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)”.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson. It is clear when students solve problems to learn and when they apply skills.

Lessons include a Warm-Up, Activities, Synthesis, and a Cool-Down. Practice Problems are in a separate section of the instructional materials, distinguishing between problems students complete and exercises in the lessons. Warm-Ups connect prior learning or engage students for learning new material in the lesson. Students learn and practice new mathematics in lesson Activities. In the Synthesis activity, students build on their understanding of the new concept. Each activity lesson ends with a Cool-Down in which students apply what they have learned from the activities, complete preliminary practice, or complete an introduction to skills they may need in the next lesson.

Practice problems are consistently found in the Practice sets that accompany each lesson. These sets of problems include problems that support students in developing mastery of the current lesson and unit concepts and review of material from previous units. When practice problems contain content from previous lessons, students apply their skills and understandings in different ways that enhance understanding or application (e.g., increased expectations for fluency, more abstract application, or a non-routine problem).

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for not being haphazard; exercises are given in intentional sequences.

Overall, clusters of lessons within units and activities within lessons are intentionally sequenced so students develop understanding. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skill and fluency, and engage with multiple activities that are sequenced from concrete to abstract or increase in complexity. Lessons end with a Cool-Down which is aligned to the daily lesson objective. Unit sequences consistently follow progressions to support students' development of conceptual understanding and procedural skills.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for having variety in what students are asked to produce.

The instructional materials prompt students to produce products in a variety of ways. Students produce solutions within Activities and Practice, as well as participating in class, groups, and partner discussions. Materials provide opportunities for students to construct viable arguments and critique the reasoning of their peers. Students use a digital platform and paper-pencil to conduct and present their work. The materials consistently prompt students for solutions that represent the language and intent of the standards. Students use representations such as tables, number lines, area diagrams, dot plots, geometric constructions, and graphs, as well as strategically choose tools to complete their work (MP5). Lesson activities and tasks are varied within and across lessons.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. The series includes a variety of virtual manipulatives and integrates hands-on activities that allow the use of physical manipulatives, for example:

• Manipulatives and other mathematical representations are consistently aligned to the expectations and concepts in the standards. The majority of manipulatives used are commonly accessible measurement and geometry tools.
• The materials provide digital applets for manipulating geometric shapes, such as GeoGebra applets, tailored to the lesson content and tasks. When physical, pictorial, or virtual manipulatives are used, they are aligned to the mathematical concepts they represent. For example, in Geometry, Lesson 5, Activity 10.2, two rectangular prisms with the same base area and the same height are used within an applet to develop Cavilieri’s Principle.
• Examples of manipulatives for Geometry include: an index card to use as a straightedge, compasses, tracing paper, blank paper, colored pencils, and scissors. Additionally, GeoGebra applets are used for constructions, to perform transformations, to explore congruence and similarity, and to visualize cross sections.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet the expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each lesson consists of a detailed lesson plan accompanied by teaching notes. Included in these teaching notes are the objectives of the lesson, suggested questions for discussion, and guiding questions designed to increase classroom discourse and foster understanding of the concepts. For example, in Geometry, Unit 4, Lesson 6.2 suggests the teacher ask, “The right triangle table is useful, but what if the angle is not a multiple of 10 degrees?” In Algebra 1, Unit 3, Lesson 7, Lesson Synthesis, the curriculum suggests the following question for discussion, “What does a scatter plot look like when its line of best fit has a correlation coefficient of -0.5? Sketch it.” The teaching notes and questions for discussion support the teachers in planning and implementing lessons effectively.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet the expectations for providing teacher supports with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

