The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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-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.
• N-RN.2: In Algebra 2, Unit 4, Lessons 3 and 4, students engage with rational exponents. During Lesson 3, students make sense of numbers and use technology to investigate how rational exponents affect the bases. 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.
• A-REI.4a: In Algebra 1, Unit 7, Lesson 12, students solve quadratic equations by completing the square. In the opener, 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-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. 
• 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. 
• 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 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 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.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 5, “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 9, “Planning a Concert” (A-CED.2, A-REI.4, N-Q), 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 8, “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 7, “So Many Flags” (G-MG.1,3, G-SRT.8), 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 4, “How Big Is That?” (A-CED.A, G-GMD.3, G-MG, N-Q), 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 6, “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.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 they 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)
• 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 11.4, 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. 
• 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 13.3, students use similarity criteria to write statements indicating why the three triangles in the picture provided are similar. Additionally, in Lesson 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.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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:

• 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. 
• 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.
• 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. 
• 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.
• 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.
• 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. 
• 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).

The instructional materials reviewed for Kendall Hunt's Illustrative Mathematics Traditional 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 1.1, 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 1.2, 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 in working the Genie problem. 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 1.2, 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 1.4, 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 13.2, 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 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.
• 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.
• 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.

The instructional materials reviewed for Kendall Hunt's Illustrative Mathematics Traditional 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, Lesson 20, 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). In Lesson 21, students determine if solutions provided for quadratic equations are rational or irrational. After completing both of these lessons, students determine if the solutions to quadratic equations are rational or irrational.
• In Geometry, Unit 6, Lessons 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 4.3, 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). 

The instructional materials reviewed for Kendall Hunt's Illustrative Mathematics Traditional 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 2, 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.

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

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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, synthetic 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. They encounter different contexts and use expressions, graphs, and tables to distinguish between the two types of functions. They gain fluency in how to compare two exponential functions, how they differ in their expressions, and what that will mean for growth in context. In Algebra 2, Unit 6, students use these skills with trigonometric functions, recognizing and discussing amplitude, frequency, and shifts in many and varied contexts over the course of four lessons.
• F-BF.3: In Algebra 1, Unit 4, Lesson 14, students analyze the type of transformation a constant value creates with an absolute value function. In the cumulative practice problem set, there are more problems for students to practice this skill.
• G-GPE.4: In Geometry, Unit 6, Lesson 10 practice, students write equations using parallel slopes and identify equations that are parallel to a specific line. In Geometry, Unit 6, Lesson 11, students have the same opportunity with perpendicular slopes.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 15.3, students solve a system of exponential equations involving a cicada population using logarithms and graphing; specifically, students explain why they can use the intersection of the two graphs 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 9.3, 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 17.3, students apply volume and density to determine the number of fish that could be housed in a tube-shaped aquarium with an open 4-foot cylinder in the middle for viewing.
• S-ID.6 and N-Q.3: In Algebra 1, Unit 3, Lesson 6, students apply residual value, line of best fit, and percent error to data related to the weight of oranges in a crate. Students also apply these concepts in practice problems 4 and 5 involving applications of car sales and temperatures.
• S-IC.1: In Algebra 2, Unit 7, Lesson 3, students evaluate the randomness in population samples. Students determine the best way to have random samples and the factors that could affect the randomness in several different scenarios. Students draw conclusions from a variety of non-routine application problems.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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. 

The following are examples of balancing the three aspects of rigor in the instructional 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. 
• 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 2.2, students represent and analyze histograms; in Lessons 3.1 and 4.1, 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 they may use to analyze a situation and describe data displays they may use 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 7, Lesson 6, 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. They 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. They 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 20.3, students write a simple rational equation about batting average to develop procedural skill, and they demonstrate conceptual understanding when working with a word problem and writing it algebraically. The extension and what-if questions about the rational equation address application of rational equations.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 some 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, as students articulate what they notice and wonder, they attend to precision in language they use to describe what they see. Students may initially propose less formal or imprecise language, then restate their observation with more precise language in order to communicate clearly. Relevant vocabulary includes the following: equidistant, congruent segments, and parabola.
• In Algebra 2, Unit 3, Lesson 6, students solve simple equations involving squares and square roots. The teacher notes state: “Students attend to precision when they reason about solutions to equations involving squares and square roots from the meaning of the √ symbol (MP6).” In this lesson, students explore the idea that every positive number has two square roots. The convention of giving only the positive root is also discussed in terms of its precise meaning. Students explore the use of the radical symbol as a tool of precision.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 a cooling coffee function. They must determine, from given intervals, the best interval to use for an appropriate “average rate of cooling” and support their choice.
• In Algebra 1, Unit 7, Lesson 1.3, 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, they 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. They then compare the two rates. Data from the whole class is collected and a discussion held as they 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 11.3, 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 23.3, 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 4.3, students answer questions about operations on polynomials. They experiment to develop reasons to support their answers. During the group discussion, they defend and critique the reasoning of their classmates as they describe events as subsets through taking turns in trading roles as they explain their thinking.
• In Algebra 2, Unit 4, Lesson 14, students practice constructing logical arguments when they justify solutions and explain why a certain value is a reasonable estimate for a given logarithm.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 20.2, 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 19.2, 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 7.2 students create an arbitrary triangle, use angle bisectors and constructions to find the incenter, and construct the triangle’s inscribed circle. The narrative states: “Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).” The narrative also states: “Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.”
• In Geometry, Unit 8, Lesson 9 students use two-way tables as a sample space to decide if events are independent and to estimate conditional probabilities. Although technology is not required, it is recommended that technology be made available as there are opportunities for students to choose to use appropriate technology to solve problems. 
• In Algebra 2, Unit 4, Lesson 5, students create an exponential function given a table of values. During this lesson, teachers make sure that students have access to a spreadsheet tool to reason about the given questions. This helps students focus on the questions rather than the calculations, and students use tools to their advantage during the lesson.
• In Algebra 2, Unit 5, Lesson 11, students apply transformations on functions to determine the best model for a data set, specifically temperature data from heating objects. The Lesson Narrative states: “This can be done by hand via experimenting, but students may also choose to use graphing technology to help choose the appropriate translations, scalings, and reflections (MP5).”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional 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 as they 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 5.3, 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 as they 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 as they generalize formulas for each.
• 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 10.3, students determine specific trigonometric values for large angles. “Students make connections between angles greater than 2$$\pi$$ and between 0 and 2$$\pi$$ that correspond to the same point on the unit circle (MP 8)”.