High School - Gateway 2

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials include tasks that address conceptual understanding throughout the series and provide opportunities for students to demonstrate conceptual understanding in lesson Warm-ups, Activities, and Cool-downs. The Course Guide, Key Structures in This Course: Principles of IM Curriculum Design, Developing Conceptual Understanding and Fluency, states, “The initial lesson in a unit activates prior knowledge and provides an easy entry point to new concepts, so that students at different levels of both mathematical and English language proficiency engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift toward procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.”

Examples include:

• Algebra 1, Unit 5, Lesson 4, Activity 4.3, students develop conceptual understanding of functions as they connect tables, verbal rules, algebraic expressions, and function notation to describe relationships between quantities. They examine tables to identify structure in input–output relationships and connect verbal rules, algebraic expressions, and function notation to represent how outputs are generated from inputs. By matching descriptions to equations and analyzing the order of operations, students determine how structure influences function behavior. Student Task Statement states, “1. A square that has a side length of 9 cm has an area of 81 cm^2. The relationship between the side length and the area of the square is a function. a. Complete the table with the area for each given side length. Then, write a rule for a function, A, that gives the area of the square in cm² when the side length is s cm. Use function notation. b. What does A(2) represent in this situation? What is its value? c. On the coordinate plane, sketch a graph of this function.” The task provides students with a table of values showing side length (cm) and area (cm^2 ), along with a first-quadrant coordinate grid labeled with side length (cm) on the x-axis and area  (cm^2 ) on the y-axis. (F-IF.2)

• Geometry, Unit 4, Lesson 4, Curated Practice Problem Set, Problem 3, students demonstrate conceptual understanding of the side ratios in 30^\circ-60^\circ-90^\circ right triangles. Student Task Statement states, “Priya says, ‘I know everything about a right triangle with a 30-degree angle and a hypotenuse with length 1 cm. Here, look.’ The other angle is 60 degrees. The leg adjacent to the 30-degree angle is 0.866 cm long. The side opposite the 30-degree angle is 0.5 cm long. Han asks, ‘What would happen if a right triangle with a 30-degree angle has a hypotenuse that is 2 cm instead?’ Help them find the unknown angle measures and side lengths in the new triangle. Explain or show your reasoning.” (G-SRT.6)

• Algebra 2, Unit 2, Lesson 5, Warm-up, students develop conceptual understanding of the relationship among factors, zeros, and the graphical features of polynomial functions. Student Task Statement states, “What do you notice? What do you wonder?” Students are given three polynomial equations written in factored form along with their corresponding graphs. Activity 5.2, Student Task Statement states, “Find all values of x  that make the equation true.” Students are given eight equations written in factored form, such as “1. (x+4)(x+2)(-1)=0.” Teacher questioning reinforces the idea that when a factor equals zero, the entire product equals zero and helps students connect algebraic structure to graphical behavior. (A-APR.3).

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters

Materials address procedural fluency across the series through structured practice and distributed problem sets. Course Guide, Key Structures of This Course, Principles of IM Curriculum Design, Developing Conceptual Understanding and Procedural Fluency states, “Warm-up routines, practice problems, and other built-in activities help students develop procedural fluency, which develops over time… As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift toward procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.”

Materials also incorporate independent work within activities to reinforce procedural application before discussion. Course Guide, Key Structures of This Course, Principles of IM Curriculum Design, Coherent Progression states, “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.” 

Examples include:

• Algebra 1, Unit 5, Lesson 15, Curated Practice Problem Set, students demonstrate procedural fluency by writing inverse functions using algebraic methods. Problem 3 states, “Each equation represents a function. For each, find the inverse function. 1. c=w+3; 2. y=x-2; 3. y=5x; 4. w=\frac{d}{7}.” (F-BF.4a)

• Geometry, Unit 6, Lesson 5, Warm-up, students develop procedural fluency by applying the distributive property to rewrite algebraic expressions. Activity Narrative states, “This Math Talk focuses on strategies for distributing pairs of binomials. It encourages students to think about algebraic strategies they have used in earlier courses and to rely on patterns for multiplying, squaring, and finding differences to mentally solve problems. The strategies elicited here will help students develop fluency and will be helpful later in the lesson when students complete the square to find the center and radius of a circle.” Student Task Statement states, “Distribute each expression mentally. 5(x+3); x(x-3); (x+4)(x+2); (x-5)(x-5).” (A-SSE.2)

• Algebra 2, Unit 3, Lesson 3, Activity 3.3, students demonstrate procedural fluency by rewriting rational expressions and identifying horizontal asymptotes using algebraic structure and contextual information. They match graphs to equations of rational functions and rewrite expressions as needed to determine horizontal asymptotes from both algebraic form and graphical features. Launch states, “Since this activity was designed to be completed without technology, ask students to put away any devices. Display the 4 graphs from the activity for all to see and ask students to identify the horizontal asymptote of each graph. After a brief quiet think time, select students to share what horizontal asymptotes they identified, and record these next to the appropriate graph in the form y= ___. Allow students to reopen their books or devices and continue with the activity. Select students who rewrite the expression in a different form to share during the Activity Synthesis.” Student Task Statement asks “1. Match each function with its graphical representation. a(x)=\frac{4}{x}-1, b(x)=\frac{1}{x}-4, c(x)=\frac{1+4x}{x}, d(x)=\frac{x+4}{x}, e(x)=\frac{1-4x}{x}, f(x)=\frac{4-x}{x}, g(x)=1+\frac{4}{x}, h(x)=\frac{1}{x}+4. 2 Where do you see the horizontal asymptote of the graph in the expressions for the functions?”(A-APR.6)

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Multiple routine and non-routine applications of mathematics are included throughout each course with single- and multi-step application problems embedded within Activities or Cool-downs. Students have opportunities to engage with these applications both with teacher support and independently. Course Guide, Key Structures in This Course, Principles of IM Curriculum Design, Applying Mathematics states, “Students make connections to real-world contexts throughout the materials. Carefully chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Many units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts. Additionally, a set of mathematical modeling prompts provide students with opportunities to engage in authentic, grade-level, appropriate mathematical modeling.”

