High School - Gateway 1

The materials reviewed for IM® Integrated Math v.360 meet expectations for materials attending to the full intent of the mathematical content contained in the high school standards for all students.

Across the series, lessons consistently include a Warm-up, one to three instructional Activities, and a Lesson Synthesis that engage students in the mathematical content of the non-plus high school standards. 

Examples include:

• G-CO.13: Integrated Math 1, Unit 1, Lesson 4, Activity 4.3, students use a straightedge and compass to construct equilateral triangles of different sizes and explain why each triangle is equilateral. Students apply construction techniques, compare side lengths using compass moves, and reason about congruence and rotational symmetry.

• S-ID.1: Integrated Math 1, Unit 3, Lesson 4, Activity 4.2, students interpret and compare multiple representations of data on a number line. In the Card Sort Activity, students match dot plots, histograms, and box plots that represent the same data set and explain their reasoning to a partner. Students use precise statistical language (e.g., symmetric, skewed, uniform, and bimodal) to describe distribution shapes across different displays.

• N-RN.3: Integrated Math 2, Unit 5, Lesson 21, Activity 21.2, students construct a general argument showing that the sum and product of two rational numbers are rational by representing the numbers as fractions, reasoning that ad+bc, ad, bc and bd are integers, and concluding that the results remain fractions and are therefore rational. In Activity 21.3, students show that the sum and product of a rational number and an irrational number (e.g., \sqrt{2} and \frac{1}{9}) are irrational. 

• A-SSE.1a: Integrated Math 3, Unit 2, Lesson 1, Cool-down, students interpret terms in a given equation to determine a reasonable domain from a real-world context. The Student Task Statement states, “Outside of the United States, the common paper size is called A4 and measures 21 by 29.7 centimeters. Let V(x)=(21-2x)(29.7-2x)(x) be the volume in cubic centimeters of a box made from A4 paper by cutting out squares of side length x in centimeters from each corner and then folding up the sides. What is a reasonable domain for V in this context? Explain or show your reasoning.” 

• F-LE.1: Integrated Math 3, Unit 4, Lesson 1, Activity 1.2, students analyze exponential growth in the context of successive scaling. In Activity 1.3, students recognize situations in which one quantity changes at a constant rate per unit interval relative to another as they examine a situation involving exponential growth. In Integrated Math 3, Unit 4, Lesson 5, Activity 5.3, students use coordinates from a graph representing radioactive mass over time to determine the value of a function for a different input.

The materials reviewed for IM® Integrated Math v.360 meet expectations for attending to the full intent of the modeling process when applied to the modeling standards.

Each course includes designated opportunities for students to engage in one or more aspects of the modeling process as well as tasks intended to address the full modeling process. Across Integrated Math 1, Integrated Math 2, and Integrated Math 3, modeling standards from multiple conceptual categories are addressed through these tasks. Each course has a set of Modeling Prompts that teachers can access from the Course or Unit landing page. The Modeling Prompts page includes a list of the prompts organized by aligned unit guidance on when to use the prompt within a unit and the standards that align to the prompt. The Course Guide section titled Key Structures in This Course, When to Use Mathematical Modeling Prompts provides implementation guidance. This structure includes modeling tasks in each course that engage students in the complete modeling cycle: defining problems, formulating and computing with mathematical models, interpreting results in context, validating conclusions, and reporting findings. The Course Guide states, “Mathematical modeling is often new territory for both students and teachers. Oftentimes, within the regular classroom lessons, activities include scaled-back modeling scenarios, for which students engage in only a part of the modeling cycle. These activities are tagged with the Aspects of Mathematical Modeling instructional routine, and the specific opportunity to engage in an aspect of modeling is explained in the Activity Narrative.”

