concept of translations. The word vector is purposely avoided because the geometric interpretation of a vector should arise as a consequence of future work with vectors, not as a definition.”

Plus standards not mentioned in this report do not appear in the materials.

## Rigor & Mathematical Practices

#### Meets Expectations

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Gateway Two Details

### Criterion 2a - 2d

Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.

8/8
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Rigor and Balance. The materials meet expectations for providing students opportunities in developing conceptual understanding, procedural skills, and application, and the materials also meet expectations for balancing the three aspects of Rigor.

### Indicator 2a

Materials support the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

2/2
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Indicator Rating Details

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.

### Indicator 2b

Materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

2/2
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Indicator Rating Details

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.

### Indicator 2c

Materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

2/2
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Indicator Rating Details

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.

### Indicator 2d

The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

2/2
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Indicator Rating Details

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.

### Criterion 2e - 2h

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

8/8
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Practice-Content Connections. The materials intentionally develop all of the mathematical practices to their full intent: make sense of problems and persevere in solving them (MP1), reason abstractly and quantitatively (MP2), construct viable arguments and critique the reasoning of others (MP3), model with mathematics (MP4), use appropriate tools strategically (MP5), attend to precision (MP6), look for and make use of structure (MP7), and look for and express regularity in repeated reasoning (MP8).

### Indicator 2e

Materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

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.

### Indicator 2f

Materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

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.

### Indicator 2g

Materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

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).”

### Indicator 2h

Materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

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)”.

## Usability

#### Meets Expectations

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Gateway Three Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Usability. The materials meet expectations for Criterion 1 (Teacher Supports), Criterion 2 (Assessment), and Criterion 3 (Student Supports).

### Criterion 3a - 3h

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

9/9
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials; contain adult-level explanations and examples of the more complex course-level concepts and concepts beyond the current courses so that teachers can improve their own knowledge of the subject; include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities.

### Indicator 3a

Materials provide 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.

2/2
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Indicator Rating Details

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.

### Indicator 3b

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

2/2
+
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Indicator Rating Details

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.

### Indicator 3c

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

2/2
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Indicator Rating Details

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.

### Indicator 3d

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

Narrative Evidence Only
+
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Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement. The printed resources do not have any of these supports in them; however, every unit in the online resources have family support sections in the Learning Resources. This section of each unit tells the stakeholder what the students will be learning in this unit and provides tasks to try with the student at home. This resource is available online, or may be printed and distributed to families. Examples include:

• In Algebra 1, Unit 2, the parent guide states, “In this unit, your student will analyze constraints on different quantities. For example, the amount you spend on groceries may be limited by your budget. To qualify for a sports team, you may need to practice at least a certain number of hours, or lift at least a certain number of pounds.” A task is also provided to complete with the student.

•  In Geometry, Unit 7, the parent guide states, “In this unit, your student will make connections between geometry and algebra by working in the coordinate plane with geometric concepts from prior units. The coordinate grid imposes a structure that can provide new insights into ideas students have previously explored.” Then a task is provided to work at home with their student.

### Indicator 3e

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

2/2
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Indicator Rating Details

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).”

### Indicator 3f

Materials provide a comprehensive list of supplies needed to support instructional activities.

1/1
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Indicator Rating Details

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.

### Indicator 3g

This is not an assessed indicator in Mathematics.

### Indicator 3h

This is not an assessed indicator in Mathematics.

### Criterion 3i - 3l

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

10/10
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Assessment. The materials indicate which standards are assessed and include an assessment system that provides multiple opportunities throughout the courses to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The materials also provide assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices.

### Indicator 3i

Assessment information is included in the materials to indicate which standards are assessed.

2/2
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Indicator Rating Details

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.

### Indicator 3j

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

4/4
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Indicator Rating Details

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.

### Indicator 3k

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

4/4
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Indicator Rating Details

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.

### Indicator 3l

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

Narrative Evidence Only
+
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Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment. Examples include:

• When printing an assessment, there are options to change the font size, font style, and short answer letter choices.

• When using an assessment digitally, there are options for text-to-speech, highlighter, notepad, line reader, protractor, calculator (simple, scientific and graphing), reference sheet. The teacher can turn any of these options on or off, but there was no guidance found to support teachers in doing this.

### Criterion 3m - 3v

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

8/8
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations to support their regular and active participation in learning course-level mathematics; extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity; 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; and manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

### Indicator 3m

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

2/2
+
-
Indicator Rating Details

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, …”

### Indicator 3n

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

2/2
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-
Indicator Rating Details

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.