• Each lesson includes the Learning Goals written for teachers and students, learning targets written for students, a list of Word/PDF documents that can be downloaded, CCSSM Standards that are “built upon” or “addressed” for the lesson, and any instructional routines to be implemented. Within the technology, there are expandable links to standards and instructional routines.
• Lessons include detailed guidance for teachers for the Warm-Up, Activities, and the Lesson Synthesis.
• Each lesson activity contains an Overview and Launch Narrative, Guidance for Teachers and Student-facing materials, Anticipated Misconceptions, “Are you ready for more?”, and an Activity Synthesis. Included within these narratives are guiding questions and additional support for students.
• The teacher materials that correspond to the student lessons provide annotations and suggestions on how to present the content. “Launch” explains how to set up the activity and what to tell students. After the activity is complete, there are often Anticipated Misconceptions in the teaching notes, which describes how students may incorrectly interpret or misunderstand concepts and includes suggestions for addressing those misconceptions.
• The materials are available in both print and digital forms. The digital format has embedded GeoGebra applets. Guidance is provided to both the teacher and the student on how to use the Geometry Toolkit and applet. For example, in Geometry, Unit 1, Lesson 1, teachers and students are provided access and time to play with the applet tools. During the Launch, teachers are encouraged to model the different tools, practice with the students, and answer questions.

The instructional materials reviewed for the Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for the teacher’s edition containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge.

The narratives provided for each unit include information about the mathematical connections of concepts being taught. Previous and future grade levels are also referenced to show the progression of mathematics over time. Important vocabulary is included when it relates to the “big picture” of the unit.

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.

The Lesson Narrative for Algebra 2, Unit 3 states, “In the next set of lessons, students connect the $$\sqrt{ }$$ and $$\sqrt[3]{ }$$ symbols with solutions to quadratic and cubic equations. Students learn that a number is a square root of c if it squares to make c. In other words, square roots of c are solutions to the equation $$x^2=c$$. Students use the graph of $$y=x^2$$ to see that all positive numbers have two square roots, one positive and one negative. They learn the convention that the positive square root is given the symbol $$\sqrt{ }$$, so the positive square root of c is written $$\sqrt{c}$$ and the negative square root is written $$-\sqrt{c}$$.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Course Guide and Narratives describe how mathematical concepts are built from previous grade-level/course and lesson material. For example, in Algebra 1, Unit 4, the Lesson Narrative states, “In grade 8, students learned that a function is a rule that assigns exactly one output to each input. They represented functions in different ways—with verbal descriptions, algebraic expressions, graphs, and tables—and used functions to model relationships between quantities, linear relationships in particular.” In addition, in Algebra 1, Unit 5, the Lesson Narrative states, “Before starting this unit, students are familiar with linear functions from previous units in this course and from work in grade 8. They have been formally introduced to functions and function notation and have explored the behaviors and traits of both linear and non-linear functions. Additionally, students have spent significant time graphing, interpreting graphs, and exploring how to compare the graphs of two linear functions to each other. In this unit, students frequently use the properties of exponents, a topic developed in grade 8. They also apply their understanding of percent change from grade 7 and use an exponent to express repeated increase or decrease by the same percentage.”

For some units, there are explanations given for how the grade-level concepts fit into future high school work. For example, in Geometry, Unit 7, the Lesson Narrative states, “Students develop fluency with radian measures by shading portions of circles and working with a double number line. This is important for the transition towards Algebra 2. In that course, students will explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels.

• Prior grade-level or course standards are indicated in the instructional materials. The lesson Warm-Up is designed to engage students' thinking about the upcoming lesson and/or to revisit previous grades' concepts or skills.
• Prior knowledge is gathered about students through the pre-unit Check Your Readiness assessments. In these assessments prerequisite skills necessary for understanding the topics in the unit are assessed. Commentary for each question provides the relevance of the questions to the topic and a list of standards assessed is provided for the teacher. For example, in Algebra 2, Unit 5, Check Your Readiness assessment problem 2 shows 7.G.1 as an aligned standard. The teacher note states, “Geometry Unit 3 Lesson 1 Activity 4, Match the Scale Factors, is an example of a brief activity that could be added to review scale factors.” The notes also state, “Vertical and horizontal stretches are not dilations because the stretch is only applied in a single direction while a dilation applies the same scale factor in all directions. Nonetheless, the idea of scaling is common to both situations.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

