High School - Gateway 3

The materials reviewed for IM® Integrated Math v.360 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The Course Guide contains sections titled What’s in an IM Lesson, Key Structures in This Course, and Scope and Sequence. Within the Scope and Sequence, there is a Pacing Guide and a Dependency Diagram. The sections within the Course Guide provide instructional guidance related to the use of student and ancillary materials. 

Examples include:

• The Course Guide, Scope and Sequence, lists each of the units, explains connections to prior learning, and describes the progression of learning throughout each unit.

• The Course Guide, Scope and Sequence, Pacing Guide, and Dependency Diagram show the interconnectedness among lessons and units within the series (Integrated Math 1, Integrated Math 2, and Integrated Math 3), as well as how the series connects to Grades 6-8.

• The Course Guide, What’s in an IM Lesson, Narratives Tell The Story states, “The story of each grade is told across the units in the narratives. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each section within a unit has a narrative that describes the mathematical work in the section. Each lesson and each activity in a unit also has narratives. The Lesson Narrative explains: The mathematical content of the lesson and its place in the learning sequence. The meaning of any new terms introduced in the lesson. How the mathematical practices come into play, as appropriate. The Activity Narrative explains: The mathematical purpose of the activity and its place in the learning sequence. What students are doing during the activity. What to look for while students are working on an activity to orchestrate an effective Activity Synthesis. Connections to the mathematical practices, when appropriate.”

• The Course Guide, Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer the opportunity to observe students’ prior understandings. Each lesson starts with a Warm-up to activate students’ prior knowledge and to set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The Lesson Synthesis at the end consolidates understanding and makes the learning goals of the lesson explicit. In the Cool-down that follows, students apply what they learned. Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.” 

• The Course Guide, Key Structures in This Course, Using the 5 Practices For Orchestrating Productive Discussions states, "Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. To facilitate these conversations, the IM curriculum incorporates the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). The 5 Practices are: anticipate, monitor, select, sequence, and make connections between students’ responses. All IM lessons support the practices of anticipating, monitoring, and selecting students’ work to share during whole-group discussions. In lessons in which students make connections between representations, strategies, concepts, and procedures, the Lesson Narrative and the Activity Narrative support the practices of sequencing and connecting as well, and the lesson is tagged so that these opportunities are easily identifiable. For additional opportunities to connect students’ work, look for activities tagged with MLR7 Compare and Connect. Similar to the 5 Practices routine, Compare and Connect supports the practices of monitoring, selecting, and making connections. In curriculum workshops and PLCs, rehearse and reflect on enacting the 5 Practices."

• The Glossary provides a visual glossary for teachers that includes both definitions and illustrations. 

The materials include annotations and suggestions that are presented within the context of specific learning objectives. Within each lesson or activity, implementation guidance includes a rationale for each strategy or activity. 

Examples include:

• Integrated Math 1, Unit 4, Unit Narrative states, “In this unit, students examine solving and graphing linear equations and systems of linear equations. The unit builds on learning from middle school when students used variables to write equations, manipulated equations using valid moves such as the distributive property, and solved basic systems of linear equations using graphs and substitution. In the first section, students recall writing equations to represent situations. In the second section, they use valid moves to write equivalent equations that can be used to solve for unknown values or to isolate variables. The third section examines solving systems of equations using graphs, substitution for variables, and elimination of variables. Students use their understanding of writing equivalent equations to understand why each of the methods works for finding the solution.”

• Integrated Math 2, Unit 2, Lesson 4, Activity 4.2, Activity Synthesis states, “The goal of the synthesis is for students to understand the connection between the process of dilating points on a line and dilating the line itself. Specifically, what happens if the line goes through the center of the dilation? Invite students to share how the definition of dilation can help them answer these questions. Students should have the opportunity to hear and articulate that because dilations, by definition, take points along rays through the center, dilating a line through the center will take all the points to points on that same line, so the line doesn’t move. It may be hard for students to put into words that the points are dilated, but due to the nature of infinity, the line is not changed, so invite several students to put their explanation into their own words. In a later activity, students will state and record a theorem about lines that do and do not pass through the center of the dilation, so it’s useful for students to be clear about why this is true.” 

• Integrated Math 3, Unit 3, Lesson 7, Activity 7.3, Activity Narrative states, “The purpose of this activity is for students to practice solving rational equations and identifying extraneous solutions, if they exist. Students are expected to use algebraic methods to solve the equations and should be discouraged from using graphing technology.” The Launch provides additional teacher guidance and states, “Arrange students in groups of 2. Tell students to complete the first problem individually and then check their work with their partner before completing the following problems together.”

The materials reviewed for IM® Integrated Math v.360 meet expectations for containing explanations and examples of grade-level/course-level concepts and/or standards and how the concepts and/or standards align to other grade/course levels so that teachers can improve their own knowledge of the subject.

The materials contain adult-level explanations and examples of the more complex course-level concepts so that teachers can improve their own knowledge of the subject. At the lesson level, the materials identify vertical alignment. Standards addressed in the lesson are listed as “Building On,” “Addressing,” and “Building Towards.” Unit Narratives, Lesson Narratives, and Activity Synthesis sections throughout the materials provide explanations and examples of how course-level concepts connect to prior grade-level and course-level content and support progression to subsequent units and courses.

