High School - Gateway 2

The materials reviewed for Math & YOU: Concepts & Connections 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.

Multiple conceptual understanding problems are embedded throughout the series within the Investigate, Discussions, In-class examples and Big Ideas Tasks. Students have opportunities to engage with these problems both independently and with teacher support. 

According to the Implementation Handbook, Foundational Beliefs, “Each lesson begins with opportunities for students to engage in investigation resulting in observations, conjectures, and discovery of informal strategies. These opportunities support students in developing an understanding of the mathematical concepts grounded in meaningful experience and connected to prior learning, building a sturdy conceptual foundation. From here, the focus turns to formalizing these ideas through explicit instruction of new mathematical terminology, formal strategies, and key concepts, while connecting back to students’ experiences during their investigation.”

Examples include:

• Algebra 1, Chapter 5, BIg Ideas Tasks, students demonstrate conceptual understanding by writing a system of linear equations with three equations that has a specified solution. Exercise 1 states, “a. Write a system of 3 linear equations that has a solution of (−2,5). b. Use a tool to mathematically justify that (−2,5) is a solution of the system. c. Let one of the equations in your system be represented by h(x). What do you know about h(−2)? Explain your thinking. d. Is it possible that the system of 3 linear equations has another solution? Use mathematics to explain why or why not.” (A-REI.11).

• Geometry, Chapter 9, Big Idea Tasks, students demonstrate conceptual understanding by connecting similarity of triangles with definitions of trigonometric ratios for acute angles. Exercise 3 states, “Determine whether each statement is true or not true. Construct a mathematical argument to justify your reasoning. a. If A and B are complementary angles, then sin⁡ A= cos ⁡B. b. For right \triangle ABC, if tan⁡A=\frac{7}{24}, then the length of the hypotenuse is 24 units. c. The side lengths of a triangle are 3 units, 6 units, and 9 units, so it is a 30\degree − 60\degree − 90\degree triangle. d. If you know the lengths of two sides of a right triangle, you can solve the triangle. e. The two legs of a right triangle are 10 units and 24 units, so the measures of the acute angles of the triangle are about 67.4\degree and about 22.6\degree. f. The expression sin^{-1}x  is equivalent to \frac{1}{sinx}.” (G-SRT.6).

• Algebra 2, Chapter 4, Lesson 4, Investigate, students develop conceptual understanding by using technology to match polynomial equations with their graphs and then write each polynomial function in factored form. Exercise 2 states, “Use technology to match each polynomial function with its graph. Then write each polynomial function in factored form. a. f(x)=x^2+5x+4 b. f(x)=x^3-2x^2-x+2c. f(x)=x^3+x^2-2x d. f(x)=x^3-x e. f(x)=x^4-5x^2+4 f. f(x)=x^4-2x^3-x^2+2x.” Students use technology to match provided polynomial graphs with equations and write each function in factored form. Exercise 3 states, “What information can you obtain from the factored form of a polynomial function?” The Instructional Guide states, “Students should recall that when a function is written in factored form, they can find the zeros of the function. Knowing the zeros is helpful in sketching a graph of the function. Exercise 2 reviews the connection between the factored form of a function and the x-intercepts of its graph. The quadratic function in Exercise 2(a) is a good place for students to start. They can use the connection between the x-intercepts, −1 and −4, and the factored form f(x)=(x+1)(x+4) to reason about the factored forms of the other polynomial functions. ‘How can you check whether your factored form of a function is correct?’ Multiply the factors and simplify, or use technology to graph the factored form of the function and compare it to the original graph. Circulate and listen to students’ reasoning about matching each function with its graph. Do any students mention end behavior?” (A-APR.3).

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

Multiple procedural skill and fluency problems are embedded throughout the series within In-Class Practice, which extends their learning from the Key Concept. The formal assessments offered at the lesson, chapter, multi-chapter, and course levels provide opportunities for students to independently demonstrate their procedural skills and fluency. 

According to the Implementation Handbook, Foundational Beliefs, “Students have opportunities to develop procedural fluency through targeted practice supported by question prompts designed to encourage reflection on the accuracy and efficiency of their strategy. Practice opportunities support students in solving tasks that incorporate procedures with connections, requiring students to think meaningfully about which strategies they are using and how they apply in the problem context, and to reason about the meaning of the resulting solution. Students regularly apply their learning in new real-world or mathematical contexts, focusing on how strategies extend to these contexts and interpreting the meaning of the solution in light of the situational context.” 

For example,

• Algebra 1, Chapter 7, Chapter Test, students demonstrate procedural skill and fluency by adding, subtracting, and multiplying polynomials. Exercise 1 states, “Find the sum. (2x^2-2x+5)+(-x^3-x^2+3)=-x^3+x^2-2x+8).” (A-APR.1).