Examples where students engage in one or more aspects of the modeling process include:

• Integrated Math 1, Unit 4, Lesson 1, Activity 1.2 and Activity 1.3, students estimate the cost of a pizza party. They formulate expressions to represent the parameters of the scenario, make assumptions, set constraints, and compute total cost based on those assumptions. Groups analyze how changes to parameters affect total cost and interpret the results to revise their estimates. (A-CED.2, A-CED.3, N-Q.2) 

• Integrated Math 2, Unit 3, Lesson 1, Activity 1.3, students consider relevant factors that affect ramp safety as part of designing a ramp to accompany a specific set of stairs (problem). They design a ramp for a school setting that meets accessibility requirements (formulate/compute). Students evaluate whether their design satisfies Americans with Disabilities Act (ADA) guidelines (interpret) and revise the design as needed to meet those guidelines (validate). (G-MG.1, G-MG.3, N-Q.2) 

• Integrated Math 3, Unit 2, Lesson 1, Activity 1.2, students construct open-top boxes from standard 8.5 × 11-inch paper and create a table of values relating the side length of the square cut from each corner to the resulting length, width, height, and volume of the box (compute). In Activity 1.3, students write an expression to model the volume of the box as a function of the side length of the square cut from each corner and use the graph of the function to approximate the dimensions that produce the greatest volume (formulate, compute, and interpret). (F-IF.4)

Examples where students engage in the full modeling process include: 

• Integrated Math 1, Modeling Prompt 9, students determine how bonuses should be distributed among five workers who contributed to a project. They develop at least two distribution methods based on defined variables relevant to the context (problem/compute/report). Students recommend one method and justify their selection using quantitative reasoning. They calculate each employee’s bonus under each method (compute), analyze advantages and disadvantages of each approach (interpret), and evaluate how the distribution may be received by different employees, providing justification for the selected method (validate). Students revise their models as needed based on their analysis (interpretation). (N-Q.A, N-Q.2)

• Integrated Math 2, Modeling Prompt 5, students are commissioned to construct a flag based on given instructions. They choose a measurement for one side and use the instructions to determine the remaining measurements (define the problem). Students apply their knowledge of congruence and the area of triangles to determine the amount of material needed to construct the flag (compute). They then adapt and scale a flag of their choice to fit a triangular banner of a specified size. Students determine the amount of material needed for each colored component of the flag and report their findings (interpret, compute, and report). (G-MG.1, G-MG, 3)

• Integrated Math 3, Modeling Prompt 8, students conduct an experiment and analyze the results. They define a real-world question, identify relevant elements such as population, response variables, treatments, and outcomes, and consider potential causal relationships (problem/formulate). Students identify possible sources of error, evaluate evidence of a treatment effect, and determine whether additional data would strengthen their conclusions (interpret). They design and implement an experiment by organizing subjects into groups, collecting measurements, and calculating summary statistics (compute/formulate). Students analyze differences between groups, assess whether observed differences are likely due to chance, examine sources of variability, and interpret results within context (interpret/validate). (S-IC.5, S-ID.2, S-ID.4).

The materials reviewed for IM® Integrated Math v.360 meet expectations for, when used as designed, allowing students to spend the majority of their time on the content from CCSSM, widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

The Course Guide includes a Scope and Sequence and a teacher-facing Pacing Guide for each course. The Pacing Guide organizes instructional time across units and lessons and identifies topics aligned to standards designated as widely applicable as prerequisites (WAPs) for postsecondary study and careers. The Lessons by Standard table lists each content standard for the course and the lessons in which it appears. The Standards by Lesson table identifies the standards addressed within each lesson. The Pacing Guide and Dependency Diagram outline the number of lessons and suggested instructional days per unit to support year-long planning. Across Integrated Math 1, Integrated Math 2, and Integrated Math 3, the Scope and Sequence and Pacing Guides allocate more than half of instructional days to lessons aligned to standards designated as widely applicable as prerequisites (WAPs), while non-WAP standards support WAP content rather than functioning as standalone units. Across courses, these resources indicate that a majority of lessons align with standards identified as WAPs.