Examples include:

• Integrated Math 1, Unit 6, Lesson 1, Warm-up, Activity Synthesis clarifies the meaning of variables and categorical variables by directing teachers to “Point out that a variable is a characteristic that can take on different values and a categorical variable represents data which can be divided into groups or categories.” An example follows, and question prompts are provided for the teacher. Materials state, “Ask students: ‘Why is this table an effective method for solving this question?’ (It is effective because it keeps track of all the information we know and don’t know in a way that is organized to show the relationship between each piece of information.) ‘Was using a table the only way to solve the questions?’ (No, there are other methods to arrive at a good solution. Reasoning from the original statements, drawing a representation of the situation, or other similar methods are also good ways to approach the problem.) ‘What does the word ‘two’ in a two-way table refer to? How is it seen in the example from the task?’ (It refers to the number of categorical variables, not the number of categories for each variable. In this example, the two variables are writing utensil preference and paper preference.) Summarize by stating: ‘A two-way table can be used to organize data from two different categorical variables.’”

• Integrated Math 2, Unit 3, Unit Narrative states, “In this unit, students build an understanding of ratios in right triangles, which leads to naming cosine, sine, and tangent as trigonometric ratios. Prior to beginning this unit, students will have considerable familiarity with right triangles and similarity. They learned to identify right triangles in grade 4. Students studied the Pythagorean Theorem in grade 8 and used similar right triangles to build the idea of slope. This unit builds on this extensive experience and grounds trigonometric ratios in familiar contexts. Several concepts build throughout the unit. Students begin by using similar triangles to create a table of ratios of the side lengths in right triangles. At first, their table includes only the bottom rows of the table shown here. Taking the time to both build and use the table helps students construct a solid foundation before they learn the names of trigonometric ratios.” An example of the sine, cosine, and tangent ratio for a 25^\circ and 65^\circ degree angle are provided. The Unit Narrative goes on to state, “Students notice patterns between the columns for cosine and sine before they first hear the terms ‘cosine’ and ‘sine.’ In a subsequent lesson, they investigate that relationship, proving the two ratios are equal for complementary angles. Finding the measures of acute angles in a right triangle follows a similar arc, where students first use the table to estimate and then, in a subsequent lesson, learn how to calculate an angle measure given the side measures by using arcsine, arccosine, and arctangent. As students measure side lengths and compute ratios, there is an opportunity to discuss precision. In this unit, students will round side lengths to the nearest tenth and angle measures to the nearest degree in most cases. When students solve problems in context, they grapple with whether or not their answer is reasonable, as well as the appropriate degree of precision to report. Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.”

• Integrated Math 3, Unit 2, Lesson 5, Lesson Narrative states, “Students investigate the connection between the factors of a polynomial, its zeros, and the horizontal intercepts of its graph. Students were introduced to these ideas with quadratics in a previous course. In this lesson, they extend that thinking into higher-degree polynomials, using structure to make connections between points on graphs and solutions to equations (MP7). Since these connections are more difficult to see from the standard form, students work primarily with the factored form in this lesson. Students begin by noticing patterns, given equations of cubic polynomials written in factored form and their graphs. They reason about the zeros of a function and the horizontal intercepts of its graph, and also use the zero product property to infer zeros from factors. Students will not be able to infer a function’s linear factors from its zeros until the Remainder Theorem is proved in a later lesson, but students can begin to see connections between factors and zeros here. Then students strengthen their understanding by matching equations with graphs or descriptions of graphs, using intercepts and other features such as input-output pairs.”

The materials contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. The Course Guide includes a “Further Reading” section that connects mathematical research and pedagogy to course content. Articles extend teacher content knowledge beyond grade-level expectations and clarify mathematical concepts in greater depth. Information regarding professional development is available through the “Professional Learning” link on the webpage.

Examples include:

• Integrated Math 1, Course Guide, Further Reading, Mathematics 1, Unit 4, Extraneous and Lost Roots states, “In this essay…McCallum characterizes solving an equation as applying a function to each side of the equation, and the conditions that result in extraneous roots.”

• Integrated Math 2, Course Guide, Further Reading, Mathematics 2, Unit 3, Making Sense of Distance in the Coordinate Plane states, “In this post, Richard shows how the distance formula is connected to an equation of a circle, which is the topic of a later unit in this course.”

• Integrated Math 3, Course Guide, Further Reading, Mathematics 3, Entire Course, Different Uses of Letters and Types of Representations in Algebra states, “In this essay…Oehrtman and Shultz explore the uses of letters (variables, constants, parameters) in high school mathematics and beyond.”

The materials reviewed for IM® Integrated Math v.360 meet expectations for including a year-long scope and sequence with standards correlation information.

The Course Guide includes multiple components that support planning and understanding of the program’s structure and standards alignment. The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. Lessons by Standard provides a table that shows each content standard for the course and the lessons in which it appears. Standards by Lesson provides a table listing the standards covered within each lesson. Course Guide, Standards for Mathematical Practice, aligns math practice standards (MPs) to lessons. The Pacing Guide and Dependency Diagram outline the number of lessons and suggested teaching days per unit. Each lesson references the standards addressed as "Building On,” "Addressing," or “Building Toward.” Within each unit, there is a Unit at a Glance document that can be accessed from Unit Downloads. The Unit At a Glance provides a table that outlines the standards addressed for each lesson as "Building On,” "Addressing," or “Building Toward.”