 Lesson Activities include teaching notes that identify where students may make a mistake or struggle. There is a rationale that explains why the mistake could have been made, suggestions for teachers to make instructional adjustments for students, and steps teachers can take to help clear up the misconceptions. For example, in Geometry, Unit 2, Lesson 6.2, the teacher notes state, “Anticipated Misconceptions: If students are searching too far back, point students toward the proof in the warm-up activity, Information Overload. The goal is for students to understand and adapt that proof to this situation, so help students find the proof relatively quickly so they can have time to engage in productive struggle as they try to understand and adapt it.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

 The lesson structure consisting of a Warm-up, Activities, Lesson Synthesis, and Cool-down provide students with opportunities to connect prior knowledge to new learning, engage with content, and synthesize their learning. Throughout the lesson, students have opportunities to work independently, with partners, and in groups where review, practice, and feedback are embedded into the instructional routine. Practice Problems for each lesson activity reinforce learning concepts and skills and enable students to engage with the content and receive timely feedback. In addition, discussion prompts provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for assessments clearly denoting which standards are being emphasized.

 Assessments are accessed through the Assessment tab for each unit and are available in two print options. For each unit, there is a Check Your Readiness and an End-Unit Assessment. Longer units also include a Mid-Unit Assessment. Assessments begin with guidance for teachers on each problem, followed by the student-facing problem, solution(s), and the standard targeted.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Assessments include an answer key, and when applicable, a rubric consisting of three to four tiers, ranging from Tier 1 (work is complete, acceptable errors) to Tiers 3 and 4 (significant errors, conceptual mistakes).

Assessments include multiple choice, multiple response, short answer, restricted constructed response, and extended response. Restricted constructed response and extended response items have rubrics that are provided to evaluate the level of student responses. The restricted constructed response includes a 3-tier rubric, and the extended constructed response includes a 4-tier rubric. For these types of questions, the teacher materials provide guidance as to what is needed for each tier as well as some sample responses.

In the Assessment Teacher Guide for each End of Unit Assessment, there are narratives about what may have caused students to choose an incorrect response before the problems are shown along with the correct responses and aligned standards. For example, in Algebra 2, Unit 5, End of Unit Assessment, Problem 2, the Assessment Teacher Guide states, “Students who select A may be confusing how to represent horizontal translations using function notation. Students who select C need more work with representing shifts, reflections, and stretches using function notation. Students who select D may be confusing how to represent horizontal and vertical reflections using function notation.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• Each lesson is designed with a Warm-Up that reviews prior knowledge and/or prepares all students for the activities that follow, and the Cool-Down reviews the concepts of the lesson.
• Within a lesson, narratives provide explicit instructional support for the teacher, including the Activity Launch, Anticipated Misconceptions, and Lesson Synthesis. This information assists teachers in making the content accessible to all learners.
• Lesson Narratives often include guidance on where to focus questions in Activities or the Lesson Synthesis.
• Optional activities are often included that can be used for additional practice or support before moving to the next activity or lesson.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

The lesson structure—Warm-Up, Activities, Lesson Synthesis, and Cool-Down—includes guidance for the teacher on the mathematics of the lesson, possible misconceptions, and specific strategies to address the needs of a range of learners. Embedded supports include:

• Mathematical Language Routines to support a range of learners to be successful are provided for the teacher throughout lessons to maximize output and cultivate conversation. For example:
• MLR1: Stronger and Clearer Each Time, in which “students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response.”
• MLR4: Information Gap, which “allows teachers to facilitate meaningful interactions by giving partners or team members different pieces of necessary information that must be used together to solve a problem or play a game...[S]tudents need to orally (and/or visually) share their ideas and information in order to bridge the gap.”
• MLR6: Three Reads, in order to "ensure that students know what they are being asked to do, and to create an opportunity for students to reflect on the ways mathematical questions are presented. This routine supports reading comprehension of problems and meta-awareness of mathematical language. It also supports negotiating information in a text with a partner in mathematical conversation.”
• Teaching notes appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These teaching notes state specific needs addressed in a recommended strategy that is relevant to the given task and includes supports for Conceptual Processing, Expressive & Receptive Language, Visual-Spatial Processing, Executive Functioning, Memory, Social-Emotional Functioning, and Fine-motor Skills. For each support, there are multiple strategies teachers can employ, for example: Conceptual Processing includes strategies to Eliminate Barriers, Processing Time, Peer Tutors, Assistive Technology, Visual Aids, Graphic Organizers, and Brain Breaks.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for embedding tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

The problem-based design engages students with complex tasks multiple times each lesson. The Warm-Up, Activities, Lesson Synthesis, and Cool-Down provide opportunities for students to apply mathematics from multiple entry points.

Specific examples of strategies found in the materials include “Notice and Wonder” and “Which One Doesn’t Belong?” The lesson and task narratives provided for teachers offer possible solution paths and presentation strategies from various levels, for example:

• In Geometry, Unit 6, Lesson 7, students consider a set of points that are equidistant from a line and a given point. Students share what they notice and what they wonder. This takes place in the Warm-Up of this lesson, and offers an opportunity for all students to access the content.
• In Algebra 1, Unit 4, Lesson 8, students compare four graphs of temperature over time. Students may realize that each of the four representations might not belong based on different criteria. This instructional routine allows students to be precise in their language and to define the parameters necessary when solving contextual problems.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for including support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The ELL Design is highlighted in the Teacher Guide and embodies the Understanding Language/SCALE Framework from the Stanford University’s Graduate School of Education, which consists of four principles: Support Sense-Making, Optimize Outputs, Cultivate Conversation, and Maximize Meta-Awareness. In addition, there are eight Mathematical Language Routines (MLR) that were included “because they are the most effective and practical for simultaneously learning mathematical practices, content, and language.” "A Mathematical Language Routine refers to a structured but adaptable format for amplifying, accessing, and developing students’ language."

ELL Enhanced Lessons are identified in the Unit Overview. These lessons highlight specific strategies for students who have a language barrier which affects their ability to participate in a given task. Throughout lessons, a variety of instructional routines are designed to assist students in developing full understanding of math concepts and terminology. These Mathematical Language Routines include:

• MLR2, Collect and Display, in which “The teacher listens for, and scribes, the student output using written words, diagrams and pictures; this collected output can be organized, revoiced, or explicitly connected to other language in a display for all students to use.”
• MLR5, Co-Craft Questions and Problems, which “[allows] students to get inside of a context before feeling pressure to produce answers, and to create space for students to produce the language of mathematical questions themselves.”
• MLR7, Compare and Connect, which “[fosters] students’ meta-awareness as they identify, compare, and contrast different mathematical approaches, representations, and language.”

Lesson Narratives include strategies designed to assist other special populations of students in completing specific tasks. Examples of these supports for students with disabilities include:

• Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.
• Conceptual Processing: Eliminate Barriers. Assist students in seeing the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.
• Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.
• Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.
• Memory: Processing Time. Provide students with a number line that includes rational numbers.
• Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

All students complete the same lessons and activities; however, there are some optional lessons and activities that a teacher may choose to implement with students. For example, in Algebra 1, Unit 3, Lesson 10 is an optional lesson intended to provide an opportunity for students to use skills and knowledge gained from other lessons in this unit.

“Are you ready for more?” is included in some lessons to provide students additional interactions with the key concepts of the lesson. Some of these tasks would be considered investigations at greater depth, while others are additional practice.

There is no clear guidance for the teacher on ways to specifically engage advanced students in investigating the mathematics content at greater depth.