High School - Gateway 1

The materials reviewed for IM® 9–12 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:

• S-ID.1: Algebra 1, Unit 1, 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: Algebra 1, Unit 8, 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 21.3 Activity, 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. 

• G-CO.13: Geometry, 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.

• F-BF.2: Algebra 2, Unit 1, Lesson 5, Activity 5.3, students identify sequences as arithmetic or geometric and define them recursively using function notation. In Unit 1, Lesson 10, 10.2 Activity, students write arithmetic and geometric equations, recursively and explicitly, to model animal population data. In Unit 1, End-of-Unit Assessment, Problem 2, students identify an explicit formula to represent a sequence defined recursively.

A-SSE.1a: Algebra 2, 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 in centimeters from each corner and then folding up the sides. What is a reasonable domain for in this context? Explain or show your reasoning.”

The materials reviewed for IM® 9–12 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 Algebra 1, Geometry, and Algebra 2, 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:

• Algebra 1, Unit 2, 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) 

• Geometry, Unit 4, 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) 

• Algebra 2, Unit 1, Lesson 9, Activity 9.2, students complete a table representing the fraction of water remaining in a cooler after n people fill their water bottles (compute). They write recursive and explicit geometric sequences to represent the scenario (formulate) and describe a reasonable domain for the function based on the context (interpret). (F-BF.2, F-IF.5)

Examples where students engage in the full modeling process include: 

• Algebra 1, Modeling Prompt 5, 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 (interpret). (N-Q.A, N-Q.2)

• Geometry, Modeling Prompt 8, students are commissioned to design a container that holds 16 fluid ounces. They identify relevant factors, including volume, dimensions, unit conversions, and geometric structure (problem). Students develop mathematical models by selecting or combining three-dimensional figures, applying volume formulas, and defining variables to determine dimensions that meet the volume requirement (formulate/compute). They build physical or digital prototypes, test their designs against the volume constraint, and revise their dimensions as needed (validate). During a gallery walk, students compare solutions and analyze differences in design approaches (compute/interpret/validate). Each student presents a final design to the company (report). (G-GMD.3, G-MG.1, G-MG.3).

• Algebra 2, Modeling Prompt 7, 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® 9–12 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 Algebra 1, Geometry, and Algebra 2, 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 to 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 too:

• Algebra 1, Unit 4, 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)

• Algebra 1, Unit 6, 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)

• Geometry, Unit 3, 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)

• Algebra 2, Unit 4, 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 b12 to b13 and evaluate expressions with fractional exponents. (N-RN.1, N-RN.2)

• Algebra 2, Unit 8, 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® 9–12 Math v.360 meet expectations for materials, 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 all of the non-plus standards include:

• Algebra 1, Unit 8, Lesson 3, Activity 3.2, students solve quadratic equations such as n^2+4=404 by taking square roots or by graphing. Algebra 1, Unit 8, Lesson 9, Activity 9.2, students solve quadratic equations by rewriting expressions in factored form and applying the zero product property. Algebra 1, Unit 8, Lesson 12, Activity 12.3, students solve quadratic equations by rewriting the equation in a form that allows completing the square. Algebra 1, Unit 8, Lesson 13, Activity 13.2, students analyze worked examples of equations solved by completing the square and identify errors in the reasoning. Algebra 1, Unit 8, Lesson 16, Activity 16.3, students apply the quadratic formula to solve quadratic equations. Algebra 2, Unit 4, Lesson 17, Activity 17.3, students solve quadratic equations that have complex solutions by completing the square. Algebra 2, Unit 4, Lesson 18, Cool-down, students determine the number of real and non-real solutions of a given quadratic equation and then solve it using the quadratic formula. (A-REI.4b)

• Algebra 1, Unit 3, 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. Algebra 1, Unit 3, Lesson 7, 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.” Algebra 1, Unit 3, 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)

• Geometry, 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). Geometry, Unit 1, 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. Geometry, Unit 1, 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)

• Geometry, Unit 4, Lesson 9, Activity 9.2, students prove the Pythagorean identity sin^2(\theta)+cos^2(\theta)=1for any acute angle \theta. In Geometry, Unit 4, Lesson 9, 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 Algebra 2, Unit 7, 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)

• Algebra 2, Unit 4, Lesson 10, Activity 10.2, 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. Algebra 2, Unit 4, Lesson 11, Activity 11.4, 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)

The materials reviewed for IM® 9–12 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:

• Algebra 1, Unit 3, 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)

• Geometry, Unit 3, Lesson 17, Activity 17.2, 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)

• Algebra 2, Unit 7, 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:

• Algebra 1, Unit 8, 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)

• Geometry, Unit 5, Lesson 6, 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)

• Algebra 2, Unit 5, Lesson 13, Curated Practice Problem Set, Problem 1, students use a given exponential function to approximate and interpret the population of a growing town over time. Student Task Statement states, ”The population of a town is growing exponentially and can be modeled by the equation f(t)=42\cdot e^{(0.015t)}. The population is measured in thousands, and time is measured in years since 1950. 1. What was the population of the town in 1950? 2. What is the approximate percent increase in the population each year? 3. According to this model, approximately what was the population in 1960?” (F-LE.1c)

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

• Algebra 1, Unit 3, 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).