Examples:

• Integrated Math 1, Course Guide, Scope and Sequence narratively outlines unit content, prior knowledge, future learning, and terminology. The Course Guide states, “The course begins with students using a compass and a straightedge to improve their logical-reasoning skills in a geometric setting. Students gradually build a toolkit of constructions that lead to rigid transformations and showing congruence of figures. In particular, they examine conditions needed to guarantee triangle congruence. Students describe the shape of data distributions, using measures of center and variability. This leads them to model how multiple variables are related, using linear equations and systems of linear equations. Students write, evaluate, graph, and solve equations, explaining and validating their reasoning with increased precision. By examining how transformations affect graphs, students connect their geometric understanding of rigid transformations to their understanding of linear equations on a coordinate plane. These insights lead into a unit on two-variable statistics in which students examine relationships between variables, using two-way tables, scatter plots, and linear models. Students continue their exploration of graphs by solving linear inequalities and systems of linear inequalities to represent constraints in situations. Shifting focus, students deepen their understanding of functions by representing, interpreting, and communicating about them, using function notation, domain and range, average rate of change, and features of graphs. They also see categories of functions, starting with linear functions (including their inverses) and piecewise-defined functions (including absolute-value functions), followed by exponential functions. For each function type, students investigate real-world contexts, look closely at the structural attributes of the function, and analyze how these attributes are expressed in different representations. Within the classroom activities, students have opportunities to engage in aspects of mathematical modeling. Additionally, modeling prompts are provided for use throughout the course, offering opportunities for students to engage in the full modeling cycle. Implement these in a variety of ways. Please see the Mathematics Modeling Prompts section of this Course Guide for a more detailed explanation.”

• Integrated Math 1, Course Guide, Standards by Lesson, Unit 7, Lesson 3 identifies A-CED.1, A-CED.3, A-REI.3.

• Integrated Math 1, Course Guide, Standards for Mathematical Practice, identifies MP1 as being present in Unit 1 Lessons 4, 5, 8, 11, 12, 14, and 18.

• Integrated Math 1, Course Guide, Lessons by Standard, A-REI.5 is identified in Unit 4, Lesson 16.

• Integrated Math 1, Course Downloads, Unit at a Glance, Unit 5, Lesson 9 identifies Building Toward standards G-GPE.4 and Addressing standards G-GPE.4, G-GPE.5, and G-GPE.7.

• Integrated Math 1, Unit 3, Lesson 4, Preparation, Standards Alignment identifies Addressing standard S-ID.1, and Building Toward standard S-ID.2.

• Integrated Math 2, Course Guide, Scope and Sequence narratively outlines unit content, prior knowledge, future learning, and terminology. The Course Guide states, “Students begin the course making observations about triangles. Building from these observations, students gather experimental information, develop conjectures, write informal justifications, and then progress to writing formal proofs using definitions, assertions, and theorems developed in Math 1. Using transformation-based definitions of congruence and similarity allows students to rigorously prove triangle similarity theorems. Students apply theorems to prove results about quadrilaterals and other figures. Students extend their understanding of similarity to right triangle trigonometry in this course and to periodic functions in future courses. Students then begin their study of quadratic functions. Students investigate real-world contexts, look closely at the structural attributes of a quadratic function, and analyze how these attributes are expressed in different representations. The unit concludes with a study of the geometry of parabolas. Next, students engage with quadratic equations. Through reasoning, writing equivalent equations, and applying the quadratic formula, students extend their ability to use equations to model relationships and solve problems. Along the way, students encounter rational and irrational solutions, deepening their understanding of the real-number system. This work leads to students developing an understanding of complex numbers and solving quadratic equations that include non-real solutions. The idea of i, a number whose square is -1, expands the number system to include complex numbers. Nearing the end of the course, students analyze relationships between segments and angles in circles and develop the concept of radian measure for angles, which will be built upon in subsequent courses. Students close the year by extending what they learned about probability in grade 7 to consider probabilities of combined events and to identify when events are independent. Within the classroom activities, students have opportunities to engage in aspects of mathematical modeling. Additionally, modeling prompts are provided for use throughout the course, offering opportunities for students to engage in the full modeling cycle. Implement these in a variety of ways. Please see the Mathematics Modeling Prompts section of this Course Guide for a more detailed explanation.”

• Integrated Math 2, Course Guide, Standards by Lesson, Unit 4, Lesson 9 identifies A-APR.A, A-SSE.2, and A-SSE.3.

• Integrated Math 2, Course Guide, Standards for Mathematical Practice, identifies MP7 in Unit 3, Lessons 2, 3, 5, and 11.

• Integrated Math 2, Course Guide, Lessons by Standard, G-CO.5 is identified in Unit 1, Lesson 8.

• Integrated Math 2, Unit 5, Unit Downloads, Unit at a Glance, Unit 5, Lesson 1 identifies Building Toward standards A-CED.3, Addressing standard A-CED.1, and Building On F-IF.4.

• Integrated Math 2, Unit 2, Lesson 1, Preparation, Standard Alignment identifies Building On standards 7.G.1, 8.G.3, Addressing standards G-CO.2, G-SRT.1, G-SRT.1b, and Building Toward standards G-SRT.1a, G-SRT.1b, and G-SRT.2