### Indicator 3o

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide varied approaches to learning tasks over time and provide variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning. Examples include:

• Ongoing reviews are provided in each practice section across the series. The reviews spiral the learning from mostly the current course. The practice problems are also a source for students to monitor their own progress.

• Learning targets for each unit are provided for student reflection at the end of each unit.

• Cool downs are provided for feedback to both the teacher and the student with regard to each day's learning targets.

• There are opportunities for students to share their thinking though a common instructional strategy of “Which One Doesn’t Belong.” For example, in Algebra 1, Unit 4, Lesson 8, students use this strategy to share their reasoning as to why a particular item does not belong as well as finding at least one reason why each item doesn’t belong.

• There are opportunities for students to compare their thinking with other students. For example, in Geometry, Unit 1, Lesson 1, students work in pairs to write a proof, and then the teacher is encouraged to have 2 different pairs of students share their work, one pair who wrote a transformational proof and another pair who wrote a proof using calculations.

• Instructional routines are provided in each lesson throughout the series that provide students with multi-modal opportunities to investigate, make sense of phenomena and problem solve using a variety of formats and methods. For example, in Geometry, Unit 6, Lesson 8, students use a card sort and a collect-and-display routine to match graphs and equations of parabolas.

### Indicator 3p

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

Narrative Evidence Only
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Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide opportunities for teachers to use a variety of grouping strategies. Notes for the teacher indicate what types of groupings can be used for each activity. Examples include:

• In Algebra 1, Unit 6, Lesson 11, there are 3 activities with different groupings. Activity 1 uses a “Think, Pair, Share” to identify possible points on the graph of a quadratic function. Activity 2 has students work with a partner to complete a table of values comparing quadratic equations in factored form. In Activity 3, teachers “arrange students in groups of 2–4. Provide access to devices that can run graphing technology. Give students a few minutes of quiet time to think about the first question. Then, ask them to discuss their response and to complete the second question with their group . . . to save time, consider asking groups to split the graphing work (each group member graphs only one function and analyze the graphs together).”

• In Geometry, Unit 2, Lesson 3, “Sometimes True” and “Invisible Triangles”, teachers arrange students in groups of 2 before the whole class discussion.

• In Algebra 2, Unit 4, Lesson 11, Warmup, students work alone before there is a whole group discussion. In Activities 2 and 3, students work in groups of 2. The lesson ends with an Activity Synthesis with a whole group discussion.

### Indicator 3q

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

2/2
+
-
Indicator Rating Details

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.”

### Indicator 3r

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide a balance of images or information about people, representing various demographic and physical characteristics. The materials have few images of people, and most are found as pictures for the unit in the Table of Contents. The few pictures do vary by names and pictures to balance between male, female and other physical and demographic characteristics. Examples include:

• In Algebra 1, Unit 5, Lesson 21, practice question 2, “Mai used a computer simulation to roll number cubes and count how many rolls…”

• In Geometry, Unit 5, Lesson 7, practice item 7, “Noah edits the school newspaper. He is planning to print a photograph of a flyer for the upcoming school play.”

• In Geometry, Unit 8, Lesson 1, activity 1, “In Elena’s Spanish class, they have a quiz every two weeks.”

### Indicator 3s

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series do not provide guidance to encourage teachers to draw upon student home language to facilitate learning. ELL supports are provided with regular use of Math Language Routines, but these are all supports for the student to understand the math and to do so in English rather than in their own language. These resources are used to help students understand the math regardless of any language barriers.

### Indicator 3t

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning. No references to cultural or social supports to facilitate learning were found.

### Indicator 3u

Materials provide supports for different reading levels to ensure accessibility for students.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide Math Language Routines that are specifically geared directly to different reading levels to ensure accessibility for students. Examples include:

• Math Language Routine 3, “Clarify, Critique, and Correct: Purpose: To give students a piece of mathematical writing that is not their own to analyze, reflect on, and develop.” This specific routine is used in Algebra 1, Unit 2, Lesson 4, Activity 1 “Weekend Earnings,” and in Geometry, Unit 1, Lesson 5, “Bisect This.”

