High School - Gateway 2

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)