• Geometry, Chapter 3, Chapter Test, students demonstrate procedural skill and fluency by finding equations of lines perpendicular to a given line through a given point. Exercise 6 states, “Write an equation of the line passing through the point (-8, 2) that is perpendicular to the line y=\frac{4}{3}x-8.” (G-GPE.5).

• Algebra 2, Chapter 4, Lesson 3, Practice, students develop procedural skill and fluency by dividing polynomials using long and synthetic division and by rewriting rational expressions using two methods. The Instructional Guide, Paul’s Notes states, “When students hear ‘shortcut,’ they tend to mentally replace long division with this ‘easier’ method. From the very beginning, it is important to stress that synthetic division cannot always be used to divide polynomials. Emphasize that this shortcut can only be used when the divisor is of the form x-k. Use Example 2(a) to explain and model the steps of synthetic division. Compare the results to the results of using long division to solve the same problem so that students see the parallel work and the same numbers. It may take several examples for students to understand why synthetic division works. Have students try Example 2(b) independently, and then discuss as a class.” Example 2a states, “Divide -x^3+4x^2+9 by x-3.” Example 2b states, “Divide 3x^3-2x^2+2x-5 by x+1.” (A-APR.6).

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Multiple routine and non-routine applications of mathematics are included throughout the series, with single and multi-step application problems embedded within lessons, including Investigate and Connecting to Real Life. Students engage with these applications both with teacher support and independently through Examples and Practice. Materials are designed to provide opportunities for students to demonstrate their understanding of grade-level mathematics when appropriate. 

For example:

• Algebra 1, Chapter 5, Performance Task, students apply their understanding by modeling ascent and descent times using systems of linear equations and inequalities, graphing feasible solutions, and determining maximum and minimum time constraints for a planned submersible expedition. Paul’s Notes, Conclude the Chapter with Career, states “In this Performance Task, students will use their understanding of systems of linear equations to compare ascent and descent times for submersibles. Then they will use a system of linear inequalities to plan an expedition. Formulate: ‘What relationships are given about the dives? How can you model these relationships? What relationships will you use to plan your expedition?’ Compute: ‘What are you trying to determine about each dive? How can you use a model to find this information? How will you use the ascent and descent times for the 1960 and 2012 expeditions to write a plan for your dive?’ Interpret: ‘How can you use units to interpret your solution? In your plan, are your ascent and descent times reasonable? What is the maximum total amount of time for your expedition?’ Validate: Encourage students to research manned dives. Ask students what the ‘unexpected circumstances’ might include and how their model could reflect these possibilities.’” Performance Task, PLAN AN EXPEDITION states, “You pilot a manned dive to the Challenger Deep. Use the ascent times and the descent times for the 1960 and 2012 expeditions to write a plan for your dive, including: • how long you will spend descending• how long you will spend on the ocean floor• how long you will spend ascending• goals you hope to accomplish. Dive schedules must be flexible to allow for unexpected circumstances. Determine the maximum total amount of time for your expedition and the minimum and maximum amounts of time you want to spend on the ocean floor. Use a graph to show the amounts of time that you can spend on the ocean floor and the amounts of time that you can spend traveling.” (A-CED.2, A-CED.3, A-REI.3, A-REI.6, F-IF.7)

• Geometry, Chapter 12, Lesson 6, Practice, students apply their understanding by using volume and density to calculate the mass of an American Eagle Silver Bullion Coin. Exercise 2 states, “The United States has minted one-dollar silver coins called American Eagle Silver Bullion Coins since 1986. Each coin has a diameter of 40.6 millimeters and is 2.98 millimeters thick. The density of silver is 10.5 grams per cubic centimeter. Find the mass of an American Eagle Silver Bullion Coin.” (G-MG.2)

• Algebra 2, Chapter 3, Lesson 5, Practice, students apply their understanding by solving a system consisting of a linear equation and a circle to determine the portion of a highway within a radio tower’s broadcast range. Exercise 20 states, "The equations shown model the range (in miles) of a broadcast signal from a radio tower and the path of a straight highway. For what length of the highway are cars able to receive the broadcast signal? Broadcast Signal: x^2+y^2=1620; Highway: y==\frac{1}{3}x+30”. (A-REI.11)

The materials reviewed for Math & YOU: Concepts & Connections meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within each course in the series as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each lesson within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. 