Examples include:

• Algebra 1, Unit 2, Lesson 17, Activity 17.2, students write equations representing two constraints in a context and solve the resulting system both algebraically and graphically. Student Task Statement states, “A recreation center is offering special prices on its pool passes and gym memberships for the summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. She paid $72. 1. Write a system of equations that represents the relationships between pool passes, gym memberships, and the costs. Be sure to state what each variable represents. 2. Find the price of a pool pass and the price of a gym membership by solving the system algebraically. Explain or show your reasoning. 3. Use graphing technology to graph the equations in the system. Make 1-2 observations about your graphs.” (A-CED.3)

• Geometry, Unit 5, Lesson 15, Curated Practice Problem Set, students use the volume formula for a pyramid to determine its dimensions. Problem 5 states, “A toy company packages modeling clay in the shape of a rectangular prism with dimensions 6 inches by 1 inch by \frac{1}{2} inch. They want to change the shape to a rectangular pyramid that uses the same amount of clay. Determine 2 sets of possible dimensions for the pyramid.” (G-GMD.3)

• Algebra 2, Unit 1, Lesson 11, Activity 11.2, students model a repeated real-world process using a geometric sequence and analyze how the accumulated sum approaches a limiting value. Student Task Statement states, “1. Tyler has a piece of paper and is sharing it with Elena, Clare, and Andre. He cuts the paper to create 4 equal pieces, then hands 1 piece each to the others and keeps 1 for himself. What fraction of the original piece of paper does each person have? 2. Tyler then takes his remaining paper and does it again. He cuts the paper to create 4 equal pieces, then hands 1 piece each to the others and keeps 1 for himself. What fraction of the original piece of paper does each person have now? 3.Tyler then takes his remaining paper and does it again. What fraction of the original piece of paper does each person have now? What happens after more steps of the same process?” Building on Student Thinking states, “Emphasize that Tyler gives away pieces of his original sheet of paper, while the other 3 students take the pieces from Tyler. This will help students recognize that the amount of paper Tyler has is decreasing, while the amount of paper the other students have is increasing. By keeping this in mind, students can understand better why adding Tyler's sequence does not make sense, while adding the sequence representing the other sheets of paper does.” (F-BF.1a)

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within each course in the series as reflected by the standards.

Across the series, lessons include opportunities for conceptual understanding, procedural fluency, and application. Tasks often require students to engage in more than one aspect of rigor within the same lesson.

Examples include:

• Algebra 1, Unit 3, Lesson 4, Activity 4.2, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application, as they create a scatter plot from data representing a real-world scenario, determine a line of best fit, interpret the slope and vertical intercept of the linear model, and use the model to make predictions. Student Task Statement states, “1. Watch the video, and record the weight for the number of oranges in the box. 2. Create a scatter plot of the data. 3. Draw a line through the data that fits the data well. 4. Estimate a value for the slope of the line that you drew. What does the value of the slope represent? 5. Estimate the weight of a box containing 11 oranges. Will this estimate be close to the actual value? Explain your reasoning. 6. Estimate the weight of a box containing 50 oranges. Will this estimate be close to the actual value? Explain your reasoning. 7. Estimate the coordinates for the vertical intercept of the line you drew. What might the -coordinate for this point represent? 8. Which point(s) are best fit by your linear model? How did you decide? 9. Which point(s) fit the least well with your linear model? How did you decide?” (S-ID.7)

• Geometry, Unit 6, Lesson 7, Activity 7.3, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application, as they use distance calculations to determine whether specific points satisfy the definition of a parabola and generalize a method for testing any point. Students apply the definition of a parabola as the set of points equidistant from a focus and a directrix. They calculate distances from given points to the focus and to the directrix using coordinate differences and the Pythagorean Theorem. Students determine whether specific points satisfy the definition of a parabola by comparing these distances. They analyze how the position of the focus and directrix affects the direction the parabola opens and describe how the vertex relates to these features. Students examine diagrams, verify whether points lie on the parabola using distance relationships, and generalize a method for determining whether any point in the coordinate plane satisfies the definition. Student Task Statement states, “The image shows a parabola with focus (6, 4) and directrix y=0 (the x-axis). 1. The point (11, 5) looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning. 2. The point (14, 10) looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning. 3. In general, how can you determine if a particular point (x, y) is on the parabola?” (G-GPE.4)

• Algebra 2, Unit 5, Lesson 5, Activity 5.2, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application, as they model exponential depreciation and periodic moon data using functions, interpret parameters in context, perform calculations to generate and refine equations, and use their models to make and evaluate predictions. Student Task Statement states, “After purchase, the value of a machine depreciates exponentially. The table shows its value as a function of years since purchase. 1. The value of the machine in dollars is a function f of time t, the number of years since the machine was purchased. Find an equation defining f , and be prepared to explain your reasoning. 2. Find the value of the machine when t is 0.5 and 1.5. Record the values in the table. 3. Observe the values in the table. By what factor did the value of the machine change: a. every one year, say from 1 year to 2 years, or from 0.5 year to 1.5 years? b. every half a year, say from 0 years to 0.5 year, or from 1.5 years to 2 years? 4. Suppose we know f(q), the value of the machine q years since purchase. Explain how we could use f(q) to find f(q + 0.5), the value of the machine half a year after that point.” (F-LE.1a, F-LE.1c)

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, for students, in connection to the high school content standards, as required by the mathematical practice standards. 

Students engage with MP1 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP1 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students are required to analyze and make sense of problems, find solution pathways, and engage in problem solving. They persevere in solving problems by monitoring and evaluating their progress, determining if their answers make sense, reflecting on and revising their problem-solving strategies, and checking their answers with different methods. Teacher materials guide teachers to pose rich problems, provide time for students to make sense of problems, and create opportunities for students to engage in problem solving.

Examples include: 

• Algebra 1, Unit 8, Lesson 9, Activity 9.2, students analyze a fully solved example problem to understand each step in the solution strategy, then solve quadratic equations by rewriting them in factored form and applying the zero product property. Activity Narrative states, “In this activity, students solve a variety of quadratic equations by integrating what they learned about rewriting quadratic expressions in factored form and their understanding of the zero product property. They begin by analyzing and explaining the steps in a solution strategy and then applying their observations to solve other equations, both of which require sense making and perseverance (MP1). As students work, look for the equations many students find challenging and those on which errors are commonly made. Discuss these challenges and errors during the Activity Synthesis.” Student Task Statement states, “1. To solve the equation n^2-2n=99, Tyler wrote out the following steps. Analyze Tyler’s work. Write down what Tyler did in each step.” The materials include Tyler’s work for steps 1–4. “2. Solve each equation by rewriting it in factored form and using the zero product property. Show your reasoning. a. x^2+8x+15=5, b. x^2-8x+12=5, c. x^2-10x-11=0, d. 49-x^2=04, e. (x+4)(x+5)-30=0.” The Activity Synthesis states, “Consider displaying the solutions for all to see and discussing only the equations that students found challenging and any common errors. The last equation is unlike most equations students have seen. Invite students to share how they solved that equation. Discuss questions such as: ‘Can we use the zero product property to write x+4=0 and x+5=0? Why or why not?’ (No. The zero product property can be applied only to products that equal 0. The expression (x+4)(x+5)-30 has a product for one of its terms, but the expression itself is a difference.) ‘How can it be solved, other than by graphing?’ (We can expand the factored expressions using the distributive property and write an equivalent equation: x^2+9x+20-30=0, or x^2+9x-10=0. This last equation can then be written as (x+10)(x-1)=0, which allows it to be solved.”)