• Integrated Math 3, Course Guide, Scope and Sequence narratively outlines unit content, prior knowledge, future learning, and terminology. The Course Guide states, “Students begin the course with a study of solid geometry. They examine cross-sections of solids and make connections between cross-sections and dilations. Students also use square root and cube root graphs to illustrate the relationship between scale factors and scaled area and volume. This work is an opportunity to revisit functions and how they can be represented in a variety of ways. This work leads students to analyze situations that are well modeled by polynomial functions before pivoting to study the structure of polynomial graphs and equations. Students do arithmetic on polynomial and rational functions and use different forms of the functions to identify asymptotes and end behavior. Students solve rational and radical equations and learn to recognize when the steps used to solve these types of equations result in solutions that are not solutions to the original equation. Students also study polynomial identities and use some key identities to establish the formula for the sum of the first n terms of a geometric sequence. Students then return to their study of exponential functions and establish that the property of growth by equal factors over equal intervals holds even when the interval has non-integer length. Students use logarithms to solve for unknown exponents and are introduced to the number e and its use in modeling continuous growth. Logarithm functions and some situations they model well are also briefly addressed. Students learn to transform functions graphically and algebraically. In previous courses and units, students adjusted the parameters of particular types of models to fit data. In this part of the course, students consolidate and generalize this understanding. This work is useful in the study of periodic functions that comes next. Students work with the unit circle to make sense of trigonometric functions, and then students use trigonometric functions to model periodic relationships. The last unit, on statistical inference, focuses on analyzing experimental data modeled by normal distributions. Students learn to use sampling and simulations to account for variability in data and estimate population means, margins of error, and proportions. Students develop skepticism about news stories that summarize data inappropriately. Modeling prompts are provided for use throughout the course. While students have opportunities to engage in aspects of mathematical modeling during class activities, modeling prompts allow students to engage in the full modeling cycle. Modeling prompts can be implemented in various ways. Please see the Mathematics Modeling Prompts section of this Course Guide for a more detailed explanation.

• Integrated Math 3, Course Guide, Standards by Lesson, Unit 6, Lesson 8 identifies F-IF.4 and F-IF.C.

• Integrated Math 3, Course Guide, Standards for Mathematical Practice, Unit 7, identifies MP2 for Lessons 1, 6, 10, 13, and 15.

• Integrated Math 3, Course Guide, Lessons by Standard, A-APR.1 is identified in Unit 2, Lessons 2, 4, and 6. 

• Integrated Math 3, Unit 5, Unit Downloads, Unit at a Glance, Unit 5, Lesson 2 identifies Addressing standards F-BF.3, F-IF.2, F-IF.7c and Building On standards F-IF.A.

• Integrated Math 3, Unit 3, Lesson 2, Preparation, Standards Alignment identifies Building On standards A-CED.4, Addressing standards A-CED.2, F-IF.2, F-IF.4, F-IF.C, F-IF.7, and Building Toward standards A-CED.2, F-BF.3, F-IF.4.

The materials reviewed for IM® Integrated Math v.360 meet expectations for explaining the program’s instructional approaches, identifying research-based strategies, and explaining the role of the standards.

  Examples include:

• Course Guide, Problem-Based Teaching and Learning states, “Illustrative Mathematics is a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, students’ thinking, and their own teaching practice. The curriculum and the professional-learning materials are designed to support students’ and teachers’ learning. This document defines the principles that guide IM’s approach to mathematics teaching and learning. It then outlines how each component of the curriculum supports teaching and learning, based on these principles.”

• Course Guide, Problem-Based Teaching and Learning, Learning Mathematics by Doing Mathematics states, “‘Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving’ (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the problem solvers learning the mathematics.”

• Course Guide, Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “The basic architecture of the materials supports all learners through a coherent progression of the mathematics, based both on the standards and on research-based learning trajectories. Activities and lessons are parts of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense.”

• Course Guide, What’s in an IM Lesson, Instructional Routines, describes Instructional Routines (IRs) as “designs for interaction that invite all students to engage in the mathematics of each lesson.” The materials describe IRs as structured opportunities for students to connect personal experience and mathematical understanding through discussion, questioning, justification, and reasoning. The materials state that IRs “have a predictable structure and flow” and are intended to provide consistent support for both teachers and students. The curriculum uses a finite set of routines across lessons to support pacing, reduce repeated explanation of directions, and increase time spent on mathematical learning. Some IRs, identified as Mathematical Language Routines (MLRs), were developed by the Stanford University UL/SCALE team. MLRs are integrated into lessons either as embedded components or as optional supports for English Learners. The first instance of each routine includes detailed guidance for implementation, while subsequent instances offer abbreviated reminders. The materials also include Digital Routines (DRs), which identify required or suggested uses of technology. These routines are flagged within activities to support lesson planning and professional development.

The materials reviewed for IM® Integrated Math v.360 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Guide includes a comprehensive Required Materials list detailing all materials needed for the grade-level content. The Unit at a Glance, located in the Unit Downloads section of each unit, provides a table that includes any required materials for each lesson within the unit. The Preparation section for each lesson lists Required Materials and Required Preparation for specific activities. Each activity specifies Materials To Gather; if an activity does not require materials, this section will be blank or state "none."

Examples include:

• Integrated Math 1, Course Guide, lists the following as required materials: “1-inch strips cut from card stock with evenly spaced holes, A collection of balls that bounce, Blank paper, Chart paper, Colored pencils, Dried linguine pasta. Dynamic geometry software, Four-function calculators, Geometry toolkits (HS) (Index cards to use as straightedges, compasses, tracing paper, blank paper, colored pencils, and scissors), Glue or glue sticks, Graphing technology, Graph paper, Highlighters, Internet-enabled device, Masking tape, Math Community Chart, Measuring tapes, Metal paper fasteners, Nut and bolt, Protractors, Rulers, Scientific calculators, Scissors, Spreadsheet technology, Statistical technology, Sticky notes, Tools for creating a visual display.”

• Integrated Math 2, Unit 7, Lesson 6, Preparation, Required Materials lists the following as materials: “Geometry toolkits (HS), Activity 2: Protractors”

• Integrated Math 3, Unit 7, Lesson 3, Preparation, Required Materials states, “Activity 3: Random number generator,” Required Preparation states, “Activity 3: All students will need a random number generator (for example, a calculator, a spreadsheet, or numbered pieces of paper that students randomly draw) that can generate whole numbers between 1 and 100. Data will need to be collected from each student to create four data sets so a dot plot can be made from each data set.”