Examples of how the materials allow students to spend the majority of their time on widely applicable prerequisites (WAPs) include, but are not limited to:

• Integrated Math 1, Unit 7, Lesson 3, Curated Practice Problem Set, Problem 3, students write and solve an inequality representing a real-world scenario. The Student Task Statement states, “A cell phone company offers two texting plans. People who use Plan A pay 10 cents for each text sent or received. People who use Plan B pay 12 dollars per month, and then pay an additional 2 cents for each text sent or received. 1. Write an inequality to represent the fact that it is cheaper for someone to use plan A than plan B. Use x to represent the number of texts they send. 2. Solve the inequality.” (A-CED.1, A-REI.3)

• Integrated Math 1, Unit 9, Lesson 21, Activity 21.3, students analyze population data presented in a table of years (1804, 1927, 1960, 1974, 1987, 1999, 2011) and corresponding world population values in billions (1, 2, 3, 4, 5, 6, 7). They determine whether a linear or exponential model best fits the data and use the selected model to make and critique predictions. Student Task Statement states, “1. Would a linear function be appropriate for modeling the world population growth over the last 200 years? Explain. If you think it is appropriate, find a linear model. 2. Would an exponential function be appropriate for modeling the world population growth over the last 200 years? Explain. If you think it is appropriate, find an exponential model. 3. From 1950 to the present day, by about what percentage has the world population grown each year? 4. From 1950 to the present day, by about how many people has the world population grown each year? 5. If the growth trend continues, what will the world population be in 2050? How long do you think the growth will continue? Explain your reasoning.” (F-LE.1, F-LE.2)

• Integrated Math 2, Unit 2, Section B Checkpoint, Problem 2, students apply triangle similarity theorems to determine whether two triangles are similar and justify their conclusions. The Student Task Statement states, “Are the triangles similar? If so, write a similarity statement and explain your reasoning. If not, what additional information is needed?” (G-SRT.5)

• Integrated Math 2, Unit 6, Lesson 3, Activity 3.2, students represent the relationship between powers of \frac{1}{2} and square roots and use exponent rules to show that b^{\frac{1}{2}} is equivalent to \sqrt{b}. In Activity 3.3, students extend this reasoning to cube roots by relating b^\frac{1}{2} to b^\frac{1}{3} and evaluate expressions with fractional exponents. (N-RN.1, N-RN.2)

• Integrated Math 3, Unit 7, Lesson 5, Activity 5.4, students interpret how the mean represents the center of a normal distribution and how the standard deviation affects the spread of the distribution. Student Task Statement states, “These curves represent normal distributions with different means and standard deviations. What do you notice?” (S-ID.2)

The materials reviewed for IM® Integrated Math v.360 meet expectations for materials and, when used as designed, allow students to fully learn each standard.

The materials engage students with the full scope of the mathematical content in the non-plus high school standards. Across the series, each lesson includes a Warm-up, one to three instructional Activities, and a Lesson Synthesis through which students work directly with the targeted standards.

Examples of how the materials allow students to fully learn the non-plus standards include:

• Integrated Math 1, Unit 1, Lesson 11, Cool-down, students analyze sample student work in which a reflection is performed incorrectly. Using the definition of a reflection, they identify one idea the student applied correctly (e.g., the image is the same distance from the line of reflection as the original figure) and one idea the student applied incorrectly (e.g., segments connecting corresponding points must be perpendicular to the line of reflection). In Lesson 12, Activity 12.3, students translate a triangle using a directed line segment and analyze the relationship between corresponding figures, determining that translated lines are parallel and translated segments have equal length. They justify these conclusions by reasoning that a translation moves every point the same distance in the same direction, consistent with the definition and properties of translations. In Lesson 14, Cool-down, students analyze sample student work in which a rotation is performed incorrectly. Using the definition of a rotation, they identify one idea the student applied correctly (e.g., each point is rotated along a circular path centered at the same point) and one idea the student applied incorrectly (e.g., all points must share a single center of rotation). (G-CO.4)

• Integrated Math 1, Unit 6, Lesson 7, Activity 7.3, students match scatter plots to numerical correlation coefficients and justify their matches by reasoning about the direction and strength of linear relationships, using the facts that the sign of the correlation coefficient matches the sign of the slope and that values closer to ±1 indicate stronger linear relationships. In the Cool-down, Student Task Statement states, “1. What information does a correlation coefficient tell us about the data in a scatter plot? 2. Which value best estimates the value for the correlation coefficient of the scatter plot: -1, -0.8, -0.2, 0.2, 0.8, or 1? Explain your reasoning.” In Lesson 8, Activity 8.2, students use technology to calculate the correlation coefficient for a data set relating distance traveled and travel time and interpret its meaning in context. (S-ID.8)