• Geometry, Unit 2, Lesson 12, Activity 12.2, students make specific statements to be proved when given general conjectures about parallelograms, draw a diagram to represent the statement, and then prove it. Lesson Narrative states, “Throughout the lesson students move from making conjectures to making specific statements to be proved, to writing a proof, to critiquing a proof. When students have to determine the given information and draw their own diagram, they are making sense of problems (MP1).” Student Task Statement states, “Here are some conjectures: All rectangles are parallelograms. If a parallelogram has (at least) one right angle, then it is a rectangle. If a quadrilateral has 2 pairs of opposite sides that are congruent, then it is a parallelogram. If the diagonals of a quadrilateral both bisect each other, then the quadrilateral is a parallelogram. If the diagonals of a quadrilateral both bisect each other and they are perpendicular, then the quadrilateral is a rhombus. 1. Pick one conjecture, and use the strips to convince yourself it is true. 2. Rewrite the conjecture to identify the given information and the statement to prove. 3. Draw a diagram of the situation. Mark the given information and any information you can figure out for sure. 4. Write a rough draft of how you might prove your conjecture is true.” Activity Synthesis states, “Focus discussion on how students went from the broad statements of the conjectures to specific statements and diagrams that showed what was given and what had to be proved. Invite students to share their thinking. Then ask other students if the statement matches the given information and defines what we want to prove. Support the class to refine one another’s less precise statements until the conjecture is written as a precise statement.”

• Algebra 2, Unit 5, Lesson 6, Activity 6.2, students engage in an Information Gap routine to write an exponential equation using information from a graph held by a partner. Activity Narrative states, “The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1).” Problem Card 1 states, “Here is a graph of an exponential function f (graph of an exponential growth function is given with an x-axis scale and labeled points A, B, C with no coordinates provided). FInd an equation defining f.” Data Card 1 states, “The y-intercept is 200. The y-coordinate of B is 300. The y-coordinate of C is 1,012.5. The rate of increase from 0 to 1.5 is 237.5%.” Activity Synthesis states, “After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Ask, ‘What information do you need to write an exponential function?’ Also highlight different methods students used to find the growth factor when the input increases by 1.”

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP2 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP2 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students engage in tasks that require them to represent situations symbolically, consider units involved in a problem, attend to the meaning of quantities, understand the relationships between problem scenarios and mathematical representations, explain what numbers or symbols in an expression or equation represent, and determine if their answers make sense. Teachers are guided to support this development by ensuring students make connections between mathematical representations and scenarios, providing opportunities for students to engage in active mathematical discourse, asking clarifying and probing questions, modeling the use of mathematical symbols and notation, supporting students in analyzing quantities and their relationships, and facilitating connections between multiple representations.

Examples include: 

• Algebra 1, Unit 6, Lesson 10, Activity 10.2, students examine data showing the number of coffee shops, determine the average rates of growth for given intervals, and support their choice. Lesson Narrative states, “The purpose of this lesson is for students to revisit the idea of average rate of change and apply it to exponential functions. While rate of change has an unambiguous meaning for linear functions, for nonlinear functions rates of change are not constant, so an interval must be specified. In this lesson, students first are invited to recall how to calculate an average rate of change for a specific interval for a function represented by a data table. By comparing the average rate of change of different intervals that all start at the same point, students can observe that while the average rate of change can describe how the data is changing with reasonable accuracy over some intervals, it is not a good predictor over larger intervals, because exponential functions do not have a constant rate of change. Contexts are used throughout this lesson to give students an opportunity to reason abstractly and quantitatively (MP2).” Student Task Statement states, “Here are a table and a graph that show the number of coffee shops worldwide that a company had in its first 10 years, between 1987 and 1997. The growth in the number of stores was roughly exponential. 1. Find the average rate of change for each period of time. Show your reasoning. a. 1987 and 1990, b. 1987 and 1993, c. 1987 and 1997. 2. Make some observations about the rates of change that you calculated. What do these average rates tell us about how the company was growing during this time period? 3. Use the graph to support your answers to these questions. How well do the average rates of change describe the growth of the company in: a. the first 3 years? b. the first 6 years? c. the entire 10 years? 4. Let be the function so that f(t) represents the number of stores years since 1987. The value of f(20) is 15,011. Find \frac{f(20)-f(10)}{20-10}, and say what it tells us about the change in the number of stores.” Activity Synthesis states, “For the third question, make sure students see that when they calculated the average rate of change for each of the three time periods, they were in effect finding the slope of the line that goes through two points that represent the starting year and the ending year. Display a graph like the one shown here to help illustrate this point: The line that connects the points for 1987 and 1990 fit the data for that period fairly well, so the slope of that line (the average rate of change between those two points) describes the growth in those three years fairly accurately. In contrast, the line that connects the points for 1987 and 1997, does not at all fit the data, so the slope of that line does not paint an accurate picture of how the company was growing that decade. Here are some questions for discussion ‘Would the average rate of change between 1994 and 1997 accurately depict how the company was growing in the last three years in the data?’ ( Yes, because the average rate of change from 1994 to 1997 is 329 shops per year because \frac{1412-425}{10-7}=329 The actual number of shops increased by 252, 338, and then 397 from 1994 to 1997, respectively.) ‘Would the average rate of change between any 2 years accurately reflect the growth for all of 1987 to 1997?’ (No, because the number of shops is increasing at an increasing rate, an average of two values is not enough to summarize the growth of the company.)”