The materials reviewed for IM® Integrated Math v.360 meet expectations for providing consistent opportunities to determine student learning throughout the school year. The assessment system provides sufficient teacher guidance for evaluating student performance and determining instructional next steps.

The assessment system provides multiple opportunities throughout the course to determine student learning throughout the school year. This includes:

• Every unit begins with a Check Your Readiness diagnostic assessment. 

• Most lessons end with a Cool-down formative assessment. 

• Each section ends with a Checkpoint of 1-3 problems that assess learning goals. 

• Each unit ends with an End-of-Unit Assessment. 

• Longer units have a Mid-Unit Assessment.

The Course Guide, Assessment Guidance states, "Assessment guidance focuses on what can be clearly observed and uses asset-based language to focus on what students understand about particular math concepts and what they show they can do as it relates to procedural skills, fluency, and application…The guidance also encourages reliance on the coherence of the math in the curriculum when considering how to address any unfinished learning.” 

The materials provide assessment guidance at the lesson, section, and unit levels throughout the course. Each End-of-Unit Assessment and the Mid-Unit Assessment includes answer keys and standards alignment to support teachers in interpreting student understanding. According to the Course Guide and Assessment Guidance, “All summative assessment problems include a complete solution and standards alignment. Multiple-choice and multiple-response problems often include a reason for each potential error that a student might make. Restricted constructed-response and extended-response items include a rubric. Unlike formative assessments, problems on summative assessments generally do not prescribe a method of solution.”

Examples Include:

• Integrated Math 1, Unit 7, End-of-Unit Assessment, Problem 4, students graph a linear inequality. The Student Task Statement states, “Graph the solution to the inequality 4x+5y<20.” The Problem Narrative states, “Students may graph the equation 4x+5y=20 either by plotting points (likely by finding the intercepts), or by rearranging the equation into slope-intercept form. Take note of their method for graphing the equation as well as the point that they test to see which side of the line to shade. It is also possible for students to decide which side of the line to shade without testing a point: Since the coefficients of x and y are positive and 4x+5y must be less than a positive number, the side containing (0,0) must contain the solutions.” The Solution states, “Minimal Tier 1 response: Work is complete and correct. Sample: See graph. Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: A boundary line that is solid rather than dashed; one mistake related to graphing the line, such as an algebra error on the way to y=mx+b form, or a line drawn with positive slope (provided that the decision about which side to shade after testing a point is consistent with this previous work); the wrong side of a correctly-drawn line is shaded. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: No shading at all; no line is graphed; two or more Tier 2 error types.” 

• Integrated Math 2, Unit 3, End-of-Unit Assessment, Problem 2, students identify trigonometric ratios for a right triangle. The Student Task Statement states, “Select all of the true equations based on the figure. A. cos(42)=\frac{b}{c} B. cos(48)=\frac{b}{c} C. sin(42)=\frac{b}{c} D. sin(48)=\frac{b}{c} E. tan(42)=\frac{b}{a} F. tan(48)=\frac{a}{b}“ The task includes an image of triangle \triangle ABC with angle A labeled 48^\circ and corresponding sides, a, b, c. The Problem Narrative states, “A student who does not select choice B or choice F may be confused about the roles that the hypotenuse, adjacent side, and opposite side play in determining trigonometric ratios. Students who do not select choice C or choice E may not recognize 42 degrees as the complement of 48 degrees. A student may select choice D if they mistake sine and cosine. A student may select choice A if they are not paying attention to the angle.”

• Integrated Math 3, Unit 4, Mid-Unit Assessment, Problem 7, students interpret a graph showing exponential decay. The Student Task Statement states, “The graph shows the amount of a certain chemical m, in milliliters (mL), left in a tank of water h hours after a filter begins running. The amount of the chemical decreases exponentially. 1. By what factor did the chemical decrease in the first hour and a half? Explain how you know. 2. By what factor did the chemical decrease in the first half hour? What about in the first hour? Explain how you know. 3. Write an equation relating m, the number of milliliters of the chemical in the tank of water, and h, the number of hours since the filtering began.” The task includes a graph of m,(mL of chemical in tank) on the y-axis and h, (hours since filtering began) on the x-axis. The Problem Narrative states, “Students interpret a graph showing exponential decay. They find, by extracting a root, the hourly decay rate given two values that are 1.5 hours apart. They use the hourly decay rate to write an equation representing the situation.” The Solution states, “Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample responses: 1. \frac{8}{27} (or equivalent) because \frac{80}{27}=\frac{8}{27}. 2. \sqrt[3]{27} (or equivalent) for the first half hour and \sqrt[3]{\frac{8}{27}}^2 for the first hour because there are 3 half hours in an hour and half, and there are 2 half hours in an hour. 3. m=270\cdot(\frac{4}{9})^h (or equivalent). Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: Correct answers to parts a and b, but uses the half-hourly rate instead of the hourly rate in the equation for part c; miscalculates answer for part a, but uses this correctly in parts b and c; incorrect answer to part b based on correct work for part a, and correct work for part c based on answer to part b. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: Incorrect answers to parts a and b; answer to part b is not justified or incorrectly justified; equation in part c, in addition to part a or b, is incorrect. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Answers to parts a and c are incorrect, and part b is not justified or incorrectly justified.”