For Example:

• Algebra 1, Chapter 3, Lesson 3, Investigate, students demonstrate all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application as they explore a linear pattern using data from a Nutrition Facts label. Using a table, students record the amount of fiber for different numbers of servings by repeatedly multiplying the grams per serving by the number of servings. Students then create a scatter plot of the data, observe that the pattern is linear, and use the pattern to predict the amount of fiber in 10 servings. Investigate states, “Work with a partner. A partial nutrition label is shown. 1. Copy and complete the table. 2. What pattern do you notice in the data? 3. Make a scatter plot of the data. What pattern do you notice in the scatter plot? 4. Predict the amount of fiber in the 10 servings.” (A-REI.10, F-IF.5, F-LE.1b)

• Geometry, Chapter 5, Lesson 8, In-Class Practice, students demonstrate conceptual understanding and procedural skill and fluency by planning and writing coordinate proofs to establish triangle congruence. In Exercise 3, students develop a plan for a coordinate proof showing that two triangles in the coordinate plane are congruent given a bisecting ray. In Exercise 4, students write and execute a coordinate proof to show that two triangles are congruent using the coordinates of their vertices. (G-GPE.4)

• Algebra 2, Chapter 5, Lesson 7, Practice, Interpreting Data, students demonstrate conceptual understanding, procedural skill and fluency, and application by analyzing a data display of projected body fat percentage in relation to body mass index (BMI), describing the relationship between the variables, and finding and interpreting an inverse function in context. Exercise 43 states, “Describe the relationship between BMI and body fat percentage.” Exercise 44 states, “The body fat percentage of a person can be modeled by f(x)=1.116x+0.71 where x is the person's BMI. Find and interpret f^{-1}(32).” (F-BF.1b).

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP1 throughout the series. MP1 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 2, Lesson 6, Student Edition, students solve a multi-step real-world problem involving hotel costs and discounts, write and solve an absolute value inequality to determine which hotels meet the given cost condition, and explain their reasoning. In-Class Practice Exercise 10 states, “A softball team will spend three nights at a hotel. Each hotel offers a 50% discount for the third night. The coach wants to keep the total cost per player at $225 with an absolute deviation of at most $25. Which hotels meet this condition?" An image of a table showing Hotels A, B, C, and D with prices per night of $80, $105, $75, and $90. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students complete their study of linear inequalities by transferring their understanding of compound inequalities, relating them to a given absolute value inequality. Once students are comfortable with this connection, they will be ready to write and solve absolute value inequalities. There are a multitude of ways linking students to SMP.1, including opportunities for students to discuss the meaning of these problems with classmates, but specifically by checking answers and making sure solutions are reasonable and make sense. In-Class Practice Exercise 10 can allow for some productive struggle. Consider prompting student thinking by asking: ‘What are you having trouble with?’ ‘What question are you seeking to answer? What information do you have? Is any information not needed?’ ‘How can you use a model to make sense of the problem?’”

• Geometry, Chapter 9, Lesson 3, Student Edition, students develop a conjecture comparing the relationship between the arithmetic mean and geometric mean of two nonnegative numbers. Practice Exercise 23 states, “The arithmetic mean and geometric mean of two nonnegative numbers x and y are shown. Write an inequality that relates these two means. Justify your answer. arithmetic mean = \frac{x+y}{2}; geometric mean = \sqrt{xy}” The Answer Guide states, “To solve the problem requires critical thinking. Help students to think about the strategies they might use to get started on this problem. Some students might create a small data set and calculate the means for comparison. Others may begin solving algebraically. Pose advancing questions such as, ‘Is there a special case that you could test? What can you deduce about conditions outside of your test case? What algebraic process could you try to relate the two expressions?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students solve mathematical and real-life problems, encourage them to engage in SMP.1 by independently working through productive struggle to make sense of problems, plan solution pathways, and solve and check their answers for reasonableness. Solving Practice Exercise 23 requires critical thinking. Ask questions to encourage students to think about possible entry points. Suggested prompts: ‘What strategies might you use to get started? Can you begin by solving algebraically? Can you create a small data set? Do you have a different solution pathway?  Is there a special case that you could test? What can you deduce about conditions outside of your test case? What algebraic process could you try to relate the two expressions?  How can you check your answer?’”

