The materials reviewed for Math Nation AGA meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. Conceptual understanding is mostly developed within Warm-Ups and Exploration Activities throughout the series.

Examples of opportunities for students to independently demonstrate conceptual understanding throughout the series include:

• Algebra 1, Unit 2, Lesson 10, 2.10.2 Exploration Activity, students work with solutions of graphs of equations in order to answer a set of questions. Students may be assigned or choose one or two equations from a list of three equations, each of which represents a relationship between the number of amusement park games and rides a student may participate in within a budgetary constraint. Students answer questions by interpreting the meaning of the equations chosen in order to graph the equation(s) in the coordinate plane and determine the slopes of the lines that represent the equations. (A-REI.10)

• Algebra 1, Unit 6, Lesson 11, 6.11.1 Warm-Up, students find the values of unknown points and explain their reasoning. “Here is a graph of a function w defined by $w(x) = (x + 1.6)(x - 2)$. Three points on the graph are labeled. Find the values of  a, b, c, d, e, and f. Be prepared to explain your reasoning.” Students are provided with a graph of $w(x) = (x + 1.6)(x - 2)$. (A-SSE.A)

• Geometry, Unit 2, Lesson 1, 2.1.4 Exploration Activity, students use their knowledge of corresponding parts in congruent triangles to justify a conjecture of a new shape. “1. Draw a triangle. 2. Find the midpoint of the longest side of your triangle. 3. Rotate your triangle 180° using the midpoint of the longest side as the center of the rotation. 4. Label the corresponding parts and mark what must be congruent. 5. Make a conjecture and justify it. A. What type of quadrilateral have you formed? B. What is the definition of that quadrilateral type? C. Why must the quadrilateral you have fit the definition?” (G-CO.5)

• Geometry, Unit 4, Lesson 1, 4.1.2 Exploration Activity, students use their knowledge of the Pythagorean Theorem to determine if they can find unknown sides length of triangles. “Find the values of x, y, and z. If there is not enough information, what else do you need to know?” Students are provided with three right triangles with various length and angles labeled. (G-SRT.6)

• Algebra 2, Unit 2, Lesson 10, 2.10.2 Exploration Activity, Questions 1 and 2, students graph polynomial functions and identify the end behavior. Question 1, students write the degree, all the zeros, and identify the end behavior of each polynomial (A-F). Students use the information they have written to sketch a graph of the polynomial and check their sketch using graphing technology. Question 2, students create their own polynomial with a degree greater than 2, but less than 8 and write an equation to represent their polynomial.  Students exchange papers with a partner. Then students write the degree, all the zeros, and identify the end behavior of the polynomial created by their partner and sketch a graph. Once they trade papers back they check their partner sketch using graphing technology. (A-APR.3 and F-IF.7c)

• Algebra 2, Unit 4, Lesson 1, 4.1.4 Exploration Activity, students determine growth of algae on a pond over specific time periods, knowing the pond area covered by the algae doubles each day. “On May 12, a fast-growing species of algae was accidentally introduced to a pond in an urban park. The area of the pond that the algae covers doubles each day. If not controlled, the algae will cover the entire surface of the pond, depriving the fish in the pond of oxygen. At the rate it is growing, this will happen on May 24. 1. On which day is the pond halfway covered? 2. On May 18, Clare visits the park. A park caretaker mentions to her that the pond will be completely covered in less than a week. Clare thinks that the caretaker must be mistaken. Why might she find the caretaker's claim hard to believe? 3. What fraction of the area of the pond was covered by the algae initially, on May 12? Explain or show your reasoning.” (F-LE.1b and F-LE.2)

The materials reviewed for Math Nation 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.

Examples of the materials developing procedural skills and students independently demonstrating procedural skills throughout the series include: 

• Algebra 1, Unit 4, Lesson 15, 4.15.6 Practice Problems, Question 3, students find the inverse of functions. “Each equation represents a function. For each, find the inverse function. A. $c=w+3$ B. $y=x-2$ C. $y=5x$ D. $w=\frac{d}{7}$ .” (F-BF.4)   

• Algebra 1, Unit 7, Lesson 16, 7.16.3 Exploration Activity, students use an example of the quadratic formula to verify solutions of six quadratic equations. “Here are some quadratic equations and their solutions. Use the quadratic formula to show that the solutions are correct. 1. $x^2+4x-5=0$. The solutions are $x=-5$ and $x=1$…6. $6x^2+9x-15=0$. The solutions are $x=-\frac{5}{2}$ and $x=1$.” (A-REI.4b) 

• Geometry, Unit 3, Lesson 6, 3.6.6 Practice Problems, Question 2, students find the scale factor of dilations for two quadrilaterals. “Quadrilaterals Q and P are similar. A. What is the scale factor of the dilation that takes P to Q? B. What is the scale factor of the dilation that takes Q to P?” Two quadrilaterals are shown. One is labeled P and has sides 4, 3, 2 and one is labeled Q and has sides 5, 2.5. (G-SRT.1)

• Geometry, Unit 6, Lesson 10, 6.10.3 Exploration Activity, Question 1, students write the equation of a line that is parallel to an equation and passes through a certain point. “Write the equation of a line parallel to $y=2x+3$, passing though -4,1.” (G-GPE.5)

• Algebra 2, Unit 2, Lesson 15, 2.15.4 Exploration Activity, Question 1, student determined which polynomials out of group would have a certain factor. “Which of these polynomials could have $x-2$ as a factor?” The materials list six polynomials (A-F) for students to select. (A-APR.2) 

• Algebra 2, Unit 7, Lesson 12, 7.12.6 Practice Problems, Question 1, students calculate the mean and margin of error of the amount spent yearly on daycare. “Technology required. The mean amount spent on daycare yearly by random samples of 10 families are listed. $7,213 $13,512 $6,543 $8,256 $9,106 $12,649 $10,256 $9,553 $7,698 $10,156 Use the values to estimate the mean amount spent on daycare yearly for the population and provide a margin of error (round to the nearest dollar).” (S-IC.4)

The materials reviewed for Math Nation 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.