Materials also provide teachers with structured guidance following formative assessments. Check Your Readiness serves as a pre-unit diagnostic formative assessment that identifies students who need additional support with prerequisite skills and highlights topics to address during the unit. Assessment Guidance highlights certain lessons and/or activities to slow down and spend extra time to ensure student understanding of the mathematical concepts. Most lessons conclude with Cool-downs that assess student thinking in relation to the lesson’s learning goal. At the conclusion of lesson sections within a unit, Checkpoints include one to three problems that assess section learning goals. According to the Course Guide, Assessment Guidance, assessment guidance for Cool-downs and Checkpoints falls into three categories with suggestions for next steps if most students struggle: “More Chances,” which signals students will revisit the concept without altering instruction; “Points to Emphasize,” which recommends minor instructional adjustments; and “Press Pause,” which suggests more time is needed before advancing. 

Examples include:

• Integrated Math 1, Unit 9, Check your Readiness, Problem 9, students interpret features of a discrete graph and make sense of statements given in function notation in terms of a real-life situation. The Student Task Statement states, “The function g represents the temperature in a city, in degrees Fahrenheit, h hours after 8:00 a.m. The graph plots some values of g (the task includes a graph of a function g with “temperature in F^\circ on the y-axis and “hours after 8 a.m.” on the x-axis). Select all statements that must be true about the situation. A. g(0)=50; b. g(12) represents the temperature at noon; C. The maximum temperature shown on the graph happens at 3:00 p.m.; D. g(11)<g(8); E. The temperature increased about 10 degrees Fahrenheit between 8:00 am and 1:00 p.m.” The Problem Narrative states, “Students interpret features of a discrete graph and make sense of statements given in function notation in terms of the situations they represent. Students may select B if they do not read carefully that the x-axis shows hours after 8:00 a.m. rather than hours on the clock. This same error could lead them to disregard A, C, D, or E. If most students struggle with this item, use Lesson 9 Activity 2 to show that f(9) represents an output and, because this is a function of time f(t), then t is the input. It is important to help students interpret correctly that t is ‘time since 1977,’ very similar to this item (time since 8 a.m.).”

• Integrated Math 2, Unit 2, Lesson 8, Cool-down, students prove theorems about triangles and other figures using similarity. The Student Task Statement states, "Noah has a conjecture about rhombi with certain properties always being similar. 1. What would you need to do to show that the conjecture is incorrect? 2. What would you need to do to show that the conjecture is true?” Responding to Student Thinking states, “Press Pause: By this point in the unit, there should be some student mastery of sufficient and necessary conditions for similarity. If most students struggle, make time to revisit related work in the practice problems referred to here (links to Integrated Math 2, Unit 2, Lesson 8 Problem 3 and Integrated Math 2, Unit 2, Lesson 18, Problem 9).”

• Integrated Math 3, Unit 2, Section A Checkpoint, Problem 2, students explain that the sum of two polynomials is a polynomial. The Student Task Statement states, “Two polynomial expressions are added together to get x+4x^2-3+7x^3. Is the sum a polynomial? Explain your reasoning.” Responding to Student Thinking states, “Points to Emphasize: If most students struggle with the definition of a polynomial, focus on this as opportunities arise in the next several lessons. For example, in the Synthesis of the activity (Integrated Math 3, Unit 2, Lesson 5 Activity 1: Notice and Wonder: Factored Form), ask students whether these are polynomial functions and to explain how they know.”

The materials reviewed for IM® Integrated Math v.360 meet expectations for providing strategies and support for students in special populations to work with grade-level content and meet or exceed grade-level standards, which support their regular and active participation in learning. 

The Course Guide includes overarching guidance on strategies and accommodations for special populations in Universal Design for Learning and Access for Students with Disabilities.

Examples include:

• Course Guide, Universal Design for Learning and Access for Students with Disabilities, Access for Students with Disabilities states, “Supplemental instructional strategies, labeled Access for Students with Disabilities, are included in each lesson. They are designed to help meet the individual needs of a diverse group of learners. Each support is aligned to one of the three principles of Universal Design for Learning to provide multiple means of engagement, representation, or action and expression, and includes a suggested strategy to increase access and eliminate barriers. Use these lesson-specific supports, as needed, to help students succeed with a specific activity, without reducing the mathematical demands of the task. Phase them out as students gain understanding and fluency.”

• Course Guide, Universal Design for Learning and Access for Students with Disabilities, Accessibility for Students with Visual Impairments states, “For students with visual impairments, accessibility features are built into the materials: 1. A palette of colors distinguishable to people with the most common types of color blindness. 2. Tasks and problems designed so that success does not depend on the ability to distinguish between colors. 3. Mathematical diagrams, presented in scalable vector graphic (SVG) format, can be magnified, without loss of resolution, and rendered in Braille. 4. Where possible, text associated with images is not part of the image file, but rather included as an image caption accessible to screen readers. 5. Alt text on all images makes interpretation easier for users accessing the materials with a screen reader. If students with visual impairments are accessing the materials using a screen reader, it is important to understand: All images in the curriculum have alt text: a very short indication of the image’s contents, so that the screen reader doesn’t skip over as if nothing is there. Some images have a longer description to help a student with a visual impairment recreate the image in their mind. Understand that students with visual impairments likely will need help accessing images in lesson activities and assessments. Prepare appropriate accommodations. Accessibility experts, who reviewed this curriculum, recommended that eligible students have access to a Braille version of the curriculum materials, because a verbal description of many of the complex mathematical diagrams is inadequate for supporting their learning.”

Examples include:

• Integrated Math 1, Unit 7, Lesson 8, Activity 8.2, Access for Students with Disabilities states “Action and Expression: Internalize Executive Function. Invite students to plan a strategy, including the tools that they will use, for checking if points are a solution. If time allows, invite students to share their plan with a partner before they begin. Recommend to students that they start with the step of plotting the point on their graph. Support them in articulating criteria that address evaluating shaded regions and boundary lines. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity. Supports accessibility for: Attention, Social-Emotional Functioning.”