• Algebra 2, Chapter 4, Lesson 5, Student Edition, students complete an equation given possible rational solutions and actual rational solutions of the equation. Practice Exercise 32 states, “All the possible rational solutions and actual rational solutions of the equation below are shown. Complete the equation. Justify your answer. Possible: x=\pm1, \pm2, \pm4, \pm8, \pm16. Actual: x=-1 2. (x+__)(x^2+__)=0.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students find possible zero(s) and write polynomials with given characteristics, you can support them in developing SMP.1. Many of these exercises do not prescribe steps for students to take. Rather, students must interpret what they know about the polynomial or its zeros and decide what additional information can be determined from various tools (e.g., The Rational Root Theorem and the Irrational Conjugates Theorem). Encourage students to choose a strategy, to monitor its effectiveness, and to verify their final solution using various tools. Practice Exercise 32 requires students to bring together their knowledge of zeros and factors with their understanding of the Rational Root Theorem. You can support their productive struggle with prompts: ‘What information can you glean from the actual roots? How does the list of possible rational solutions help you identify the missing information? What additional constraints are provided by the information in the problem? How can you verify that your result satisfies all the conditions?’”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP2 throughout the series. MP2 is found in lessons where they focus on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra 1, Chapter 4, Lesson 2, Student Edition, students compare and interpret multiple representations of linear relationships to analyze changes in three bank accounts over time. Practice Exercise 11 states, “Three bank accounts are opened and then have fixed amounts of money withdrawn each month. The graph shows Account A, the table represents Account B, and the equation y=-22.5x+90 represents Account C, where y represents the amount of money (in dollars) left after x months. a. Which account has the greatest initial value? the least initial value? b. Which account has the most amount of money withdrawn each month? the least amount? c. Which account runs out of money first? last?” Students are provided with a graph labeled Money Left (dollars) on the y-axis and Month on the x-axis, showing two points on a line, (2, 80) and (4, 40). They are also provided with a table showing Month (x) with values 1, 2, 3, and 4, and Money Left (y) with corresponding values of $100, $75, $50, and $25. The Answer Guide states, “Students must decontextualize to analyze the intercepts and rates of change and then contextualize to compare the bank accounts. Ask, ‘What information is given in each representation? What key aspects of each can be used to indicate how the money in the accounts are changing?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students continue to use mathematical symbols to represent situations, further enabling SMP.2. Provide students with a variety of problems in contexts that are interesting to them. This allows each student to pursue solutions in different ways.”

• Geometry, Chapter 1, Lesson 3, Digital Teaching Experience, students are given a real-life context to explore the meaning of midpoint. Paul’s Notes, Launch states, “‘Imagine that you live far away from a relative and want to meet them halfway so that neither of you has to drive the entire distance between your locations.  How would you determine where you should meet? What do you need to know to figure this out?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students use midpoint and distance formulas, they will have the opportunity to engage in SMP.2 by reasoning abstractly and quantitatively to apply these formulas in real-life contexts. Use the Launch to encourage abstract and quantitative thinking. Suggested prompts: ‘Think of a distant relative. How might you determine a fair meeting location?’ Listen for students to think abstractly as they generalize an approach for finding an equidistant meeting point. ‘How can you be sure that you are meeting in the middle? What aspects of the real-life context do you need to consider?’”

• Algebra 2, Chapter 8, Lesson 5, Student Edition, students calculate margin of error and estimate population means for a real-world context. Practice Exercise 16 states, “A survey reports that 47% of the voters surveyed, or about 235 voters, said they voted for Candidate A and the remainder said they voted for Candidate B. a. How many voters were surveyed? b. What is the margin of error for the survey? c. For each candidate, find the interval that is likely to contain the exact percentage of all voters who voted for the candidate. d. Based on your intervals in part (c), can you be confident that Candidate B won? If not, how many people in the sample would need to vote for Candidate B for you to be confident that Candidate B won?” The Answer Guide states, “Ask students to share their thinking about part (d). What do your intervals indicate about the entire population of voters? Allow students to share their understandings about each candidate’s possibility of winning based on the survey. ‘Would you come to a different conclusion if the same results had come from a survey that had double the sample size? triple the sample size?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students explore inferences, they consider both the quantitative results stemming from surveys and the meaning of those results based on the context that the data represents. Engage students in using SMP.2 as they move fluidly between the mathematical results and the contextual conclusions, helping them to provide meaning for the mathematical values Support students in reasoning abstractly and quantitatively as they solve Practice Exercise 18. Suggested prompts: ‘What value did you find for the margin of error? What does the margin of error mean in the context of this problem? What does it help you to conclude? Based on your margin of error, what is the population mean likely to be? How would the conclusions change if the survey sample doubled? Tripled? Explain why the sample size impacts your conclusions.’”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP3 throughout the series. MP3 is found in lessons where they focus on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 10, Lesson 2, Student Edition, students match functions to their corresponding graphs and justify their reasoning by explaining how transformations affect each graph. In-Class Practice states, “Match the function with its graph. Exercise 3. m(x)=\sqrt[3]{x-2} Exercise 4. g(x)=\sqrt[3]{x}+2 Exercise 5. h(x)=\sqrt[3]{x}-2.” Students are provided three graphs they match to the functions. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will graph and describe cube root functions, eventually using the cube root function in the context of real-life problems. Students continue to apply their knowledge of the patterns and rules used to perform transformations on a parent graph. They will engage with SMP.3 by explaining and justifying their thinking using mathematical terms and concepts when engaging in peer discussions. Use In-Class Practice Exercises 3-5 to encourage debate among students. Consider using these questions to stimulate the dialogue: ‘Your friend says that subtracting 2 within a cube root indicates a vertical translation. Do you agree? Explain your reasoning. How are these equations and graphs similar? Different?’”