Examples of lessons that include multiple opportunities for students to engage in routine applications of mathematics throughout the series include:

• Algebra 1, Unit 5, Lesson 7, 5.7.2 Exploration Activity, students write an equation to represent the structure of a coral over a certain time interval and explain what values means in the context of the situation. “A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year. 1. Write an equation of the form $y = a \cdot b^t$ representing the relationship, where t is time in years since the coral was measured and y is the volume of the sea coral in cubic centimeters. (You need to figure out what  a and b are in this situation.) 2. Find the volume of the coral when t is 5, 1, 0, -1, and -2. 3. What does it mean, in this situation, when t is -2? 4. In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.” (A-CED.2)

• Geometry, Unit 4, Lesson 7, 4.7.4 Exploration Extension, students use trigonometry to calculate the height of a building. “You're sitting on a ledge 300 feet from a building. You have to look up 60 degrees to see the top of the building and down 15 degrees to see the bottom of the building. How tall is the building?”(N-Q.2 and G-SRT.8)

• Algebra 2, Unit 4, Lesson 1, 4.1.2 Exploration Activity, students use an example of a passport photo to look at a situation of real-life exponential decay. “The distance from Elena’s chin to the top of her head is 150 mm in an image. For a U.S. passport photo, this measurement needs to be between 25 mm and 35 mm. 1. Find the height of the image after it has been scaled by 80% the following number of times. Explain or show your reasoning. A. 3 times B. 6 times 2. How many times would the image need to be scaled by 80% for the image to be less than 35 mm? 3. How many times would the image need to be scaled by 80% to be less than 25 mm?” (F-LE.1c and F-LE.2)

Examples of lessons where the materials include multiple opportunities for students to engage in non-routine applications of mathematics throughout the series include:

• Algebra 1, Unit 2, Lesson 1, 2.1.2 Exploration Activity, students work in a group to plan a pizza party. “Imagine your class is having a pizza party. Work with your group to plan what to order and to estimate what the party would cost. 1. Record your group’s plan and cost estimate. What would it take to convince the class to go with your group's plan? Be prepared to explain your reasoning. 2. Write down one or more expressions that show how your group’s cost estimate was calculated. 3. A. In your expression(s), are there quantities that might change on the day of the party? Which ones? B. Rewrite your expression(s), replacing the quantities that might change with letters. Be sure to specify what the letters represent.” (N-Q.2, ACED.2, and A-CED.3)

• Geometry, Unit 5, Lesson 17, 5.17.2 Exploration Activity, students use concepts of volume and unit conversion to enhance their understanding of density. “The feathers in a pillow have a total mass of 59 grams. The pillow is in the shape of a rectangular prism measuring 51 cm by 66 cm by 7 cm. A steel anchor is shaped like a square pyramid. Each side of the base measures 20 cm, and its height is 28 cm. The anchor's mass is 30 kg . 1. What’s the density of feathers in kilograms per cubic meter? 2. What’s the density of steel in kilograms per cubic meter? 3. What’s the volume of 1,000 kg of feathers in cubic meters? 4. What’s the volume of 1,000kg of steel in cubic meters?” (N-Q.1, and G-MG.2)

• Algebra 2, Unit 7, Lesson 16, 7.16.2 Exploration Activity, students collect and summarize data from an experiment and compare their results with another group. Students perform an experiment to determine if counting out loud while exercising affects the heart rate. Students are divided into two groups. One group will count out loud while exercising and the other group will remain silent while exercising. Immediately following the exercise, students in each group will measure and record their heart rates, and find the difference between their heart rates after exercising and their previously recorded resting heart rates. (S-IC.5)

The materials reviewed for Math Nation AGA meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. The three aspects are balanced with respect to the standards addressed.

Examples of where the materials independently engage aspects of rigor include:

• Algebra 1, Unit 2, Lesson 8, 2.8.6 Practice Problems, Question 2, students develop procedural skills and fluency as they solve a linear equation in two variables for each variable. “Here is a linear equation in two variables: $2x+4y-31=123$. Solve the equation, first for x and then for y. (A-CED.4)

• Geometry, Unit 7, Lesson 2, 7.2.4 Exploration Activity, students use their conceptual understanding of circles, congruence, and similarity to prove two inscribed triangles are similar. “The image shows a circle with chords CD, CB, ED, and EB. The highlighted arc from point C to point E measures 100 degrees. The highlighted arc from point D to point B measures 140 degrees. Prove that triangles CFD and EFB are similar.”(G-SRT.5, and G-C.2)

• Algebra 2, Unit 4, Lesson 16, 4.16.1 Warm-Up, students use their understanding of functions and function notation to answer an application of mathematics involving two different accounts. “A business owner opened two different types of investment accounts at the start of the year. The functions f and g represent the values of the two accounts as a function of the number of months after the accounts were opened. 1. Here are some true statements about the investment accounts. What does each statement mean? A. $f(3) > g(3)$ B. $f(6) < g(6)$ C. $f(m) = g(m)$ 2. If the two functions were graphed on the same coordinate plane, what might it look like? Sketch the two functions.” (A-REI.11, and F-IF.2)