• Geometry, Unit 2, Lesson 1, Activity 1.3, students extend their work with using transformations to verify congruence (8.G.1) by applying properties of rotation to identify corresponding parts of congruent triangles (G-CO.5).

• Algebra 2, Unit 1, Lesson 5, Warm-up, students apply function notation (8.F.1) to define a sequence recursively (F-BF.2) when given the first five terms of the sequence.

The materials reviewed for IM® 9–12 Math v.360 meet expectations for 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: 

• Algebra 1, Unit 6, 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).

• Geometry, Unit 2, Lesson 7, requires students to prove the Angle-Side-Angle Triangle Congruence Theorem. In Unit 3, 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 Unit 3, Lesson 13, 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).

• Algebra 2, Unit 1, Lesson 2, Activity 2.2, requires students to generate two geometric sequences, represent them using tables and graphs, and describe how the variables change across representations. In Unit 5, Lesson 1, Warm-up, students write the first three terms of arithmetic and geometric sequences given a real-world context involving a bank account, distinguishing between additive and multiplicative growth (F-LE.2).

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

• Algebra 1, Unit 2, Lesson 17, Activity 17.2, requires students to analyze systems of equations with no solutions by representing them graphically as parallel lines and identifying that such lines have the same slope but different y-intercepts (A-CED.3, A-REI.6). In Geometry, Unit 6, Lesson 10, Activity 10.3, students prove that a quadrilateral formed by pairs of lines with equal slopes is a parallelogram (G-GPE.4, G-GPE.5).

• Geometry, Unit 6, Lesson 5, Activity 5.3, requires students to 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 (G-GPE.1). In Algebra 2, Unit 4, Lesson 17, Activity 17.3, students use completing the square to solve quadratic equations with complex solutions (A-REI.4b). Students apply the distributive property and recognize structure in expressions (A-SSE.2) as they rewrite quadratic and circle equations to reveal key features. They rewrite perfect square trinomials as squared binomials and express circle equations in standard form to identify the center and radius (A-SSE.3, G-GPE.1). These tasks require students to connect algebraic manipulation with geometric interpretation by determining circle parameters from equivalent forms of an equation.

• Algebra 1, Unit 1, 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 Algebra 2, Unit 8, Lesson 5, Warm-up, students analyze histograms and distribution shapes to identify characteristics of a normal distribution (S-ID.4).

The materials reviewed for IM® 9–12 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 Algebra 1, Geometry, and Algebra 2, the structure connects prior learning to new content rather than presenting high school standards independently. 

Examples include:

• Algebra 1, Unit 6, 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. 1. Find the quotient \frac{h(x+2)}{h(x)}. 2. What does this tell you about how the value of h changes when the input is increased by 2? 3. Find the quotient \frac{h(x+3)}{h(x)}. 2. What does this tell you about how the value of h changes when the input is increased by 3?”

• Geometry, Unit 3, Lesson 16, Cool-down, students apply their understanding of the Pythagorean Theorem to determine unknown side lengths in right triangles (8.G.7) and extend this reasoning by using triangle similarity criteria, including ratios of corresponding side lengths, to find unknown side lengths (G-SRT.5). Student Task statement states, “Triangle ABC is similar to triangle A’B’C’. Calculate the length of side A’B’. Then check your answer using a different method.” 

• Algebra 2, Unit 4, Lesson 11, Activity 11.4, 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.”

The materials reviewed for IM® 9–12 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: 

• Algebra 1, Unit 5, 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. 1. If the car has gone 140 miles, how much gas was used? 2. Write an equation that represents the inverse function: the gallons of gas used as a function of distance in miles.”

• Geometry, Unit 1, Mid-Unit Assessment, Assessment Teacher Guide answer key specifies the standards addressed for each problem, such as Problem 1, which aligns with G-CO.1: “This diagram is a straightedge and compass construction. C is the center of both circles. Select all statements that must be true by construction. Segments AB and AD have the same length. Segments AC and AD have the same length. Segments AC and CD have the same length. Triangle BCE is isosceles. Triangle CDE is isosceles.”

• Algebra 2, Unit 7, End-of-Unit Assessment, Assessment Teacher Guide answer key specifies the standards addressed for each problem, such as Problem 4, which aligns with F-TF.5: “Here is the graph of a trigonometric function. 1. What are the midline and amplitude of this function? 2. What is the period of the function? 3. Write an equation of a function that has this graph.”

According to the Algebra 1, Geometry, and Algebra 2 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 Algebra 1, Geometry, and Algebra 2 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 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:

• Algebra 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.”

• Geometry, “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 can use a variety of math tools to solve a problem.”

• Algebra 2, “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.”