• Integrated Math 2, Unit 3, Lesson 9, Activity 9.2, students prove the Pythagorean identity sin^2(\theta)+cos^2(\theta)=1for any acute angle \theta. In Activity 9.3, students apply the identity to determine sin(\theta) given cos(\theta), and vice versa, and use this relationship to decide whether given statements must be true, could be true, or cannot be true. In Unit 6, Lesson 6, Activity 6.3, students sort two sets of cards displaying the value of the sine, cosine, or tangent of an unknown angle along with the quadrant of the angle on the unit circle. Students then select one matched pair and calculate the remaining two trigonometric values. (F-TF.8)

• Integrated Math 2, Unit 6, Lesson 7, Activity 7.2, students analyze the graph of y=x^2 for values such as y=9, 0, -1, and reason that squaring any real number cannot produce a negative value. They conclude that the equation y=x^2 has no real solutions when y is negative. Students then define a new number, i, as a solution to x^2=-1, and represent it relative to the real number line, extending the number system. In Lesson 8, Activity 8.4, students plot complex numbers in the complex plane and represent them in the form a+bi where a and b are real numbers. (N-CN.1)

• Integrated Math 3, Unit 2, Lesson 2, Activity 2.2, students begin working with polynomial notation by evaluating polynomial functions for specific values of x and making connections between parts of a polynomial expression and parts of an integer written in base 10. In Unit 2, Lesson 4, Activity 4.3, students experiment with adding, subtracting, and multiplying polynomials to determine whether the results are always polynomials. Student Task statement states, “Here are some questions about polynomials. You and a partner will work on one of these questions. 1. If you add or subtract two polynomials, will you always get a polynomial? Explain your reasoning. 2. If you multiply two polynomials, will you always get a polynomial? Explain your reasoning. a. Try combining some polynomials to answer your question. Use the ones given by your teacher or make up your own. Keep a record of what polynomials you tried and the results. b. When you think you have an answer to your question, explain your reasoning using equations, graphs, visuals, calculations, words, or in any way that will help others understand your reasons.” (A-APR.1)

While there are opportunities within the materials that allow students to fully learn some non-plus standards, the materials do not allow students to fully learn the following non-plus standards. For example:

• Integrated Math 1, Unit 9, Lesson 1, Warm-up, students create a pictorial representation of exponential growth. The Activity Synthesis states, “Introduce sequence notation with this sequence. Tell students that, for this sequence, f(0)=1, f(1)=3, f(2)=9 and f(3)=27. In general, f(n) represents the number in the nth place of the sequence and f(n+1) represents the next number in the sequence.” In Activity 1, students compare the amount of two purses of reward money as one purse grows by $200 each day and the other purse doubles each day. The Launch states, “Display the values in sequence notation for all to see. For Purse A: f(0)=1000, f(1)=1200, f(2)=1400. For Purse B: g(0)=0.01, g(1)=0.02, g(2)=0.04. Ask students, ‘How can you write an equation relating f(n+1) and f(n)? What about g(n+1) and g(n)?” f(n+1)=f(n)+200 and g(n+1)=2\cdot g(n)’ ‘How much money will be in each purse after a week? That is, what are f(7) and g(7)? Explain your reasoning.” f(7)=2400 because Purse A starts with $1,000 and $200 is added each day, so we added 200 seven times to get 1000+200\cdot 7=2400. g(7)=1.28 because Purse B starts with $0.01 and doubles each day, so we multiply $0.01 by 2 seven times to get 0.01\cdot 2^7=1.28’” In Lesson 2, Activity 2.2, students make comparisons between tables representing two plans for expanding a food company’s chain of stores. The Activity Synthesis states, “How can you write the patterns using sequence notation to relate f(n+1) to f(n) and g(n+1) to g(n)?” (f(n+1)=f(n)+20 and g(n+1)=2\cdot g(n).” While notes in the Teacher Guide direct teachers to prompt students to write arithmetic and geometric sequences recursively and with explicit formulas to model these situations, the Student Task Statements do not explicitly ask students to represent the sequences recursively, nor are students required to represent sequences recursively in other lessons across the series. Additionally, opportunities for students to translate between explicit and recursive forms are limited. (F-BF.2)