• Geometry, Chapter 4, Lesson 1, Student Edition, students graph a triangle and its translated image. In-Class Exercise 3 states, “Graph \triangle RST with vertices R(2,2), S(5,2), and T(3,5) and its image after the translation (x, y)(x+1, y+2).” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Use In-Class Practice Exercise 3 as an opportunity to have students discuss their mathematical thinking. Suggested prompts: ‘Turn to a partner and explain your process. Then listen to your partner explain their process. Did you use the language of mathematics in your explanations? Do you have any questions to clarify your partner's thinking?  Did you both get the same results? If not, discuss where one of you went wrong and how you can correct your thinking. If yes, where might a student go wrong when translating this way?’ Students may mention transposing the horizontal and vertical components resulting in a triangle that is (x+2,y+1) rather than (x+1,y+2).”

• Algebra 2, Chapter 8, Lesson 6, Student Edition, students evaluate a student claim about resampling differences. Practice Exercise 6 states, “Your friend states that the mean of the resampling differences should be close to 0, as the number of resamplings increase. Explain whether your friend is correct.” The Answer Guide states, “After students have worked independently, have them explain their reasoning to a partner using mathematical arguments. Then the listening partner decides whether the explanation makes sense and asks questions to clarify or improve the arguments. Ask the pairs to share with the class. ‘What did you know about resampling that helped justify your reasoning? What are resampling differences and how did you use the definition in your argument? Was your answer different than your classmate’s?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “This lesson engages students in analyzing data from an experiment and making inferences from experiments. This lesson supports students’ use of SMP.3 as they look holistically at an experiment to interpret the results and to draw conclusions about how likely it is that the treatment had an effect. Encourage students to critique the experimental design and think critically about the types of conclusions it supports. Students also have opportunities to describe changes to the experimental design that might support stronger conclusions. Support students in evaluating claims and constructing arguments as they solve Practice Exercise 6. Consider engaging the class in a whole-group discussion after they have solved the problem. Ask students to pick a stance, with students who agree with the friend going to one side of the room and those who disagree with the friend going to the other side of the room. Depending on how evenly distributed the students are, you could either pair up students from opposite sides of the room to share their arguments for and against the statement, or you could choose a spokesperson from each side to talk to the whole group. Ask students to listen critically to the argument given by their classmates. ‘Did any of the evidence provided change your mind in any

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP4: Model with Mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP4 throughout the series. MP4 is found in lessons where the lessons focus on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 3, Lesson 4, Student Edition, students interpret real-world data about engine size and fuel economy by analyzing a graph to describe the relationship between the two variables, writing a function that models fuel economy as a function of engine size, and using the function to estimate the engine size for a given fuel economy. Practice Interpreting Data states, “ENGINE SIZE AND FUEL ECONOMY: Fuel economy is an important consideration when choosing a car. Fuel economy describes the number of miles a vehicle can travel using a specific amount of fuel. Engine size is measured in liters. Fuel economy is measured in miles per gallon. Exercise 17. What does the graph show you about the relationship between engine size and fuel economy? Exercise 18. Write a function that approximates the fuel economy f(x) (in miles per gallon) for an engine size of x liters. Then use your function to estimate the size of an engine that gets 25 miles per gallon.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students can engage with SMP.4 by making connections between different representations when interpreting functions with function notation. This is especially helpful when students see tables and graphs concurrently with connections to real life. In the Interpreting Data feature, students can explore data about the fuel economy of vehicles. You can support student engagement by asking: ‘Can you write a function, using correct notation, that relates fuel economy to engine size?’ ‘What connections do you observe in the data?’”