• Geometry, Unit 7, Lesson 11, Activity 11.2, students create a diagram to represent walking around a circular pond and use arc length to calculate the distance. Activity Narrative states, “As students create a diagram to represent a situation, perform calculations, and interpret the meaning of the results in terms of the context, they engage in quantitative and abstract reasoning (MP2).” Student Task Statement states, “There is a circular path around a pond with a fountain in the center. The path is 2 miles long. When Tyler enters the path, he is east of the fountain. He starts walking counterclockwise. 1. A sign says Tyler has walked half a mile. Sketch an image that shows the path, the fountain, the arc Tyler has walked, and the central angle defined by that arc. 2. Tyler is now northwest of the fountain. a. What is the total distance Tyler has walked? b. What is the central angle defined by the arc Tyler has walked? 3. There is a duck statue 1\frac{1}{3} miles around the path. What is the central angle defined by the arc Tyler has walked when he reaches the statue?” 

• Algebra 2, Unit 5, Lesson 3, Activity 3.2, students construct an exponential function to model population growth, graph it, and use it to make a prediction. Activity Narrative states, “In this activity, students construct an exponential function to model population growth. Then they interpret the function when evaluated at certain rational numbers. They do so quantitatively, in terms of the population, and abstractly, in terms of an expression with a fractional exponent (MP2). As students work, identify those who express the population of a given year as an exponential expression and those who use a calculator to estimate a numerical value. Both approaches are important: the first, because it gives an opportunity to interpret a fractional power, and the second, because it gives a concrete sense of the population growth.” Student Task Statement states, “In a video game, Jada is building a moon base to support a growing population and to deal with challenges. Jada’s base has a population of 54,500 in the year 2240, and between 2240 and 2270 the population of the base grows exponentially by about 60% per decade. 1. Find the population of Jada’s moon base in 2250 and in 2260 according to this model. 2. The population is a function f of the number of decades d after 2240. Write an equation for f. 3. a. Explain what f(0.5) means in this situation. b. Graph your function using graphing technology, and estimate the value of f(0.5). c. Explain why we can find the value of f(0.5) by multiplying 54,500 by \sqrt{1.6}. Find that value. 4. Based on the model, what was the population of Jada’s base in 2258? Show your reasoning.” 

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP3 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP3 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

Examples include: 

• Algebra 1, Unit 8, Lesson 13, Activity 13.2, students solve quadratic equations by completing the square, then examine worked-out solutions to the same equations and identify errors. Activity Narrative states, “In this activity, students analyze worked examples of equations solved by completing the square. The work allows them to further develop their understanding of the method when used with rational numbers and to notice common errors. Identifying errors in worked examples is an opportunity to attend to precision (MP6) and to analyze and critique the reasoning of others (MP3).” 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=124; 2. x^2-10x+16=0; 3. x^2+2.4x=-0.8; 4. x^2-\frac{6}{5}x+\frac{1}{5}=0. (The materials provide worked-out solutions for all four quadratic equations that include errors.)“ Activity Synthesis states, “Display the four solutions in the Task Statement for all to see. Select students to share the errors they spotted and their proposed corrections. To involve more students in the discussion, after each student presents, consider asking students to classify each error by type (not limited to one type per error) and explain their classification. Display these examples of types of errors. Then, ask if students have any additional categories they would like to add to the list: Careless errors—for example, writing the wrong number or symbol, repeating the same step twice, leaving out a negative sign, or not following directions. Computational errors—for example, adding or subtracting incorrectly or putting a decimal point in the wrong place. Gaps in understanding—for example, misunderstanding of the problem or the rules of algebra, applying an ineffective strategy, or choosing the wrong operation or step. Lack of precision or incomplete communication—for example, missing steps or explanations, incorrect notation, or forgetting labels or parentheses.”

• Geometry, Unit 3, Lesson 8, Activity 8.2, students analyze a faulty proof for why all rectangles are similar. Activity Narrative states, “In this activity, students analyze a faulty proof for why all rectangles are similar. Although it is incorrect, Tyler’s proof gives students a chance to see the structure of a similarity proof. In addition, it gives students a chance to correct the reasoning of others and provides another way to explain why not all rectangles are similar (MP3). Students should recognize many of the moves and justifications from proofs about congruence in a previous unit, and proofs about dilations earlier in this unit.” Student Task Statement states, “Tyler wrote a proof that all rectangles are similar. Make the image Tyler describes in each step in his proof. Which step makes a false assumption? Why is it false? (The materials include a seven-step proof.)” Activity Synthesis states, “Begin the discussion by inviting students to explain what Tyler did well, and what makes a good proof that two figures are similar. (Tyler gave reasons why each of his transformations worked. Tyler used rigid transformations and dilations.) Then, discuss where Tyler went wrong. (When Tyler did the second dilation, it changed the lengths from the first dilation, so the first pair of corresponding sides aren’t congruent anymore.) Ask students what Tyler could have done before he even wrote the proof to make sure that he didn’t waste time proving something that wasn’t true. Help students see that experimenting, looking for examples and counterexamples, and drawing pictures is part of the proof process.”

• Algebra 2, Unit 6, Lesson 6, Activity 6.3, students work with a partner to determine whether a function is even, odd, or neither from its equation. Activity Narrative states, “In this activity, students take turns with a partner to determine if a function whose equation is given is even, odd, or neither. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3). While students may begin the activity by considering particular points or sketching the graph (by hand or using graphing technology), after an initial work period, a discussion will direct students toward verifying that functions are even, odd, or neither, algebraically.” Student Task Statement states, “Take turns with your partner to decide if the function is even, odd, or neither. If it’s your turn, explain to your partner how you decided. If it’s your partner’s turn, listen carefully to their reasons and decide if you agree. If you disagree, discuss your thinking and work to reach an agreement. 1. f(x)=3x^4-2x^2+1 2. g(x)=x^3-x; 3. h(x)=(x^2-1)(x^2-4); 4. j(x)=2^x+2^{-x}; 5. k(x)=(x^3-1)x; 6. m(x)=(x-0.9)x(x+1.1); 7. n(x)=x(x^2-1)(x^2-4); 8. p(x)=(x^2+4)(x^2-3); 9. q(x)=\frac{1}{x}+x; 10. r(x)=\frac{1}{x}-x” Activity Synthesis states, “Much discussion takes place between partners. Invite students to share how they identified a function as even, odd, or neither, focusing on functions m,n,and p. Here are some questions for discussion: ‘What were some ways you handled functions that were neither even nor odd?’ ‘What were some ways you handled functions that were neither even nor odd?’ ‘Is showing that m(2)\neq m(-2) enough to prove m is not even?’ (Yes, because if m is even then m(x)=m(-x) is true for all possible inputs x.)’’Is showing that n(2)=n(-2) enough to prove n is even?’ (No, a function is even only if that is true for all possible inputs, not for just a specific input. n is actually an odd function because n(x)=-n(-x) is true for all possible inputs x.”