• Integrated Math 1, Unit 9, Lesson 8, Activity 8.2, students represent a situation involving exponential growth using a table of values, a graph, and an equation. In Lesson 11, Activity 11.2, students consider whether the height of a tennis ball after each successive bounce can be modeled by a linear or exponential function and then write an equation. In Lesson 19, Activity 19.2, students consider two investment options portrayed in two tables - one representing a linear relationship and one representing an exponential relationship - and write an equation for each option to represent the amount of money over time. In Integrated Math 3, Unit 4, Lesson 1, Warm-up, students are given the balance of an account at the start of the year and write a context in which the account balance for each succeeding month can be represented by either a geometric sequence or an arithmetic sequence. In Lesson 3, Activity 3.3, students write an exponential equation given a graph that models the amount of a chemical remaining in a lake after it has been cleaned for a certain amount of time. While students are asked in many instances across the series to construct linear and exponential functions from a graph, a description of a relationship, or two input-output pairs, they are provided with limited opportunities to construct arithmetic or geometric sequences from a graph, a description of a relationship, or two input-output pairs. (F-LE.2)

The materials reviewed for IM® Integrated Math v.360 meet expectations for requiring students to engage in high school mathematics by focusing on problem contexts and attending to various types of real numbers.

Students work with course-level problems that reference prior mathematical knowledge. Activities introduce new concepts using simpler numerical values and later require students to perform operations and apply concepts using the full real number system. Across courses, activities require students to apply concepts from Grades 6-8, including proportional relationships, systems of equations, and irrational numbers, within high school-level problems.

Examples of problems that allow students to engage in age-appropriate contexts include:

• Integrated Math 1, Unit 6, Lesson 5, Activity 5.3, students create a scatterplot and sketch a line of best fit for a data set relating the weight of ice cream sold in a day at a small store to the average outdoor temperature. Students then use technology to compute the line of best fit and interpret the slope and y-intercept in context. (S-ID.6, S-ID.7)

• Integrated Math 2, Unit 2, Lesson 18, Activity 18.2, students use similar triangles and the angle of reflection to determine where to bounce a cue ball off a rail so that it strikes another ball into a pocket. (G-SRT.5)

• Integrated Math 3, Unit 6, Lesson 19, Curated Practice Problem Set, Problem 1, students determine the period and radius of a Ferris wheel given an equation modeling its vertical position over time. (F-TF.5)

Examples of problems that allow students to engage in the use of various types of real numbers include:

• Integrated Math 1, Unit 9, Lesson 4, Activity 4.3, students make sense of a graph representing a situation characterized by exponential decay. Student Task Statement states, “Once a glow stick begins to glow, it can glow for hours. The graph shows the luminescence, in lumens, of a glow stick over time, in hours. (Students interpret labeled points (1, 6.3), (2, 4.4), (3, 3.1) on a graph representing an exponential decay situation.) 1. Scientists have found that glow stick luminescence decreases exponentially. How can you check if the graph supports the scientists’ claim? 2. How much less bright is the glow stick after the first hour? What fraction of the original luminescence is that? 3. How much less bright is the glow stick after the second hour? What fraction is that of the luminescence 1 hour earlier? 4. What fraction of luminescence stays for each hour that passes? Explain your reasoning. 5. Complete the table to show the predicted luminescence 4 and 5 hours after beginning to glow. 6. Describe how you would find how many lumens the glow stick produces after 10 hours. After h hours?” (F-IF.4, F-LE.2)