• Geometry, Chapter 5, Lesson 8, Student Edition, students compute the area of the The Pentagon building in Washington D.C. Practice, Exercise 14 states, “The Pentagon is the world's largest office building. It has 6.6 million square feet of floor space. Each side of the regular pentagon is about 920 feet long. Typically, more than 20,000 people work in the building. Use coordinate geometry to approximate the area of the building.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will use coordinates represented by variables to prove geometric theorems algebraically. In doing so, students are modeling with mathematics and engaging in aspects of SMP.4. Using variables enables the results to be true for all figures of that type. There are multiple ways to view the Pentagon as composite shapes. ‘How did you create a model to compute the area? What characteristics of the Pentagon did you capture by your model?’ ‘Did breaking the problem into smaller parts help you identify a solution pathway? What formulas did you call upon to make the necessary computations?’ Make sure that students are engaged with the modeling process by interpreting and validating their answer. ‘Have you considered the units in your answer? How did you choose to validate your answer? Did anyone refine their model during the validating process?’”

• Algebra 2, Chapter 1, Lesson 3, Student Edition, students describe a real-life situation that can be modeled by linear graphs. Investigate Exercise 2 states, “Describe a real-life situation that can be modeled by each graph. (Four linear graphs provided with scales on axes, but no labels.)” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this section, students engage with SMP.4 as they explore modeling with linear functions. During the Investigate, students focus on using a given model (i.e., a linear graph) to interpret the real-world scenario it represents. Later in the lesson, students are given the real-world context, and they formulate a linear model to represent the situation. Students then use their model to compute, interpret, validate, and report solutions in light of the context. During the Investigate, help students consider important elements of a real-world context that could be modeled by each graph: ‘What important features of the graph does your real-world context need to align with? How would you “see” that feature in a context? How are graphs a and c the same (different)? How are the contexts they modeled the same (different)?’”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP5 throughout the series. MP5 is found in lessons where each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 4, Lesson 5, Student Edition, students use a graphing calculator or online software to create a line of best fit for data showing the number of movie tickets sold from January to June, use the model to predict ticket sales for August, and explain their choice of tool and its limitations when making the prediction.  In-Class Practice, Exercise 3 states, “The table shows the approximate numbers y (in thousands) of movie tickets sold from January to June at a theater. Use the line of best fit to predict the number of tickets sold in August." A table shows Month (x) with values 1, 2, 3, 4, 5, and 6, and Ticket Sales (y) with corresponding values of 27, 28, 36, 28, 32, and 35. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Previously, students examined lines of fit when looking at a scatter plot. In this lesson, they will enable SMP.5 by using technology such as a graphing calculator or online software to deepen their understanding of linear relationships. This is particularly true when students learn to distinguish between interpolation and extrapolation. Use In Class Practice Exercise 3 to emphasize student autonomy in deciding which tool is the most appropriate tool for finding a line of best fit. Extend their thinking with the following prompts: ‘Of the tools we have used, which do you feel is most appropriate?  Are there any limitations with the tool you chose when predicting the number of tickets sold in August? If so, would you change tools?’”

• Geometry, Chapter 5, Lesson 4, Student Edition, students draw isosceles triangles inscribed in circles to generate a conjecture about angle measures of an isosceles triangle. Investigate states, “1. Construct several circles. For each circle, draw a triangle with one vertex at the center of the circle and the other two vertices on the circle. Explain why the triangles are isosceles. 2. Measure the angles of the triangle. What do you notice? Make a conjecture about the angle measures of an isosceles triangle.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students can investigate concepts by using dynamic geometry software or a compass, protractor, and straightedge. Encourage SMP.5 by having students notice and articulate the benefits of certain tools over others. They can achieve the same results using by-hand methods, but students should be able to see the advantages of technology when repeating several cases. Suggested prompts: ‘In Exercises 1 and 2, what tool will you choose to construct and explore your triangles? How does the tool you chose help with your conjecture?’ If using technology, students may say they can easily manipulate the triangle and see that the two angles opposite the congruent sides of the triangle are congruent. If constructing by hand, they will need to measure all three angles with a protractor.”