Examples of where the materials engage multiple aspects of rigor simultaneously include:

• Algebra 1, Unit 1, Lesson 15, 1.15.4 Exploration Activity, students use their conceptual understanding,procedural skills, and fluency as they work with groups of data sets to determine the best measure of center and variability based on the shape of the distribution. Students are given seven groups of data sets, varying in form (dotplots, box plots, and verbal descriptions). For each group of data sets, students must do the following: “Determine the best measure of center and measure of variability to use based on the shape of the distribution. Determine which set has the greatest measure of center. Determine which set has the greatest measure of variability. Be prepared to explain your reasoning.” (S-ID.2)

• Geometry, Unit 4, Lesson 8, 4.8.6 Practice Problems, Question 3, students engage their conceptual understanding, procedural skills, and fluency within the application of using trigonometric relationship to find the height of a building. “Technology required. Jada is visiting New York City to see the Empire State building. She is 100 feet away when she spots it. To see the top, she has to look up at an angle of 86.1 degrees. How tall is the Empire State building?” (G-SRT.7)

• Algebra 2, Unit 5, Lesson 7, 5.71 Warm-Up, students use their conceptual understanding, procedural skills, and fluency to complete a table that translates a given function into words, function notation, and expression. “Let  $g(x)=\sqrt{x}$. Complete the table. Be prepared to explain your reasoning.” The columns of the table are “words (the graph of $y=g(x$) is…)”, “function notation”, and “expression” respectively. An example of a row: “translated left 5 units”, "$g(x+5)$”, “___” respectively. (F-BF.3)

The materials reviewed for Math Nation AGA meet expectations for supporting the intentional development of overarching, mathematical practice (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

The Teacher Guide for each lesson contains a Lesson Narrative. The Standards for Mathematical Practices that are connected to each lesson are identified within each of the Lesson Narratives and within some of the individual lesson activities.  

Examples of lessons where MP1 and/or MP6 are intentionally developed to connect to course-level content across the series include:

• Algebra 1, Unit 1, Lesson 13, 1.13.2 Exploration Activity, students work in pairs to determine the information needed to answer questions about African and Asian elephant populations. Students use Problem Cards and Data Cards to discuss the information needed to investigate and interpret measures of center and variability related to the elephant populations. Students make sense of the problem and persevere to answer the questions via discussions with their partners. Through effective communication with their partners, students determine the appropriate and precise measures of center and variability needed to answer the questions found on the problem cards based on the information found on the data cards. (MP1 and MP6)

• Geometry, Unit 6, Lesson 12, 6.12.2 Exploration Activity, students work in pairs to discuss the information needed to graph and write equations of lines. Each student is given a Problem Card or a Data Card. Students who receive the Problem Card need to make sense of the problem and ask the student who received the Data Card for the information needed to solve the problem. After the student with the Problem Card has obtained all the information needed to solve the problem, the Problem Card is shared with the student who received the Data Card. Then each student solves the problem independently. Students have opportunities to use precise mathematical language as they communicate with their partners the information needed to solve the problem and discuss the reasoning to support the solutions they have found. (MP1 and MP6)

• Geometry, Unit 7, Lesson 2, 7.2.2 Exploration Activity, students develop the relationship between an inscribed angle (QBC) and its corresponding central angle (QAC). Students have access to an applet in which they can move each point (A, B, C, and Q) around the circle to determine what happens to the measures of angles (QBC) and (QAC). Students make sense of the problem by experimenting with the different inscribed and corresponding central angles that are formed and make a conjecture about the relationship of the measures between an inscribed angle and its corresponding central angle. Students attend to precision as they record the measure of the angles QAC and QBC as the movement of points occurs. (MP1 and MP6)

• Algebra 2, Unit 1, Lesson 3, 1.3.2 Exploration Activity, students are given the first five terms of three sequences. Students have opportunities to use precise mathematical language when describing ways to produce the next term by using the previous term, determining which sequence has the second greatest value for the 10th term, and determining which of the three sequences could be a geometric sequence. (MP6)

• Algebra 2, Unit 4, Lesson 8, 4.8.2 Exploration Activity, Questions 1 and 2, students are given a sequence showing a trapezoid being successively decomposed into four similar trapezoids at each step. In Question 1, students determine the relationship between the step number (n) and the number of smallest trapezoids formed at each step. In Question 2, students need to find the value of the step number, when 262,144 small trapezoids are formed. For Part A, students write an equation to represent the relationship between the step number (n) and the number of small trapezoids formed. For Part B, students explain to a partner how they might find the value of that step number. Students make sense of this problem by determining the relationship between the step number and the number of trapezoids formed, writing an equation to represent the relationship, and reasoning how to find the value of the step number when a specific number of trapezoids are formed. (MP1 and MP6)

The materials reviewed for Math Nation AGA meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

The Teacher Guide for each lesson contains a Lesson Narrative. The Standards for Mathematical Practices that are connected to each lesson are identified within each of the Lesson Narratives and within some of the individual lesson activities.  