• Integrated Math 2, Unit 5, Lesson 13, Activity 13.2, students solve quadratic equations with fractional and decimal coefficients by completing the square. Student Task Statement states, “Here are four equations, followed by worked solutions of the equations. Each solution has at least one error. Solve one or more of these equations by completing the square. Then, look at the worked solution of the same equation as the one you solved. Find and describe the error or errors in the worked solution. 1. x^2+14x=-24; 2. x^2-10x+16=0; 3. x^2+2.4x=-0.8; 4. x^2-\frac{6}{5}x+\frac{1}{5}=0.” (A-REI.4b)

• Integrated Math 3, Unit 1, Lesson 7, Curated Practice Problem Set, Problem 2, students determine the volume of dilated solids using scale factors of \frac{1}{4},0.4,1,1.2,\frac{5}{3}, resulting in volumes of \frac{3}{16} cubic units, 0.768 cubic units, 12 cubic units, 20.736 cubic units, and \frac{500}{9} cubic units. (G-GMD.1)

Examples of problems that provide opportunities for students to apply key takeaways from Grades 6-8 include:

• Integrated Math 1, Unit 6, Lesson 4, Activity 4.2, students extend their work with constructing and interpreting scatter plots (8.SP.1) by creating a scatter plot from contextual data, informally drawing a line of best fit, interpreting the slope and y-intercept of the linear model, and using the model to make predictions (S-ID.6).

• Integrated Math 2, Unit 2, Lesson 3, Activity 3.2, students use dilations of two-dimensional figures using coordinates (8.G.3) to predict and determine the side lengths of a figure after it has been dilated by a given scale factor and compare the effects of their scale factor with those of other students (G-CO.2).

• Integrated Math 3, Unit 3, Lesson 9, Activity 9.2, students use their understanding of square and cube root solutions (8.EE.2) to investigate the number of real solutions of equations in the form y = x3 and use a graph to estimate the solutions (A-REI.2).

The materials reviewed for IM® Integrated Math v.360 meet expectations for being mathematically coherent by making meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

The materials include connections within individual courses and across the series. The Course Guide, Scope and Sequence, and Dependency Diagram identify these connections for teachers, and lessons incorporate them within Warm-ups, Instructional Activities, Lesson Syntheses, and Cool-downs.

Examples where the materials foster coherence and make meaningful mathematical connections within a single course: 

• Integrated Math 1, Unit 9, Lesson 21, Activity 21.2, requires students to distinguish between linear and exponential growth and create equations to model population growth for three cities (F-LE.1, F-LE.2). Students evaluate how well their models fit the data, use the models to make predictions, and assess the reasonableness of those predictions in context (S-ID.6).

• Integrated Math 2, Unit 6, Lesson 13, Activity 13.3, requires students to use completing the square to solve quadratic equations with complex solutions (A-REI.4b). In Unit 7, Lesson 3, Activity 3.3, students use completing the square to translate between the standard form of a circle, (x-h)^2+(y-k)^2=r^2, and the general form, x^2+y^2+ax+by+c=0, to determine the center and radius of a circle (G-GPE.1). Students apply the distributive property and recognize structure in expressions (A-SSE.2) as they rewrite quadratic and circle equations to reveal key features. Students connect algebraic manipulation with geometric interpretation by determining the center and radius from equivalent forms of a circle equation.

• Integrated Math 3, Unit 5, Lesson 9, Activity 9.2, requires students to compare the effects of scaling the inputs and outputs of a function using a Ferris wheel context by calculating the scale factors needed to model a wheel that is twice the height of and a wheel that rotates at half the speed of the original function (F-BF.3). In Unit 6, Lesson 14, Activity 14.2, students revisit vertical scale factors as they learn about the amplitude of a trigonometric function by modeling a point on a spinning windmill (F-IF.7c, F-TF.5).

Examples where the materials foster coherence and make meaningful mathematical connections across courses in the series:

• Integrated Math 1, Unit 3, Lesson 4, Activity 4.2, requires students to match histograms and dot plots that represent the same data set and use precise vocabulary to describe the shape of the distribution (S-ID.A). In Integrated Math 3, Unit 7, Lesson 5, Warm-up, students analyze histograms and distribution shapes to identify characteristics of a normal distribution (S-ID.4).