• Algebra 2, Chapter 9, Lesson 2, Student Edition, students draw a circle and plot angles that correspond to specified radian values. Investigate Exercise 2 states, “Draw any circle. a. Plot points that correspond to angles of 1, 2, 3, 4, 5, and 6 radians. b. Estimate the angle measures (in degrees) that correspond to the points you plotted in part (a).” The Answer Guide states, “Students can choose to use technology, a compass and a rule, or they may ask for string, or paper they can bend, to measure along the circumference of the circles that they draw.’Ask volunteers to share and compare their methods. ‘What are the benefits and drawbacks of each method’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “This lesson introduces students to radian angle measure. As students explore the meaning of radians, they use tools to plot points on a circle that correspond to angles of various radian measures. Students use SMP.5 as they reason about the types of tools that allow them to find points to satisfy the conditions and engage in dialogue about if the tool worked as expected. Support students in thinking about their tool selection and its meaningful use as they draw a circle and plot points that correspond to angles of 1, 2, 3, 4, 5, and 6 radians on the Investigate activity. Suggested prompts, ‘What tool(s) can you use to help you identify each point? How do you know these will provide the information you need? Compare your approach with a friend. Did you use the same tool(s)? In what ways did your tool(s) lead to similar results? Different results?’”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP6: Attend to precision, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP6 throughout the series. MP6 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 6, Lesson 2, Student Edition, students solve real-world problems involving rational exponents. In-Class Practice Exercise 8 states, “The volume of a beach ball is 17,000 cubic inches. How much greater is the radius of the beach ball in Example 4?” In-Class Practice Exercise 9 states, “The average tuition cost of a 4-year college increases from $20,125 to $25,900 over a period of 6 years. The average tuition cost of a 2-year college increases from $8,540 to $10,950 over the same period. Which has a greater annual inflation rate?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “The process of reaching mastery in evaluating expressions with rational exponents necessitates students engage with SMP.6 in deciding when to estimate or give an exact answer. For instance, many of these expressions yield an exact answer, however, some real-life problems (such as those involving inflation) dictate that an estimate is perfectly acceptable. Use In Class Practice Exercises 8 and 9 to help students decide when it is appropriate to give an exact answer, and when an estimate is acceptable. Ask students: ‘Does the question imply that you should give an exact answer? What is the standard approximation for quantities such as inflation rate, which are typically expressed as decimal representations of a percent?’”

• Geometry, Chapter 1, Lesson 2, Student Edition, students estimate the distance between two cities when given a map. In-Class Practice Exercise 8 states, “The cities shown on the map lie approximately in a straight line. Find the distance from Albuquerque, New Mexico, to Provo, Utah.” (A map with distances from Albuquerque to Carlsbad and Provo to Carlsbad are given.) Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Using tools and articulating thinking about definitions allow students to attend to precision. Encourage SMP.6 by asking students to communicate with one another using precise mathematical terms. Use In-Class Practice Exercise 8 to encourage attending to precision. Suggested prompts: ‘To what degree of precision should you express your answer? Why? Do you suppose the driving distance from Albuquerque to Carlsbad is exactly 231 miles? Why or why not?’ Listen for students to recognize that the distance may have been rounded to the nearest mile and that the roads connecting the two cities are not likely to form a perfect line. ‘Can you think of a situation in which it would be important to have more precise measurements?’”

• Algebra 2, Chapter 3, Lesson 1, Student Edition, students use correct mathematical language as they describe the relationship among zeros, x-intercepts, and roots. Practice Exercise 36 states, “Describe the relationship among zeros, x-intercepts, and roots.” The Answer Guide states, “Use this opportunity for students to practice communicating precisely. Ask students to first compare informally, using their own words to articulate the comparison. ‘How are the three terms related? What are the differences?’ Then ask students to summarize in a statement that uses mathematical precision. After students share, ask them to consider different types of functions. ‘Would the relationships change for different types of functions? Why or why no?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout this lesson, students have opportunities to engage in SMP.6 as they learn to solve quadratic equations both by graphing and algebraically. As students work across representations, they learn to interpret what it means to solve a quadratic equation, recognizing how finding roots of an equation relate to finding the x-intercepts of the graph. Students also learn the importance of writing an equation in a precise form before using a particular rule, such as the Zero-Product Property. Encourage students to be more precise than simply saying zeros, x-intercepts, and roots are the ‘same.’ Ask, ‘How are the three terms related? How are they different? When is it appropriate to use each one?’ Challenge students to use precise notation and move between representations as they solve quadratic equations. On Practice Exercise 36, encourage students to be more precise than simply saying zeros, x-intercepts, and roots are the ‘same.’ ‘How are the three terms related? How are they different? When is it appropriate to use each one?’ Encourage students to choose an exercise from class and to use each of the three terms in a sentence to describe the example.”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP7: look for and make use of structure, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP7 throughout the series. MP7 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 7, Lesson 5, Student Edition, students analyze the structure of the polynomial x^2+bx+c where b = c to determine binomial factors, identify patterns between the coefficients and factors, and write the polynomial in factored form. Practice Exercise 25 states, “Write a polynomial of the form x^2+bc+c where b = c that can be written as (x+p)(x+q), where p and q are nonzero integers. Then write the polynomial in factored form.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will factor polynomials of the form x^2+bx+c. First, however, students must learn to explain how to use the coefficients b and c to find binomial factors of a polynomial of this form. This is where students engage with SMP.7 to find the structure and pattern of using these two quantities to find the proper binomial factor. Use Practice Exercise 25 to help students uncover patterns within the factoring process. Consider asking students: ‘What patterns from earlier examples can help you with this problem? What pattern are you looking for when you analyze the value of c to help you solve this problem?’”