Examples of MP2 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:

• Algebra 1, Unit 2,  Lesson 2, 2.2.7 Practice Problems, Question 4, students practice writing equations to represent the relationship between distance in miles and speed in miles per hour for a problem that involves a biking exercise. In Parts A and B, students use specific speeds and times to write equations that can be used to calculate biking distances. In Part C, students write an equation using variables in place of specific number quantities to represent the miles per hour. Students make sense of quantities by attending to the meaning of the quantities and considering the units involved in the problem situation. (MP2)

• Geometry, Unit 4, Lesson 9, 4.9.3 Exploration Activity, students need to reason quantitatively and abstractly as they use side lengths and arctan, arccos, and arcsin to find angles in a problem involving leaning a ladder against a wall. Students measure their own physical measurements and translate those in relation to determining the angle at which a ladder can be placed safely against a wall. (MP2)

• Algebra 2, Unit 1, Lesson 9, 1.9.2 Exploration Activity, Questions 1 and 2, students reason with quantities as they determine how much cake is left after people take slices and write a recursive and non-recursive function in context. “A large cake is in a room. The first person who comes in takes $\frac{1}{3}$ of the cake. Then a second person takes $\frac{1}{3}$ of what is left. Then a third person takes $\frac{1}{3}$ of what is left. And so on.”  In Question 1, students complete a table to represent the fraction of the original cake left after n people take some. In Question 2, students attend to the meaning of the quantities to determine and write the recursive and non-recursive functions that represent the problem. (MP2)

Examples of MP3 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:

• Algebra 1, Unit 2, Lesson 16, 2.16.2 Exploration Activity, students critique a student’s initial process for solving a system of equations by determining the steps she used to begin the process and the possible reasons for those steps. Then students complete the process of solving the system of equations algebraically to show the complete solution set for the system. (MP3)

• Geometry, Unit 4, Lesson 6, 4.6.6 Practice Problems, Question 3, using their knowledge about right triangles students construct arguments for whether they agree with either of the scenarios that the students in the problem are suggesting, and explain their reasoning.  “Andre and Clare are discussing triangle ABC that has a right angle at C and hypotenuse of length 15 units. Andre thinks the right triangle could possibly have legs that are 9 and 12 units long. Clare thinks that angle B could be 20 degrees and then side BC would be 14.1 units long. Do you agree with either of them? Explain or show your reasoning.” (MP3)

• Algebra 2, Unit 2, Lesson 23, 2.23.2 Exploration Activity, Question 1, students critique the reasoning of a student who thinks the difference between the squares of two consecutive integers will always be the sum of the two integers. Students determine if this assumption is correct or incorrect by explaining or showing their reasoning. (MP3)

The materials reviewed for Math Nation AGA meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

The Teacher Guide for each lesson contains a Lesson Narrative. The Standards for Mathematical Practice that are connected to each lesson are identified within each of the Lesson Narratives and within some of the individual lesson activities.

Examples of where and how there is intentional development of MP4 and/or MP5 to enrich the mathematical content within the materials include:

• Algebra 1, Unit 2, Lesson 4, 2.4.2 Exploration Activity, Questions 1 and 2, students write equations that represent a person’s weekend earnings in relation to their hourly earning rate and the amount spent on bus fare. “Jada has time on the weekends to earn some money. A local bookstore is looking for someone to help sort books and will pay $12.20 an hour. To get to and from the bookstore on a work day, however, Jada would have to spend $7.15 on bus fare. 1. Write an equation that represents Jada’s take-home earnings in dollars, E, if she works at the bookstore for h hours in one day. 2. One day, Jada takes home $90.45 after working h hours and after paying the bus fare. Write an equation to represent this situation.” (MP4)

• Algebra 1, Unit 5, Lesson 11, 5.11.2 Exploration Activity, Questions 1-4, students model the relationship between the number of bounces and the height of the rebound of a tennis ball, while selecting appropriate tools to aid in representing the model. In Question 1, students use a table that contains the maximum heights of a tennis ball after bouncing several times on a concrete surface to determine which type of function, linear or exponential, is more appropriate for modeling the relationship shown in the table (MP4 and MP5). In Question 2, students use regulations that say a tennis ball, dropped on concrete, should reach a height between 53% and 58% of the height from which it is dropped to determine if the tennis ball in the problem meets this requirement. Students have to use appropriate tools to calculate the percentage of the heights reached after successive bounces of the ball, as shown in the table and explain their reasoning (MP5). In Question 3, students write an equation that models the bounce height (h) after n bounces of the tennis ball (MP4). In Question 4, students approximate how many bounces it will take before the rebound height of the tennis ball is less than 1 centimeter. Students have opportunities to answer this question by using appropriate tools such as the table, the equation they wrote to represent the relationship between the number of bounces and the ball heights, and/or a graph that represents the relationship (MP5).

• Geometry, Unit 1, Lesson 9, 1.9.2 Exploration Activity, Questions 1-4, students are given the diagram of a square city that contains points that represent the locations of three stores.  Students select appropriate tools from an applet containing geometry software to solve problems involving partitioning the city diagram into regions, each of which contains one of the three store locations. In Question 1, students use the applet to partition the city diagram into regions so that whenever someone orders from an address, their order is sent to the store closest to their home. Using the applet, students are able to construct perpendicular bisectors to partition the city into the desired regions (MP4 and MP5). In Question 2,  students use the area tool to help determine what percentage of the diagram is covered by each region. Students use the percentages to determine how 100 store employees should be distributed among the three store locations (MP4 and MP5). In Question 3, students use the diagram they have created to determine the point equidistant from all three store locations (MP5). In Question 4, a fourth store is added to the original diagram. Students use the applet to partition the city again, this time into four regions, each containing one of the store locations (MP4 and MP5).