• Integrated Math 2, Unit 1, Lesson 7, Activity 7.3, Access for Students with Disabilities states, “Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code pairs of congruent angles and corresponding variables or expressions. Supports accessibility for: Visual-Spatial Processing”

• Integrated Math 3, Unit 5, Lesson 5, Activity 5.2, Access for Students with Disabilities states, “Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory.”

The materials reviewed for IM® Integrated Math v.360 meet expectations for regularly providing extensions and/or opportunities for advanced students to engage with grade-level mathematics at greater depth. 

Are You Ready for More? activities available in some lessons offer extension opportunities for advanced students to engage with course content at a higher level. Course Guide, Key Structures in This Course, Are You Ready For More?, states, “Select classroom activities offer differentiation for students ready for a greater challenge…Every extension problem is made available to all students, with the heading Are You Ready for More? These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some problems extend the work of the associated activity, while others involve work from prior grades, prior units in the course, or reflect work that is related to the K–12 curriculum but is a type of problem not required by the standards. The problems are not routine or procedural, and they are not just ‘the same thing again but with harder numbers.’ They are intended for use on an opt-in basis by students—if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems, and it is not expected that any single student works on all of them.” 

Building on Student Thinking guidance embedded in the Teacher Guide provides teachers with ways to deepen student understanding as they engage with course materials. Course Guide, Key Structures in This Course, Teacher Learning through Curriculum, Building on Student Thinking states, “Certain activities within each lesson plan include ways to provide guidance, based on students’ understandings and ideas. Building on Student Thinking offers look-fors and questions to support students as they engage in an activity…Building on Student Thinking section provides questions that advance students’ understanding of mathematical concepts, strategies, or connections between representations.”

Throughout the series, there are many lessons and/or activities that the materials identify as optional. The materials do not indicate that advanced students are required to complete additional assignments beyond those assigned to their classmates.

Modeling Prompts available in each course include tasks that allow students to engage in grade-level mathematics at varying levels of depth. The materials provide multiple entry points while maintaining grade-level expectations.

Examples include:

• Integrated Math 1, Unit 8, Lesson 5, Activity 5.2, students use rules to evaluate functions, interpret the input and output values within the context of the price for two different video game consoles, and solve for an unknown input value. Building on Student Thinking states, “Students may question if A is a function at all because unlike B or other function rules they have seen so far, A(x) is defined with a constant instead of an expression containing the dependent variable. Or they may wonder why A(x) has the same value no matter what the input value is. Ask students to recall the definition of function and to consider whether each input value gives only one output value. Because it does, even though it is always the same output value, A is still a function.” Are You Ready for More? states, “Describe an option that, for any amount of time used, would cost no more than one of the given options and no less than the other given option. Explain or show how you know this option would meet these requirements.”

• Integrated Math 2, Modeling Prompts, Modeling Prompt 6, Designing a Fountain, students create a two-dimensional model of the path of water in a fountain based on characteristics of its layout. The materials provide three Student Task Statement options with varying degrees of scaffolding and open-endedness. Task Statement 1, Student-Facing Statement states, “A nearby park has a decorative pool. The board of commissioners of the park is considering installing water jets in the pool to make it into a fountain. They have asked you to design the water jets. Here is a diagram of the pool from above: Statues A and B are 4 feet tall each. Statues C and D are wider and 5.5 feet tall each. Statue E is 8 feet tall, with a vase on top. The diagram is drawn to scale, and the pool is 15 feet wide. The jets are at water level. You need to figure out the number of jets, where they should go, and what path the water should make. Your design needs to meet these criteria: The jets are placed only around the edge of the pool. There are at least two jets. The water doesn't hit any of the statues. Some water goes into the vase. The water doesn't go higher than 10 feet. When you have your final design, create a presentation to explain it to the commissioners. The presentation should show the paths of the water and include your mathematical descriptions of the paths.” Task Statement 2, Student-Facing Statement states, “A nearby park has a decorative pool. The board of commissioners of the park is considering installing water jets in the pool to make it into a fountain. They have asked you to design the water jets. Here is a diagram of the pool from above: Statues A and B are 4 feet tall each. Statues C and D are wider and 5.5 feet tall each. Statue E is 8 feet tall, with a vase on top. The diagram is drawn to scale, and the pool is 15 feet wide. The dots along the edge show where to place the jets. You don’t have to put jets in all five places. The jets are at water level. You need to figure out the number of jets, where they should go, and what path the water should make. Your design needs to meet these criteria: The jets are only in the places indicated by the dots. There are at least two jets. The water doesn’t hit any of the statues. Some water goes into the vase. The water doesn’t go higher than 10 feet. When you have your final design, create a presentation to explain it to the commissioners. The presentation should show the paths of the water and include your mathematical descriptions of the paths. Before you begin your design, sketch a side view of the pool, showing the heights of the statues and the vase. This will help you figure out the paths of the water.” Task Statement 3, Student-Facing Statement states, “A nearby park has a decorative pool. The board of commissioners of the park is considering installing water jets in the pool to make it into a fountain. They have asked you to design the water jets. Here is a diagram of the pool from above: Statues A and B are 4 feet tall each. Statue C is wider and 5.5 feet tall. Statue D is 8 feet tall, with a vase on top. The diagram is drawn to scale, and the pool is 15 feet wide. The dots along the edge show where to place the jets. You don’t have to put jets in all four places. The jets are at water level. You will need to figure out the number of jets, where they should go, and what path the water should make. Your design needs to meet these criteria: The jets are only in the places indicated by the dots. There are at least two jets. The water doesn’t hit any of the statues. Some water goes into the vase. The water doesn’t go higher than 12 feet. When you have your final design, create a presentation to explain it to the commissioners. The presentation should show the paths of the water and include your mathematical descriptions of the paths. Before you begin your design, sketch what the pool would look like from each side. Make sure to show the heights of the statues and the vase. This will help you figure out the paths of the water. To help you get started, here is part of one sketch, showing the widths of Statues A, B, and D. It shows what the pool would look like if you were standing near the bottom of the diagram that shows the pool from above. It also shows the jet locations next to Statues A and B. Figure out the heights of Statues A, B, and D, according to the scale of this sketch, and then finish the sketch by representing the heights. Make your own sketches for the other sides of the pool.”