• Integrated Math 1, Unit 2, Lesson 7, requires students to prove the Angle-Side-Angle Triangle Congruence Theorem. In Integrated Math 2, Unit 2, Lesson 9, students use the Angle-Side-Angle Triangle Congruence Theorem and properties of dilations to prove that two triangles with two pairs of corresponding congruent angles are similar, establishing the Angle-Angle Triangle Similarity Theorem (G-SRT.2). In Lesson 15, Activity 15.2, students apply Angle-Angle Triangle Similarity to determine missing side lengths when an altitude from the hypotenuse of a right triangle creates three similar triangles (G-SRT.5).

• Integrated Math 2, Unit 4, Lesson 10, Activity 10.3, requires students to identify the x-intercept(s) and y-intercept of quadratic functions given their graphs, standard form equations, and factored form equations. Students generalize their findings to identify the x-intercept(s) and y-intercept of a quadratic function expressed in factored form (A-SSE.3). In Integrated Math 3, Unit 2, Lesson 5, Activity 5.3, students complete a card sort by matching polynomials written in factored form with their corresponding graphs or descriptions of the graphs' x-intercepts (A-APR.3). Students apply their understanding of x-intercepts in factored-form quadratic functions to identify the zeros of polynomial functions from their factored forms.

The materials reviewed for IM® Integrated Math v.360 meet expectations for the materials to explicitly identify and build on knowledge from Grades 6-8 to the high school standards.

Lessons build from knowledge students learned in Grades 6-8 and require application of prior concepts to new grade-level content aligned to high school standards. Teacher-facing materials, including Lesson Narratives, Unit Overviews, and Course Guides, reference relevant Grades 6-8 standards. These references identify prerequisite knowledge and explain how current tasks extend earlier understandings. Across Integrated Math 1, Integrated Math 2, and Integrated Math 3, the structure connects prior learning to new content rather than presenting high school standards independently. 

Examples include:

• Integrated Math 1, Unit 9, Lesson 20, Curated Practice Problem Set, Problem 4, students apply properties of operations and integer exponent rules (7.EE.A, 8.EE.A) to generate equivalent numerical expressions when working with complex expressions arising from exponential functions (F-LE.1c). Student Task Statement states, “The function h is given by h(x)=5^x. a. Find the quotient \frac{h(x+2)}{h(x)}. b. What does this tell you about how the value of h changes when the input is increased by 2? c. Find the quotient \frac{h(x+3)}{h(x)}. d. What does this tell you about how the value of h changes when the input is increased by 3?”

• Integrated Math 2, Unit 6, Lesson 8, Activity 8.4, students apply their understanding of a rational number as a point on the number line (6.NS.6) when they combine real and imaginary numbers through addition and plot complex numbers in the complex plane (N-CN.1). Student Task Statement states, “1. Label at least 8 different imaginary numbers on the imaginary number line. 2. When we add a real number and an imaginary number, we get a complex number. The diagram shows where 3-2i is in the complex plane. What complex number is represented by point A?” 3. Plot these complex numbers in the complex number plane and label them. a. -2-i; b.-6+3i; c. 5+4i; d. 1-3i.”

• Integrated Math 3, Unit 1, Lesson 13, Warm-up, students apply their understanding of problems involving area, surface area, and volume (7.G.B) and the formulas for the volumes of cones, cylinders, and spheres (8.G.9) when they determine the missing height of a cone and a slanted cylinder using the Pythagorean Theorem (8.G.7) and trigonometric ratio definitions (G-SRT.6). In Activity 13.3, students calculate the volume of the cylindrical core of the Cayan Tower in Dubai and determine what percentage of the building's total volume is occupied by its core (G-GMD.3).

The materials reviewed for IM® Integrated Math v.360 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

Formal assessments include Checkpoints, Check Your Readiness, Mid-Unit Assessments, End-of-Unit Assessments, and Cool-downs as part of the assessment system. Check Your Readiness, Mid-Unit Assessments, End-of-Unit Assessments, and Cool-downs align with course-level content standards.