• Algebra 2, Unit 6, Lesson 19, 6.19.2 Exploration Activity, Questions 1-4, students apply trigonometric functions to model moon data from January 2018, making predictions about the amplitude, midline, period, and horizontal translation for a trigonometric model. “The data from the warm-up is the amount of the Moon that is visible from a particular location on Earth at midnight for each day in January 2018. A value of 1 represents a full moon in which all of the illuminated portion of the moon's face is visible. A value of 0.25 means one fourth of the illuminated portion of the moon's face is visible. 1. What is an appropriate midline for modeling the Moon data? What about the amplitude? Explain your reasoning. 2. What is an appropriate period for modeling the Moon data? Explain your reasoning. 3. Choose a sine or cosine function to model the data. What is the horizontal translation for your choice of function? 4. Propose a function to model the Moon data. Explain the meaning of each parameter in your model and specify units for the input and output of your function.” (MP4) 

• Algebra 2, Unit 7, Lesson 5, 7.5.2 Exploration Activity, students use hand-span data to determine how many students in the class have a hand-span wide enough to reach two notes that are nine keys apart on a piano keyboard, using only one hand. Students use appropriate tools strategically to complete this activity, as they collect, represent, and interpret data in order to answer the question of how many students can reach two notes nine keys apart. (MP5)

The materials reviewed for Math Nation AGA meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

The Teacher Guide for each lesson contains a Lesson Narrative. The Standards for Mathematical Practice that are connected to each lesson are identified within each of the Lesson Narratives and within some of the individual lesson activities.

Examples of where and how there is intentional development of MPs 7 and/or 8 to enrich the mathematical content within the materials include:

• Algebra 1, Unit 3, Lesson 3, 3.3.2 Exploration Activity, Question 1, students complete and analyze a two-way relative frequency table to determine if there is a possible association between coral health and the levels of particular chemical concentrations. (MP7)

• Algebra 1, Unit 6, Lesson 7, 6.7.5 Practice Problems, Question 4, students make an equation to represent the relationship in a pattern and explain the relationship. In Part A, students look for structure in a given pattern of squares to determine the relationship between the step number (n) and the number of small squares (y) and write an equation to represent that relationship. Students describe how each part of the equation relates to the pattern  In Part B, students determine if the relationship is quadratic, and if so, explain how they know.In this problem students use repeated reasoning to explore and understand the pattern in the sequence. (MP7 and MP8)

• Geometry, Unit 7, Lesson 8, 7.8.2 Exploration Activity, students make use of structure as they combine fraction operations with the area and circumference formulas to find areas of shaded sectors and lengths of arcs that outline the sectors. Students are provided an image of three circles with part of each circle shaded with a degree and radii provided. (MP7)

• Geometry, Unit 8, Lesson 8, 8.8.2 Exploration Activity, students use repeated reasoning as they work with different scenarios to determine the conditional probability of events linked to a standard deck of cards. (MP8)

• Algebra 2, Unit 5, Lesson 5, 5.5.2 Exploration Activity, students are given cards which display different types of graphs. Students make use of repeated reasoning as they sort the cards into two categories of their choosing and explain the meaning of their categories. (MP8)

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

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Course Overview: A Course Overview (Unit 0) is found at the beginning of each course.  Within each Course Overview, there is a Course Narrative, which contains a summary of the mathematical content contained in each course. The Course Overview contains two main components: Course Guide and Modeling Prompts. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Materials, and Cool Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the “About These Materials” section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool Down. The “How to Use These Materials” section contains guidance about utilizing instructional routines and digital routines, which include applications of technology, and in the Scope and Sequence section, teachers will find a Pacing Guide, which contains time estimates for coverage of each of the units. The Modeling Prompts component is divided into two sections, Modeling Prompts- Overview and Mathematical Modeling Prompts. Within these sections, teachers can find guidance on introducing the modeling cycle to students, selection of modeling prompts related to units, and lessons contained in the course.

• Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessment (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Teacher Guide, Assessments, Unit at a Glance, Blackline Masters, Lesson Plans, and Teacher Presentation Materials, all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Supports, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Teacher Guide downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.

• Teacher Guide: Within each Teacher Edition lesson component, teachers can find a Teacher Guide that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.

Evidence for materials including sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives to engage and guide student mathematical learning include:

• Algebra 1, Unit 5, Lesson 11, Teacher Guide, 5.11.2 Exploration Activity, “In this activity, students examine the successive heights that a tennis ball reaches after several bounces on a hard surface and 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…Encourage those who do not think an exponential model is appropriate to look for an exponential model that best fits the given data. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).” 

• Geometry, Unit 2, Lesson 6, Teacher Guide, 2.6.2 Exploration Activity, “Activity Synthesis The goal of this discussion is to continue to emphasize that proofs using transformations are generalized statements that work for all triangles that match the given criteria, rather than just one specific drawing. Select students whose triangles require translation and rotation but not reflection to share their drawings and the steps in their transformations. Record a proof these triangles are congruent…Point out the concluding statement. Add this to the display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. You will add to the display throughout the unit. An example template is provided in the blackline masters for this lesson.”

• Algebra 2, Unit 3, Lesson 6, Teacher Guide, 3.6.3 Exploration Activity, “Anticipated Misconceptions, Since students usually see x-values on the horizontal axis and y-values on the vertical, they may look for a or s values on the wrong axis. Encourage students to annotate the graph by drawing horizontal or vertical lines that will intersect the curves at the point that represents the solution, or using some other method that is helpful for them. For example, when solving $s^2=25$, students can draw a line representing $t=25$ and see where it hits the graph of $t=s^2$, since these points represent the solutions.”