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP4: Model with mathematics, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP4 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP4 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students participate in tasks that support key components of MP4. These include engaging in the modeling cycle, applying prior knowledge to new problems, identifying important relationships and mapping relationships with tables, diagrams, graphs, and rules, and drawing conclusions from solutions as they pertain to a situation. Teachers are guided to pose problems connected to previous concepts, provide a variety of real-world contexts, offer meaningful, real-world, authentic tasks, and promote discourse and investigation that could lead to refining and revising models.

Examples include: 

• Algebra 1, Unit 6, Lesson 11, Activity 11.2, students model successive bounce heights with an exponential function and use the model to answer questions in context. Lesson Narrative states, “In this lesson, students study the successive bounce heights of balls, model the relationship between the number of bounces and the bounce heights, and use that model to answer questions about the ball’s bounciness. There are options for how much of the modeling cycle (MP4) students undertake. In one optional activity, students collect data for the bounce heights of different balls, while in two activities the data are provided. In all cases, the data are deliberately ‘messy’ to mirror data that students would gather through experimentation, and students are left to decide what kind of model to use. Though the data are not perfectly exponential or linear, an exponential model fits much better than a linear model does.” Activity Narrative states, “In this activity, students examine the successive heights that a tennis ball reaches after several bounces on a hard surface, and they consider how to model the relationship between the number of bounces and the height of the rebound. To do so, they need to determine the growth factor of successive bounce heights. Because some data is provided here, students engage in only some aspects of mathematical modeling. To engage students in the full modeling cycle that includes data gathering, consider asking students to measure the bounce heights of a ball, as suggested in the next optional activity. Real-world data is often messy, and that is the case for the data provided here. While each successive bounce height is about half of the preceding height, there is variation in the data, with the largest factor being a little more than 0.55 and the smallest a little less than 0.47. Monitor for students who: Decide that an exponential model is not appropriate because the growth factor is different from bounce to bounce. Make a rough approximation for the growth factor, for example, observe that each bounce height, , is about half the preceding bounce height: h=150\cdot(\frac{1}{2})^n. Find and use the growth factor from the first two points: h=150\cdot(\frac{8}{15}^n Take an average of the successive quotients: h=150\cdot(0.53)^n. Use graphing technology to generate a regression equation: h=150.389\cdot(0.527)^n. Have students present in this order to support more precise methods for finding a model function to fit data.” Student Task Statement states, “Here are measurements for the maximum height of a tennis ball after bouncing several times on a concrete surface. 1. Which is more appropriate for modeling the maximum height, h, in centimeters, of the tennis ball after n bounces: a linear function or an exponential function? Use data from the table to support your answer. 2. Regulations say that a tennis ball, dropped on concrete, should rebound to a height between 53% and 58% of the height from which it is dropped. Does the tennis ball here meet this requirement? Explain your reasoning. 3. Write an equation that models the bounce height, h, after n bounces for this tennis ball. 4. About how many bounces will it take before the rebound height of the tennis ball is less than 1 centimeter? Explain your reasoning.”

• Geometry, Unit 7, Lesson 14, Activity 14.2, students model a sector of a circle using a pizza slice and use given pizza diameters and per-slice costs from four vendors to determine which vendor offers the best value. Activity Narrative states, “In this activity, students are building skills that will help them in mathematical modeling (MP4). They formulate a model of a pizza slice as a sector of a circle. They compute unit costs per square inch of pizza to compare the value of several different vendors’ pizza deals. During the Activity Synthesis, students report their conclusions and the reasoning behind them, and they have an opportunity to consider how to quantify variables beyond price and area.” Student Task Statement states, “Elena was researching offers for the upcoming Pizza Palooza festival. She wants to get a good deal on a single slice of pizza. Your teacher will give you cards that show the deals offered by four vendors. Which vendor should Elena choose? Explain or show your reasoning.” Activity Synthesis states, “Invite previously selected students to share their approaches for choosing a vendor. Sequence the discussion of the approach in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see. Connect the different responses to the learning goals by asking questions, such as: ‘The photos didn’t show the actual sizes of the pizza slices—the images were scaled-down versions of the real thing. Why could we use them to gather information about the areas of the slices?’ (Scaling preserves angle measures, so we could use the photos to find the measures of the sectors’ central angles.) ‘Are there any other variables that affect which vendor you would choose? How could you quantify these variables?’ (Students could calculate the volume of the pizza if crust thickness is important to them. They could try to figure out a calorie count if they want the most energy for their money. They could create an expression that assigns relative importance to toppings, size, and crust type.)”

• Algebra 2, Unit 7, Lesson 14, Activity 14.2, students model a point on a spinning windmill and determine what trigonometric function best models the scenario. Activity Narrative states, “As students determine what type of model best describes the situation, adjust the trigonometric function to account for the amplitude, and check their predictions using technology, students are engaging in aspects of mathematical modeling (MP4).” Student Task Statement states, “Suppose a windmill has a radius of 1 meter, and the center of the windmill is at (0,0) on a coordinate grid. 1. Write a function describing the relationship between the height, h, of W and the angle of rotation, \theta. Explain your reasoning. 2. Describe how your function and its graph would change if: a. the windmill blade has a length of 3 meters. b. the windmill blade has a length of 0.5 meter. 3. Test your predictions using graphing technology.”

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP5 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP5 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students participate in tasks that support key components of MP5. These include choosing appropriate tools, using multiple tools to represent information in a situation, creating and using models to represent, and reflecting on whether the results make sense, possibly improving or revising the model. Students are provided opportunities to use technological tools to explore and deepen their understanding of concepts. Teachers are guided to support this development by making a variety of tools available, allowing students to have choice when selecting tools, modeling tools effectively including their benefits and limitations, and encouraging the use of multiple tools for communication, calculation, investigation, and sense-making.