• Math Language Routine 6, “Three Reads” supports reading comprehension, sense-making, and meta-awareness of the mathematical language as well as negotiating information in a text with a partner through mathematical conversation. This routine is used in Algebra 1, Unit 5, Lesson 1, Activity 2 “A Genie in a Bottle,” and in Algebra 2, Unit 4, Lesson 2, Activity 2, “Climbing Costs.”

### Indicator 3v

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

2/2
+
-
Indicator Rating Details

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.

### Criterion 3w - 3z

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

0/0
+
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Criterion Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in course-level standards. The materials include, but do not reference, digital technology that provides opportunities for teachers to collaborate with each other. The materials have a visual design that supports students in engaging thoughtfully with the subject, and the materials provide teacher guidance for the use of embedded technology to support and enhance student learning.

### Indicator 3w

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series integrate technology, such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software, in ways that engage students in the course-level standards, when applicable. The print materials do not integrate interactive tools. When technology is needed in the print materials, they refer to spreadsheets and graphing calculator technology to support student understanding and engagement. All dynamic software is embedded in the digital version of the materials and is unique to McGraw-Hill. Examples include:

• A-CED.4; A-REI.10: In Algebra 1, Unit 2, Lesson 10, students use the draw tool to sketch the graph of equations representing a real world situation.

• F-BF.1b,3; S-ID.6a: In Algebra 2, Unit 5, Lesson 11, students use the online drawing tools to consider which model is best to model given situations. “In addition to fitting the data well, they also consider end behavior of the functions and of the temperature of the water bottle.”

• F-BF.3 & S-ID.6a: In Algebra 2, Unit 5, Lesson 8, students “use graphing technology to identify an appropriate scale factor to use with a given expression.” The print materials indicate the need for graphing technology while the online materials include the embedded graphing calculator to support students in this lesson.

• S-ID.1,2: In Algebra 1, Unit 1, Lesson 12, students use the digital statistics tool to find the mean and standard deviations of dot plots.

• S-ID.4: In Algebra 2, Unit 7, Lesson 7, students calculate areas under a normal curve using the embedded technology. The print materials recommend using software or other statistical technology.

### Indicator 3x

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series include, but do not reference, digital technology that provides opportunities for collaboration among teachers. No evidence of opportunities for collaboration among students or between the teacher and the student was found in the materials. The digital platform syncs to Google Classroom to allow collaboration through that platform. Teacher-to-Teacher Collaboration is available via the online Community Hub.

### Indicator 3y

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic. There is a consistent layout and structure across and within all courses in that each lesson has a Warm-Up, one or more Activities, Cool Down, Lesson Synthesis and Practice. Examples of images and graphics that clearly communicate information to support student understanding include:

• In Geometry, Unit 5, Lesson 1, the Warm-Up for Which One Doesn’t Belong includes pictures of a multi-colored slinky, pool noodle, yurt, and a rubiks cube. The pictures allow students to use precise language to discuss characteristics of all the items.

• In Algebra 2, Unit 6, Lesson 10, a picture of a windmill is shown to support students making sense of graphs of sine and cosine in the context of a windmill blade as it rotates counterclockwise.

### Indicator 3z

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable. There is no direct guidance to use any of the embedded technology; however, in using the online search tool, the Math Digital Teachers Tools videos were found. The videos assist teachers in navigating the platform. Some of the videos include supporting teachers in: maximizing the impact of the digital resources, using Google Classroom alongside the Illustrative Mathematics resources, and the structure of the assessments. The eToolkit “provides students with a digital way to explore concepts and teachers with a way to create problem-based learning opportunities.”

abc123

Report Published Date: 2021/05/20

Report Edition: 2021

## Math High School Review Tool

The high school review criteria identifies the indicators for high-quality instructional materials. The review criteria supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our review criteria evaluates materials based on:

• Focus and Coherence

• Rigor and Mathematical Practices

• Instructional Supports and Usability

The High School Evidence Guides complement the review criteria by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

McGraw-Hill Illustrative Mathematics High School AGA, LearnZillion Illustrative Mathematics High School Traditional, and Kendall Hunt's Illustrative Mathematics High School Traditional draw upon the same mathematics content and therefore the scores and evidence for Gateways 1 and 2 are the same in all three programs, albeit with differences in navigation. There are differences in usability as McGraw-Hill Illustrative Mathematics High School AGA, LearnZillion Illustrative Mathematics High School Traditional, and Kendall Hunt's Illustrative High School Traditional do not have the same delivery platforms for the instructional materials.

The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.

Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways.

Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?

Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom.

In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.