The materials reviewed for Math Nation AGA meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Course Guide, About These Materials sections, states the following note about standards alignment, “There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as ‘building on.’ When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as ‘building towards.’ When a task is focused on the grade-level work, the alignment is indicated as ‘addressing.’” All lessons in the materials have this correlation information. Examples include:

• Algebra 1, Unit 3, Lesson 5, Teacher Guide, Alignments, Building on 8.SP.2; Addressing S-ID.6, S-ID.7; Building Towards S-ID.6, S-ID.6b.

• Geometry, Unit 5, Lesson 10, Teacher Guide, Alignments, Building on G-GMD.4; Addressing G-GMD.1, G-GMD.4, G-MG.1; Building Towards G-GMD.1.

• Algebra 2, Unit 6, Lesson 9, Teacher Guide, Alignments, Building on F-BF.1b; Addressing F-IF.7, F-TF.A; Building Towards F-IF.7.

Explanations of the role of the specific course-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative, and Full Lesson Plan. Examples include:

• Algebra 1, Unit 2, Lesson 6, Lesson Narrative, “In middle school, students learned that two expressions are equivalent if they have the same value for all values of the variables in the expressions. They wrote equivalent expressions by applying properties of operations, combining like terms, or rewriting parts of an expression. In this lesson, students learn that equivalent equations are equations with the exact same solutions…The emphasis of this lesson is on equations in one variable. Students will have many opportunities to study equivalent equations in two variables in future lessons.”

• Geometry, Unit 3, Unit Introduction, “Before starting this unit, students are familiar with dilations and similarity from work in grade 8. They have experimentally confirmed properties of dilations, and informally justified that figures are similar by finding a sequence of rigid motions and dilations that takes one figure onto the other…In a previous unit, students used rigid transformations to justify the triangle congruence theorems of Euclidean geometry: Side-Side-Side Triangle Congruence Theorem, Side-Angle-Side Triangle Congruence Theorem, and Angle-Side-Angle Triangle Congruence Theorem. In this unit, students use dilations and rigid transformations to justify triangles that are similar... This unit previews many of the important concepts that students rely on to make sense of trigonometry in later units. The latter part of the unit focuses on similar right triangles. In addition, students are introduced to some of the applications of right triangles that they will explore in more depth in the trigonometry unit, such as finding the heights of objects through indirect measurement.”

• Algebra 2, Unit 5, Unit Introduction, “Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. The purpose of this unit is for students to consider functions as a whole and understand how they can be transformed to fit the needs of a situation, which is an aspect of modeling with mathematics (MP4). An important takeaway of the unit is that we can transform functions in a predictable manner using translations, reflections, scale factors, and by combining multiple functions…The unit begins with students informally describing transformations of graphs, eliciting their prior knowledge and establishing language that will be refined throughout the unit…In a future unit, students use their knowledge of transformations to transform trigonometric functions to model a variety of periodic situations. By saving the introduction of trigonometric functions until after a study of transformations, students have the opportunity to revisit transformations from a new perspective which reinforces the idea that all functions, even periodic ones, behave the same way with respect to translations, reflections, and scale factors.”

The materials reviewed for Math Nation AGA meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials as well as the Mathematical Modeling Prompts- Overview sections. 

The About These Materials section states the following about the instructional approach of the program, “What is a Problem Based Curriculum? In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student's education. Families, students, teachers, and administrators may need support making this shift. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.”

Examples of materials including and referencing research-based strategies include:

• “The Five Practices Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem…”

• “Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”

• “Mathematical Modeling Prompts - Overview Mathematics is a tool for understanding the world better and making decisions. School mathematics instruction often neglects giving students opportunities to understand this, and reduces mathematics to disconnected rules for moving symbols around on paper. Mathematical modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions (NGA 2010). This mathematics will remain important beyond high school in students' lives and education after high school (NCEE 2013).”

Within the Modeling Prompts, Mathematical Modeling Prompts - Overview, a References section is included.

The materials reviewed for Math Nation AGA meet expectations for providing a comprehensive list of supplies needed to support instructional activities. Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Material(s)), Teacher Editions, in each lesson under Lesson Preparation (Required Materials), and in Teacher Guides for specific lessons. Examples include:

• Algebra 1, Unit 4, Lesson 16, Lesson Preparation, Required Material(s): “Pre-printed slips, cuts from copies of the blackline master”

• Geometry, Unit 3, Lesson 1, Lesson Preparation, Required Material(s): “Geometry toolkits (HS), Protractors, Rulers marked with centimeters”

• Algebra 2, Unit 2, Lesson 1, Lesson Preparation, Required Material(s): “Blank paper, Graph technology, Rulers, Scissors, Tape”

The materials reviewed for Math Nation AGA meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses. 

Suggestions to teachers for following-up with students are provided throughout the materials via the Check-Your-Readiness, Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.

Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:

• Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses. 

  • Restricted Constructed Response

    • Tier 1 response: Work is complete and correct.

    • Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.

    • Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.

  • Extended Response

    • Tier 1 response: Work is complete and correct, with complete explanation or justification. 

    • Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. 

    • Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. 

    • Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”

• Algebra 2, Unit 7, End-of-Unit Assessment, Question 6, “A scientist is worried about a new disease being spread by mosquitoes in an area. The scientist captures a random sample of 137 mosquitoes from the area and tests them to find out whether they carry the disease. In this sample, 22 of the mosquitoes were carrying the disease. Using this information, the scientist uses 100 simulations of additional samples with the same proportion of disease-carrying mosquitoes from the sample to determine that the standard deviation for the proportion of mosquitoes carrying the disease is approximately 0.029. 1. Estimate the proportion of mosquitoes in the area that carry the disease, and provide a margin of error. Explain or show your reasoning. 2. Why did the scientist run the simulations to find additional possible proportions of mosquitoes in the area that carry the disease? Solution 1. The proportion of mosquitoes in the area that carry the disease is approximately $0.16(\frac{22}{137}\approx 0.16)$ with a margin of error of $0.058(0.029 \cdot 2=0.058)$. 2. Sample: With the additional simulated proportions, a sense of the variability expected among samples from the population can be used to estimate a margin of error for the sample proportion. Minimal Tier 1 response: Work is complete and correct. Sample: 1. $\frac{22}{137}=.16$ The margin of error is $2 \cdot .029=0.058$. 2. They needed to know the standard deviation to get the margin of error. Tier 2 response: Work shows general conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: minor calculation mistakes in the proportion of mosquitoes or margin of error; formula used to calculate margin of error is incorrect but based on the variability of the sample proportions (for instance, using standard deviation rather than twice the standard deviation); correct answers to part a with no work shown; explanation in part b does not connect to margin of error (in name or concept) but does refer to variability or standard deviation in some way.  Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: errors in part a are more serious than Tier 2 types; response to part b does not refer to variability or standard deviation in some way.”

Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following-up with students include:

• Course Guide, Cool-Down Guidance states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”

• Algebra 1, Unit 1, Lesson 5, Cool-down Guidance, “Support Level 3. Press Pause. Notes Use the results from the Check Your Readiness Assessment to anticipate student struggle with MAD. Consider using Algebra 1 Supports Lesson 5 before this lesson if students need substantial support calculating MAD. Students will have more opportunities with IQR and the concept of variability.” 

• Geometry, Unit 1, Check-Your-Readiness, Question 8, “Students should be comfortable drawing transformations on the grid. During this unit, they will extend that understanding to the plane without the structure of a grid. If students miss the first part of the question, then you need to pay particular attention to the isometric grid translations in lesson 10. If they miss the second part of the question, then you need to pay particular attention to the isometric grid rotations in lesson 13.”

The materials reviewed for Math Nation AGA meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

All assessments regularly demonstrate the full intent of course-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in the classroom. Examples Include:  

• Algebra 1, Unit 2, Mid-Unit Assessment, Question 3, demonstrates the full intent of A-REI.10. “Tickets to the zoo cost $12 for adults and $8 for children. The school has a budget of $240 for the field trip. An equation representing the budget for the trip is $240 = 12x + 8y$. Here is a graph of this equation. (graph displayed in the problem).  Select all the true statements. A. If no adult chaperones were needed, 30 children could go to the zoo. B.  If ten children go to the zoo, then 15 adults can go. C. If four more adults go to the zoo, that means there will be room for six fewer children. D. If two more children go to the zoo, that means there will be room for three fewer adults. E. If 16 adults go to the zoo, then 6 children can go.”

• Geometry, Unit 1, End-of-Unit Assessment, Question 5, demonstrates the full intent of G-CO.9, MP1, and MP4. “Lines x and y are parallel. Write an equation that represents the relationship between b and e. Explain how you know this equation is always true.” A diagram is shown of a transversal crossing parallel lines, with multiple angles labeled.

• Algebra 2, Unit 5, Lesson 5, Cool Down, demonstrates the full intent of F-BF.3. “Let h be a function where $y = h(x)$. 1. What is the value of a if h is an even function? 2. What is the value of a if h is an odd function?” A table is provided with the columns x and y with the corresponding entries (-4, 0, 4) and (a, 0, $\frac{1}{2}$).

The materials reviewed for Math Nation AGA meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning series mathematics.

Throughout the materials, strategies, supports, and resources for students in special populations can be found in the Teacher Guides and Student Editions. In the Teacher Guide, teachers can find guidance for supporting students in special populations within various lesson activities. These supports are highlighted in green and labeled “Access for Students with Disabilities”. Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.

Examples of where and how materials regularly provide strategies, supports, and resources for students in special populations to support their regular and active participation in learning series mathematics include:

• Algebra 1, Unit 5, Lesson 1, Teacher Guide, 5.1.2 Exploration Activity, “Access for Students with Disabilities Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. Encourage students to sketch diagrams that show how the amount of money grows over the first few days. Students may benefit from this visual as they transition to the use of a table or other representation to track growth. Supports accessibility for: Conceptual processing; Visual-spatial processing.”

• Geometry, Unit 4, Lesson 7, Teacher Guide, 4.7.3 Exploration Activity, “Access for Students with Disabilities Action and Expression: Internalize Executive Functions. Provide students with a four-column table to organize. Use these column headings: angle, adjacent side, opposite side, and hypotenuse. The table will provide visual support for students to identify ratios. Supports accessibility for: Language; Organization.”  

• Algebra 2, Unit 5, Lesson 7, Teacher Guide, 5.7.2 Exploration Activity, “Access for Students with Disabilities Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of vertical translations, horizontal translations, and the vertex. Include equations and graphs that demonstrate these translations. Supports accessibility for: Conceptual processing; Language.” 

There are several accessibility options (accessed via the wrench icon in lower left hand corner of the screen) available to students to help naviagate the materials. Examples include:

• Tools Menu allows students to change the language and access a Demos Scientific and a Desmos Graphing Calculator.

• Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, and screen reader support. 

• UserWay allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), and Saturation (4 settings).

The materials reviewed for Math Nation AGA meet expectations for providing extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity.

Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. We think of them as the ‘mathematical dessert’ to follow the ‘mathematical entrée’ of a classroom activity. Every extension problem is made available to all students with the heading ‘Are You Ready for More?’ These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some of these problems extend the work of the associated activity, but some of them involve work from prior grades, prior units in the course, or reflect work that is related to the K-12 curriculum but a type of problem not required by the standards. They are not routine or procedural, and they are not just ‘the same thing again but with harder numbers.’ They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems, and it is not expected that any student works on all of them. Are You Ready for More? problems may also be good fodder for a Problem of the Week or similar structure.” If individual students were to complete these optional activities, then they might be doing more assignments than their classmates.

Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:

• Algebra 1, Unit 4, Lesson 5, 4.5.3 Exploration Extension: Are You Ready for More?, “Describe a different data plan that, for any amount of data used, would cost no more than one of the given plans and no less than the other given plan. Explain or show how you know this data plan would meet these requirements.” This extension follows an exploration activity in which students work with two functions.

• Geometry, Unit 1, Lesson 14, 1.14.4 Exploration Extension: Are You Ready for More?, “You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a regular polygon: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.”

• Algebra 2, Unit 2, Lesson 10, 2.10.4 Exploration Extension: Are You Ready for More?, “What is a possible equation of a polynomial function that has degree 5, but whose graph has exactly three horizontal intercepts and crosses the 𝑥-axis at all three intercepts? Explain why it is not possible to have a polynomial function that has degree 4 with this property.”

The materials reviewed for Math Nation AGA meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

In the Course Guide, there is a list of the Mathematics Language Routines (MLRs) that are used in the materials. The Mathematics Language Routines are instructional routines developed to support students with emerging English language proficiency. The Mathematics Language Routines included in the material are the following:  

• MLR1: Stronger and Clearer Each Time

• MLR2: Collect and Display

• MLR3: Clarify, Critique, Correct

• MLR4: Information Gap Cards

• MLR5: Co-Craft Questions

• MLR6: Three Reads

• MLR7: Compare and Connect

• MLR8: Discussion Supports

These routines are referenced under Instructional Routines in the Teacher Guide for units and lessons to assist teachers with lesson planning. The “Access for English Language Learners” section within the Teacher Guide contains explanations of how to implement the MLRs.  

Examples of where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Algebra 1, Unit 2, Lesson 10, Teacher Guide, 2.10.3 Exploration Activity, “Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to provide opportunities for students to analyze how different mathematical forms and symbols can represent different situations. Display only the problem statement without revealing the questions that follow. Invite students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that address quantities of each type of coin. Design Principle(s): Maximize meta-awareness; Support sense-making Representation: Internalize Comprehension. Activate or supply background knowledge about generalizing a process to create an equation for a given situation. Some students may benefit by first calculating how many nickels Andre would have if there were 0, 1, 5, or 10 dimes in the jar, and then how many dimes if there were 1, 5, or 10 nickels in the jar. Invite students to use what they notice about the processes they used to create an equation. Supports accessibility for: Visual-spatial processing; Conceptual processing.“

• Geometry, Unit 3, Lesson 9, Teacher Guide, 3.9.2 Exploration Activity, “Access for English Language Learners Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof that triangles with two pairs of congruent corresponding angles are similar. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, ‘What is the definition of similarity?’, ‘What is the center and scale factor of the dilation?’, and ‘How do you know that triangle A’B’C’ is congruent to PQR?’ Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why triangles with two pairs of congruent corresponding angles are similar. Design Principle(s): Optimize output (for justification); Cultivate conversation.”

• Algebra 2, Unit 1, Lesson 5, Teacher Guide, 1.5.3 Exploration Activity, “Access for English Language Learners Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: ‘I agree because . . .’ or ‘I disagree because . . .’ If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. Design Principle(s): Support sense-making.”

In the Student Edition for each lesson activity, students have access to videos which contain lesson explanations in both English and Spanish.

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

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Algebra 1, Unit 7, Lesson 1, 7.1.2 Exploration Activity, students cut a piece of material to create a frame around a picture. “Your teacher will give you a picture that is 7 inches by 4 inches, a piece of framing material measuring 4 inches by 2.5 inches, and a pair of scissors. Cut the framing material to create a rectangular frame for the picture. The frame should have the same thickness all the way around and have no overlaps. All of the framing material should be used (with no leftover pieces). Framing material is very expensive! You get 3 copies of the framing material, in case you make mistakes and need to recut.” 

• Geometry, Unit 5, Lesson 2, 5.2.2 Exploration Activity, students use an applet to cut a three-dimensional solid to identify the cross sections. “The triangle is a cross section formed when the plane slices through the cube. 1. Sketch predictions of all the kinds of cross sections that could be created as the plane moves through the cube. 2. The 3 red points control the movement of the plane. Click on them to move them up and down or side to side. You will see one of these movement arrows appear. Sketch any new cross sections you find after slicing.”

• Algebra 2, Unit 5, Lesson 1, 5.1.2 Exploration Activity, students use an applet to compare two functions and determine which function better fits the model. “A bottle of soda water is left outside on a cold day. The scatter plot shows the temperature 𝑇, in degrees Fahrenheit, of the bottle ℎ hours after it was left outside. Here are 2 functions you can use to model the temperature as a function of time:  $f(h)=45+\frac{20}{h+0.5}$  $g(h)=45+33(0.5)^{h+0.5}$ 1. Which function better fits the shape of the data? Explain your reasoning. 2. Use the applet to zoom out on the graphs. Does this change your opinion about which function fits better? 3. Where do you see the 45 in the expression for each function on the graph? 4. For the function you thought didn't fit the shape of the data as well, how would you change it to fit better?” The applet has the two functions already graphed on it and allows students to add additional functions and points to the graph.