Examples include: 

• Algebra 1, Unit 6, Lesson 19, Activity 19.2, students create linear and exponential models, use graphing technology to compare them over an appropriate domain, and revise their conclusions based on the graphical results. Lesson Narrative states, “This lesson revisits the fact that exponential functions grow more quickly, eventually, than do linear functions. The first activity compares simple interest (linear growth) with compound interest (exponential growth). Students examine tables and graphs and see that the exponential function quickly overtakes the linear function. In the second activity, the exponential function is deliberately chosen to grow slowly over a large domain, making it less clear whether or not the exponential function will overtake the linear function. The second activity provides an opportunity for students to strategically use technology (MP5), whether they make a graph (for which they will need to think carefully about the domain and range) or continue to tabulate explicit values of the two functions (likely with the aid of a calculator for the exponential function).” Activity Narrative states, “In this activity, students compare linear and exponential growth in a context involving simple and compound interest. The initial balances are chosen so that the two options stay pretty close for small values of time. The intersection of their graphs is far enough from 0 that it is not easily noticed with only a few calculations. The graph for simple interest is linear. The graph for compound interest is exponential, but it is relatively flat for small values of time. As the domain values increase, students may notice that the values of the two options get closer and closer, or they may notice from the graph that the gap between the two graphs gets smaller. Look for students who make different choices for the investment option, as the better option would depend on the length of investment, which is unspecified. As in the case of the fish's offers in an earlier lesson, simple interest is better if the investment term is short, but compound interest becomes increasingly more favorable as time passes. Some students may not recognize this until they graph the two functions in the last question. Identify students who decide to change their earlier choice and can defend the change. Ask them to share later.” Student Task Statement states, “A family has $1,000 to invest and is considering two options: investing in government bonds that offer 2% simple interest, or investing in a savings account at a bank, which charges a $20 fee to open an account and pays 2% compound interest. Both options pay interest annually. Here are two tables showing what they would earn in the first couple of years if they do not invest additional amounts or withdraw any money. 1. Bonds: How does the investment grow with simple interest? 2. Savings account: How are the amounts $999.60 and $1,019.59 calculated? 3. For each option, write an equation to represent the relationship between the amount of money and the number of years of investment. 4. Which investment option should the family choose? Use your equations or calculations to support your answer. 5. Use graphing technology to graph the two investment options and show how the money grows in each.”

• Geometry, Unit 7, Lesson 7, Activity 7.2, students create an arbitrary triangle, use angle bisectors and constructions to find the incenter, and construct the triangle’s inscribed circle. The Required Materials section lists the HS Geometry Toolkit. The Course Guide lists the following materials in the HS Geometry Toolkit: index cards used as straightedges, compasses, tracing paper, blank paper, colored pencils, and scissors. Lesson Narrative states, “Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.” Activity Narrative states, “In this activity, students create an arbitrary triangle and use what they know about angle bisectors and constructions to find the incenter. Then they construct segments measuring the distance from the incenter to the sides of the triangle. Finally, students construct the triangle’s inscribed circle, or the circle that is tangent to all three sides of the triangle. Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).” Student Task Statement states, “1. Mark 3 points and connect them with a straightedge to make a large triangle. The triangle should not be equilateral. 2. Construct the incenter of the triangle. 3. Construct the segments that show the distance from the incenter to the sides of the triangle. 4. Construct a circle centered at the incenter using one of the segments you just constructed as a radius. 5. Would it matter which of the three segments you use? Explain your thinking.”

• Algebra 2, Unit 6, Lesson 15, Activity 15.2, students apply a sequence of transformations to determine which function best represents a given data set. Activity Narrative states, “Monitor for groups using different strategies to determine the parameters for their functions. For example, some groups may use technology to quickly try many options for a vertical scale factor while others may first make a quick calculation using a specific point…In the digital version of the activity, students use an applet to graph their functions. The applet allows students to quickly adjust the parameters of their functions. The digital version may be helpful if there are time constraints or if students would benefit from seeing the function changes in a dynamic way. Making digital tools available gives students an opportunity to use appropriate tools strategically (MP5).” Student Task Statement states, “Here is a graph of data showing the temperature of a bottle of water after it has been removed from the refrigerator (scatterplot provided). For the function types assigned by your teacher: 1. Apply a sequence of transformations to your function so that it matches the data as well as possible. 2. How well does your model fit the data? Make adjustments as needed.”

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP6: Attend to precision, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP6 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP6 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students participate in tasks that support key components of MP6. These include using accurate and precise mathematical language, specifying units of measure, stating the meaning of symbols, formulating clear explanations, calculating accurately and efficiently, using and labeling tables, graphs, and other representations appropriately, and introducing and using definitions accurately. Teachers are guided to ensure students know and use clear definitions, model accurate and precise mathematical language, and provide feedback to students on the accurate use of mathematical language.

Examples include: 

• Algebra 1, Unit 5, Lesson 6, Activity 6.2, students use precise mathematical language and notation to describe and interpret key features of functions and their graphs. Lesson Narrative states, “In this lesson, students use the term maximum (or minimum) to talk about the value of a function that is greater than or equal to (or less than or equal to) all the other values. They also use the terms horizontal intercept and vertical intercept to describe the points where a graph crosses the horizontal and vertical axes. Students also interpret graphs more holistically using the terms increasing and decreasing to describe where a function’s values get greater or lesser as the graph is read left to right. Students relate these features of graphs to features of the functions represented. For instance, they look at an interval in which a graph shows a positive slope and interpret that to mean an interval in which the function’s values are increasing. Students also use statements in function notation, such as h(0) and h(t)=0, to talk about key features of a graph. By now, students are familiar with the idea of intercepts. Note that in these materials, the terms “horizontal intercept” and “vertical intercept” are used to refer to intercepts more generally, especially when a function is defined using variables other than x and y. If needed, clarify these terms for students who may be accustomed to using only the terms ‘x-intercept’ and ‘y-intercept.’ Using mathematical terms and notation to describe features of graphs and features of functions calls for attention to precision (MP6).” Activity Narrative states, “In this activity, students examine graphs of functions, identify and describe their key features, and connect these features to the situations represented. These key features include the horizontal and vertical intercepts, maximums and minimums, and intervals in which a function is increasing or decreasing (or where a graph has a positive or a negative slope). Neither the features nor the terms are likely new to students. The idea of intercepts was introduced in middle school and further developed in earlier units. Graphical features such as maximums and minimums have been considered intuitively in various cases. They are simply more precisely defined here. In a later activity, students will distinguish between a maximum of a graph and the maximum of a function.” Student Task Statement states, “A toy rocket and a drone were launched at the same time. Here are the graphs that represent the heights of the two objects as a function of time since they were launched. Height is measured in meters above the ground, and time is measured in seconds since launch. 1. Analyze the graphs and describe—as precisely as you can—what was happening with each object. ​​Your descriptions should be complete and precise enough that someone who is not looking at the graph could visualize how the objects were behaving. 2. Which parts or features of the graphs show important information about each object’s movement? List the features, or mark them on the graphs.”