• Integrated Math 3, Unit 4, Lesson 23 is identified as optional and allows students to extend their understanding of logarithmic functions as they encounter real-world applications. Lesson Narrative, Learning Goal states, “Use logarithmic relationships to solve problems about acidity and earthquakes.”

The materials reviewed for IM® Integrated Math v.360 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Course Guide, Key Structures in This Course, Purposeful Representations states, “Several considerations guide the choice of representations, the timing of their introduction, and the sequence in which they are presented, such as the extent to which they: Serve the mathematical learning goals; help to build conceptual understanding; support a coherent progression of mathematical ideas; are relevant across cases and contexts. The principle of ‘concrete before abstract’ also guides the use of representations. This principle comes into play in two ways: First, more concrete representations are introduced before those that are more abstract. For example, to develop an understanding of ratios, students first use physical objects to represent quantities related by a ratio. Next, they create discrete diagrams to represent such quantities, followed by double-number-line diagrams and tables of equivalent ratios. This thinking eventually bridges into representing non-proportional linear relationships and the multiple ways to represent functions that start in IM Grade 8 and continue throughout high school. Each representation is less concrete, more abstract, and offers greater flexibility and efficiency than the one before it. Second, a representation can show concrete concepts and numbers in earlier courses, and abstract ideas and quantities in later courses. For example, starting in IM Grade 3, students use rectangular diagrams to represent the multiplication of whole numbers. In IM Grade 6, they use such diagrams to represent the product of multi-digit decimals. In these early uses, students relate the discrete factors and product to concrete attributes of the rectangle: its side lengths and area. Later, students use rectangles to represent the multiplication of expressions that include variables, such as 5\cdot(3x+8) and a(2b+c). The same diagram is now used in a more abstract sense: to represent and organize decomposable factors and their partial products. In high school, students use the same diagram to reason about the multiplication and division of binomials and higher-degree polynomials.”

The Course Guide, Required Materials section, the Unit at a Glance documents, and the Preparation section of each lesson list suggested manipulatives. For Integrated Math 1 and Integrated Math 2, the materials recommend that students use a geometry toolkit throughout the course, including index cards or straightedges, compasses, tracing paper, blank paper, colored pencils, and scissors. The Math Tools menu, located in the top right corner of each course, unit, and lesson, provides access to virtual manipulatives, including a four-function calculator, scientific calculator, graphing calculator, geometry tool, spreadsheet tool, probability calculator, and constructions tool.

Examples include:

• Integrated Math 1, Unit 1, Lesson 5, Activity 5.2, students use a straightedge and compass to construct a line perpendicular to a given line through a point on the line. Activity Narrative states, “This activity invites students to play with straightedge and compass moves to construct a line perpendicular to a given line through a point on the given line. Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).” Student Task Statement states, “Here is a line l with a point labeled C. Use straightedge and compass moves to construct a line perpendicular to l that goes through C.”

• Integrated Math 2, Unit 4, Lessons 8 and 9 build on students’ use of the area model to represent the distributive property and multiply binomials. Lesson 8, Lesson Narrative states, “In this lesson, students begin connecting different forms of quadratics by representing multiplication as an area. Students make use of this structure to write equivalent equations in expanded form (MP7), and then make the connection to the abstract distributive property to multiply factors (MP2).” Warm-up, Student Task statement states, “1. Explain why the diagram shows that 6(3+4)=6\cdot3+6\cdot4. 2. Draw a diagram to show that 5(x+2)=5x+10.” The task includes an image of a rectangle partitioned into two smaller rectangles. Students use the area model to represent multiplication of polynomials. Lesson 9, Activity 9.2, Student Task Statement, Problem 1 states, “Show that (x-1)(x-1) and x^2-2x+1 are equivalent expressions by drawing a diagram or applying the distributive property. Show your reasoning.”

• Integrated Math 3, Unit 2, Lesson 1, Activity 1.2, students use scissors and paper to construct a box and write expressions representing its volume. Activity Narrative states, “This activity offers a hands-on introduction to the mathematical work of modeling the volume of a box using a polynomial function. Students do not need to develop an equation with variables for the volume of the box or identify the greatest possible volume at this time, as that is the focus of the following activity.” Launch states, “Demonstrate how to construct a box from an 8.5-inch by 11-inch sheet of paper by cutting out identical squares, with a side length of 1 inch, from each corner. Then measure to fill in the length, width, and height columns of the table. Calculate the volume of the box and add a point to the displayed axes that represents the relationship between the side length of the cutout square and the volume of the box. Assign each group a side length between 0.5 inch and 4 inches, in half-inch increments. Adjust these side lengths as needed if using different-sized paper. Tell groups to construct a box by cutting squares out of each corner using their assigned side length, and then to add a point to the graph representing their box.”