Examples include: 

• Integrated Math 1, Unit 8, End-of-Unit Assessment, Assessment Teacher Guide answer key specifies the standards addressed for each problem, such as Problem 4, which aligns with F-BF.4a: “The equation m=40g gives the distance in miles, m, that a car can travel using g gallons of gas. a. If the car has gone 140 miles, how much gas was used? b. Write an equation that represents the inverse function: the gallons of gas used as a function of distance in miles.”

• Integrated Math 2, Unit 2, End-of-Unit Assessment, Assessment Teacher Guide answer key specifies the standards addressed for each problem, such as Problem 7, which aligns with G-SRT.5: “Rectangle ABCD is shown with 5 triangles inside it: ABE, ADE, AEF, CDE, and DEF. a. Which of these triangles are similar to triangle AED? Write similarity statements for these triangles. Explain or show your reasoning. b. The length of EF segment is 8 units, and the length of segment ED is 10 units. Find the length of segment FA. Explain or show your reasoning.”

• Integrated Math 3, Unit 7, Mid-Unit Assessment, Assessment Teacher Guide answer key specifies the standards addressed for each problem, such as Problem 5, which aligns with S-IC.3 and S-IC.6: “An advertisement for a lawn service claims that 90% of people will like their new fertilizer treatment! When you read the small print at the bottom of the ad, you see that they asked 20 of their long-term customers whether they liked the new treatment. Why is this advertisement misleading?“

According to the Integrated Math 1, Integrated Math 2, and Integrated Math 3 Course Guides, 9. Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. Some instructional routines are generally associated with certain MPs. For example, the Card Sort routine often asks students to reason abstractly and quantitatively (MP2) and to look for and make use of structure (MP7). The Information Gap routine often requires students to make sense of problems and persevere in solving them (MP1) as well as attend to precision (MP6) in their language as they ask questions of their partner. The Math Talk routine offers opportunities to look for and make use of structure (MP7) and look for and express regularity in repeated reasoning (MP8) as students explain the strategies they use and apply strategies as they develop fluency. The Which Three Go Together? Routine also offers opportunities for attending to precision when describing why something doesn’t belong (MP6). The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs… Since the MPs in action take many forms, a list of learning targets for each MP supports teachers and students in recognizing when engagement with a particular MP is happening.”

According to the Integrated Math 1, Integrated Math 2, and Integrated Math 3 Course Guides, 9. Standards for Mathematical Practice, How to Use the Mathematical Practices Chart, “No single task is sufficient for assessing students’ engagement with the Standards for Mathematical Practice. Consider these options for assessing students:

• Provide students with the list of learning targets to self-assess their use of the MPs.

• Assign students to create and maintain a portfolio of work that highlights their progress in using the MPs throughout the course.

• Monitor collaborative work and note students’ engagement with the MPs.

Assess the MPs formatively, because students’ use of them is part of a process for engaging with mathematical content. Since the MPs in action take many forms, a list of learning targets for each MP supports teachers and students in recognizing when engagement with a particular MP is happening.”

Examples include:

• Integrated Math 1, “MP3 I Can Construct Viable Arguments and Critique the Reasoning of Others. I can recognize the information that will contribute to an argument for a problem. I can explain my reasoning for why something is true. I can listen to and read the work of others and offer feedback to help clarify or improve their work. I can make conjectures and build a logical argument that supports a conclusion.”

• Integrated Math 2, “MP5 I Can Use Appropriate Tools Strategically. I can select a tool that will help me make sense of a problem. These tools might include a protractor, a ruler, tiles, patty paper, a spreadsheet, a computer algebra system, dynamic geometry software, a calculator, a graph, a table, or external resources. I can use or experiment with tools to help explain my thinking. I can recognize when a tool is producing an unexpected result. I know how to use a variety of math tools to solve a problem.”

• Integrated Math 3, “MP8 I Can Look for and Express Regularity in Repeated Reasoning. I can identify and describe patterns and regularities. I can notice what changes and what stays the same when performing calculations, examining graphs, or interacting with geometric figures. I can use patterns and regularities to express a general rule.”