• Geometry, Unit 1, Lesson 1, Activity 1.2, students construct circles and line segments using a straightedge and compass. Lesson Narrative states, “This lesson establishes the straightedge and compass moves that students will use to perform various constructions. They begin by copying a figure and use that construction to develop an inventory of construction moves. Students build on their previous understanding of a circle as a set of points all equidistant from the center and of a line segment as a set of points on a line with two endpoints. Students attend to precision when they discuss why straightedge and compass moves communicate geometric information consistently, as opposed to eyeballing (MP6).” Activity Narrative states, “The purpose of this activity is for students to explore why straightedge and compass constructions can be used to communicate geometric information precisely and consistently.” Student Task Statement states, “Complete these steps with a straightedge and compass: 1. Draw a point and label it A. 2. Draw a circle centered at point A with a radius of length PQ. 3. Mark a point on the circle and label it B. 4. Draw another circle centered at point B that goes through point A. 5. Draw a line segment between points A and B.” Activity Synthesis states, “The goal of this discussion is to make sure students understand the straightedge and compass moves that will be allowed during activities that involve constructions. Ask students, ‘What makes this construction more precise than the sketch you made in the Warm-up?’ (The compass makes exact circles. The straightedge makes straight lines. The compass keeps the right length for the radius.) Tell students that using these moves guarantees a precise construction. Conversely, eyeballing where a point or segment should go means that there is no guarantee someone will be able to reproduce it accurately.”

• Algebra 2, Unit 6, Lesson 5, Activity 5.2, students work with partners to sort graphs into categories based on common features and refine their mathematical language during the activity synthesis as they develop definitions for even and odd functions. Activity Narrative states, “The purpose of this activity is for students to identify common features in graphs of functions that are the same when reflected across the vertical axis and those that look the same when reflected across both axes. This leads to defining these types of functions as ‘even’ or ‘odd,’ respectively. Monitor for different ways groups choose different categories, but especially for categories that distinguish between graphs of even functions and graphs of odd functions. As students work, encourage them to refine their descriptions of the graphs using more precise language and mathematical terms (MP6).” Student Task Statement states, “Your teacher will give you a set of cards that show graphs (total of eight cards are included in the activity). 1. Sort the cards into categories of your choosing. Be prepared to describe your categories. Pause here for a class discussion. 2. Sort the cards into new categories in a different way. Be prepared to describe your new categories.” Activity Synthesis states, “Direct students’ attention to the reference created using Collect and Display. Ask students to share their categories and how they sorted their graphs. Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between graphs of even functions and graphs of odd functions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. It is possible students will think of graphs of odd functions as ones where a 180^\circ rotation using the origin as the center of rotation results in the same graph. While it is true that this type of rotation appears the same as successive reflections of the graph across both axes, focus the conversation on thinking in terms of reflections since the function notation students will use to describe odd functions, g(x)=-g(-x), algebraically describes reflections. Next, display the graphs of the even functions next to the odd functions. Tell students that functions whose graphs look the same when reflected across the y-axis are called even functions. Functions whose graphs look the same when reflected across both axes are called odd functions.”

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP7: Look for and make use of structure, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP7 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP7 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students participate in tasks that support key components of MP7. These include looking for patterns and making generalizations, explaining the structure of expressions, decomposing complicated problems into simpler parts, and analyzing problems to look for more than one approach. Teachers are guided to provide tasks and problems with patterns, prompt students to look for structure and patterns, prompt students to describe what they see in the structure or pattern, and provide a variety of examples that explicitly focus on patterns and repeated reasoning.

Examples include: 

• Algebra 1, Unit 2, Lesson 17, Activity 17.3, students complete a card sort by grouping systems according to the number of solutions (one solution, infinitely many solutions, or no solution). Activity Narrative states, “In earlier activities, students gained some insights into the structure of equations in systems that have infinitely many solutions and those that have no solutions. In this activity, they apply those insights to sort systems of equations based on the number of solutions (one solution, many solutions, or no solutions). Students could solve each system algebraically or graphically and sort afterward, but given the number of systems to be solved, they will likely find this process to be time consuming. A more productive way would be to look for and make use of the structure in the equations in the systems (MP7), for example, by looking out for equivalent equations, equations with the same slope but different vertical intercepts, variable expressions with the same or opposite coefficients, and so on. As students discuss their thinking in groups, make note of the different ways they use structure to complete the task. Encourage students who are solving individual systems to analyze the features of the equations and see if they could reason about the solutions or gain information about the graphs that way. In this activity, students are analyzing the structure of equations in the systems, so technology is not an appropriate tool.” Student Task Statement states, “Your teacher will give you a set of cards (nine cards in total). Each card contains a system of equations. Sort the systems into three groups based on the number of solutions each system has. Be prepared to explain how you know where each system belongs.” Building on Student Thinking states, “Some students may not know how to begin sorting the cards. Suggest that they try solving 2–3 systems. Ask them to notice if there's a point in the solving process when they realize how many solutions the system has or what the graphs of the two equations would look like. Encourage students to look for similarities in the structure of the equations and to see how the structure might be related to the number of solutions.” Activity Synthesis states, “Invite groups to share their sorting results, and record them. Ask the class if they agree or disagree. If there are disagreements, ask students who disagree to share their reasoning. Display all the systems - sorted into groups - for all to see, and discuss the characteristics of the equations in each group. Ask students questions such as: ‘How can we tell from looking at the equations in cards 2 and 5 that the systems have no solution?’ ‘What about the equations in cards 3, 7, and 8? What features might give us a clue that the systems have many solutions?’ We can reason that all the other systems have one solution by a process of elimination - by noticing that they don’t have the features of systems with many solutions or systems with no solutions.”

• Geometry, Unit 3, Lesson 5, Warm-up, students analyze a triangle with segments connecting midpoints, identify patterns in side lengths and parallel relationships, and form conjectures about the structural relationship between the smaller triangle and the original triangle, including recognizing dilation structure. Activity Narrative states, “The purpose of this Warm-up is to elicit conjectures about lines connecting the midpoints of sides in triangles, which will be useful when students formalize and prove these conjectures in later activities in this lesson. While students may notice and wonder many things about these images, lengths, angles, and parallel lines are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that triangles formed by segments connecting midpoints are dilations of the larger triangle, and have several of the properties of dilations.” Student Task Statement states, “Here’s a triangle ABC with midpoints L, M, and N. What do you notice? What do you Wonder?” Activity Synthesis states, “Ask students to share the things they noticed and wondered. Record and display their responses, without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, ‘Is there anything on this list that you are wondering about now?’ Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement. If the possibility of the segments connecting midpoints being parallel to a side of the triangle does not come up during the conversation, ask students to discuss this idea.”

• Algebra 2, Unit 8, Lesson 7, Warm-up, students compose and decompose areas under a normal curve, use the structure that the total area is one and that the distribution is symmetric about the mean, and explain how those structural relationships determine unknown probabilities. Lesson Narrative states, “The mathematical purpose of the lesson is to find areas under a normal curve and interpret the area as a proportion of values in the interval using different contexts. When students find areas under a normal curve and interpret the proportion of values in certain intervals for a given context, they are looking for and making use of structure (MP7).” Activity Narrative states, “The mathematical purpose of this activity is for students to reason about the area under a normal curve, using what they know about composition and decomposition of areas. If students are using a table of values with z-scores for this lesson, this work will be essential to finding the desired areas in many cases.” Student Task Statement states, “The images show a normal curve with mean of 40 and standard deviation of 2. The area under the curve to the left of 39 is 0.3085. The area under the curve to the left of 43 is 0.9332. Since it is a normal curve, we know that the total area under the curve is 1. Use the given areas to find the areas in question. Explain your reasoning for each. 1. Find the area under the curve to the right of 39. 2. Find the area under the curve between 39 and 43. 3. Find the area under the curve to the left of 40. 4. Find the area under the curve between 39 and 40.” Activity Synthesis states, “The purpose of this discussion is to make sure students can compose or decompose the given areas to find additional areas. For each question, ask a student to explain how they reasoned about their answer. Highlight places where students used the symmetry of the distribution or cases where they could add or subtract known areas to find new areas.”

The materials reviewed for Illustrative Mathematics® v.360 AGA meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, for students, in connection to the high school content standards, as required by the mathematical practice standards.

Students engage with MP8 throughout the series. The Course Guide (Standards for Mathematical Practice) and specific lessons, including Preparation Narratives and Lesson Activity Narratives, reference MP8 for teachers. According to the Course Guide, 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. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

Across the series, students participate in tasks that support key components of MP8. These include looking for shortcuts and general methods when calculations or processes are repeated, describing general formulas, processes, or algorithms, and evaluating the reasonableness of their answers and thinking. Teachers are guided to provide time for students to look for patterns, structure, shortcuts, and generalizations, ask probing questions, provide situations in which students can use a strategy to develop understanding of a concept, and prompt students to make generalizations.

Examples include: 

• Algebra 1, Unit 7, Lesson 12, Activity 12.2, students experiment with changing the coefficient of the squared term and the constant term in a quadratic expression in standard form to see how the graph is impacted. Activity Narrative states, “This activity enables students to see how the coefficient of the squared term and the constant term in a quadratic expression in standard form can be seen on the graph. Students start by graphing y=x^2 using technology. They then experiment with adding positive and negative constant values to x^2and multiplying it by positive and negative coefficients. They generalize their observations afterward. Along the way, students practice looking for and expressing regularity through repeated reasoning (MP8).” Student Task Statement states, “Using graphing technology, graph y=x^2, and then experiment with each of the following changes to the function. Record your observations (include sketches, if helpful). 1. Add different constant terms to x^2 (for example: x^2+5, x^2+10, or x^2-3). 2. Multiply x^2 by different positive coefficients greater than 1 (for example: 3x2 or 7.5x2). 3. Multiply x2 by different negative coefficients less than or equal to -1 (for example: -x^2 or -4x^2). 4. Multiply x^2 by different coefficients between -1 and 1 (for example: \frac{1}{2}x^2 or -0.25x^2).” 

• Geometry, Unit 7, Lesson 8, Activity 8.2, students find the area of a shaded sector and the arc length for three circles. Activity Narrative states, “In this activity, students get experience calculating areas of circle sectors and lengths of arcs. This will help them define general methods for these calculations in a subsequent activity, as they leverage the repeated reasoning here (MP8). In the Activity Synthesis, students observe that sectors can be used to create an informal argument for the area of a circle.” Student Task Statement states, “A sector of a circle is the region enclosed by two radii. For each circle, find the area of the shaded sector and the length of the arc that outlines the sector. All units are centimeters. Give your answers in terms of \pi.” In Activity 8.3, Exercise 1, students use repeated calculations from Activity 8.2 to generalize their findings and develop formulas for the area of a sector and the arc length of a circle. Activity Narrative states, “In this activity, students generalize their earlier experiences with calculating sector areas and arc lengths.” Student Task Statement states, “Mai says, ‘I know how to find the area of a sector or the length of an arc for central angles like 180 degrees or 90 degrees. But I don’t know how to do it for central angles that make up more complicated fractions of the circle.’ 1. In the diagram, the sector’s central angle \theta measures degrees, and the circle’s radius is r units. Use the diagram to tell Mai how to find the area of a sector and the length of an arc for any angle and radius measure.” Activity Synthesis states, “Connect the different responses to the learning goals by asking questions, such as: ‘How are the formula and the verbal description expressing the same process?’ ‘What does the \frac{\theta}{360} represent?’ (It represents the fraction of the circle taken up by the sector.) ‘What does the \pi r^2 represent?’ (That’s the area of the whole circle.) If no student created a formula, invite students to rewrite their method using symbols instead of words.”

• Algebra 2, Unit 1, Lesson 8, Activity 8.2, students perform repeated calculations to complete a table and then write an explicit definition for a sequence. Activity Narrative states, “Building on their thinking from the Warm-up, students work with non-recursive definitions of two different sequences, one geometric and one arithmetic, based on different cuts of a sheet of paper with an 8-by-10 grid. Students express regularity in repeated reasoning (MP8) by using their understanding of how the values of specific terms are calculated before explaining or expressing how the nth term is calculated.” Student Task Statement states, “2. Kiran takes a piece of paper with length 8 inches and width 10 inches and cuts away 1 inch of the width. He keeps repeating this cut. a. Complete the table for the area of Kiran’s paper K(n), in square inches, after n cuts. b. Kiran says the area after 6 cuts, in square inches, is 80-8\cdot6. Explain where the different terms in his expression came from. c. Write a definition for K(n) that is not recursive.”