• Geometry, Chapter 5, Lesson 8, Student Edition, students generalize the coordinates of each endpoint of any line segment whose midpoint is the origin. Practice Exercise 11 states, “Write algebraic expressions for the coordinates of each endpoint of a line segment whose midpoint is the origin.” The Answer Guide states, “Examining the structure of the given information is key to solving this problem. Students may first want to sketch several examples and look for similarities, trends, or relationships. ‘Consider several test cases. How can you represent each case symbolically to help reveal the underlying structure of the endpoints? What do they have in common?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will engage in SMP.7 as they represent situations in the coordinate plane. For example, if it is known that a point C lies on the y-axis, then its x- coordinate must be 0 and the point can be represented as (0, c). Encourage students to look for patterns and structures that will simplify computations. Consider the following prompts to get students thinking. ‘Are there test cases you can sketch that will enable you to look for similarities, trends, or relationships? How can you represent each case symbolically to help reveal the structure of the endpoints? What do they have in common?’ If students are still stumped, have them consider the point (1,1). ‘In order for the origin to be the midpoint, what would the other coordinate have to be?’”

• Algebra 2, Chapter 2, Lesson 2, Student Edition, students are given information about a parabola and determine whether its vertex is the highest or lowest point of that parabola. Practice Exercise 18 states, “A quadratic function is increasing whenx<2and decreasing when x>2. Explain whether the vertex is the highest or the lowest point in the parabola.” The Answer Guide states, “Encourage students to create a sketch to aid in visualizing the structure of the quadratic. Students must see the structure of the graph as composed of two halves of two distinct shapes, and then visualize how those elements fit together.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout this lesson, students engage with SMP.7 as they learn to recognize characteristics of a quadratic function in algebraic and graphical form. Students begin to see the structure of a parabola based on its vertex, axis of symmetry, and intercepts. Students also learn about maximum and minimum values and can relate the leading coefficient of a quadratic equation in standard form with indicating whether the parabola opens upward (has a minimum) or downward (has a maximum). As students work through Practice Exercise 18, help them to focus on the structure of the parabola rather than relying on rules to answer the question. For example, help them to think about the parabola as two distinct parts, one where x<2and the other where x>2. ‘Where will the vertex of the parabola occur? How do you know? How can you use the two parts of the parabola to decide if the vertex is a maximum or a minimum? Will that always work?’”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students engage with MP8 throughout the series. MP8 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

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

For Example:

• Algebra I, Chapter 1, Lesson 5, Student Edition, students solve absolute value problems, recognize patterns in their solutions, and make generalizations to apply to similar equations. Practice Exercise 5 states, “Solve the equation. 2\left|=8w+6\right|=762” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will write and solve equations involving absolute value. Students will see equations that may have one absolute value expression, or in some cases, two absolute value expressions. As with the prior lesson, there are special solutions to some of these equations. Many of the examples and exercises in this section point directly to SMP.8 wherein students can look for patterns when working with tables, or work towards finding and using shortcuts or generalizations. Practice Exercise 5 may disrupt student reasoning since the absolute value function is multiplied by a constant. This is a good place to ask the following: What generalizations have you made in the lesson thus far? How does this exercise change your approach? How could this problem help you solve other equations involving absolute value?’”

• Geometry, Chapter 7, Lesson 4, Student Edition, students generalize diagonals of rectangles, rhombuses, and squares after repeatedly constructing quadrilaterals meeting specified constraints. Investigate states, “2. Use technology to construct two congruent line segments that bisect each other. Draw a quadrilateral by connecting the endpoints. 3. Is the quadrilateral you drew a parallelogram? a rectangle? a rhombus? a square? 4. Repeat Exercises 2 and 3 for several pairs of congruent line segments that bisect each other. Make conjectures based on your results. 5. Use technology to construct two line segments that are perpendicular bisectors of each other. Draw a quadrilateral by connecting the endpoints. 6. Is the quadrilateral you drew a parallelogram? a rectangle? a rhombus? a square? 7. Repeat Exercises 5 and 6 for several other line segments that are perpendicular bisectors of each other. Make conjectures based on your results. 8. What are some properties of the diagonals of rectangles, rhombuses, and squares?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students explore rhombuses, rectangles, and squares. These special parallelograms have additional properties related to diagonals and opposite angles. Students are familiar with the definitions of rhombus, rectangle, and square from middle school, but they may confuse properties of these quadrilaterals with their definitions. Students will engage in SMP.8 as they experiment with technology to discover these properties. In the Investigate, ask questions about students’ repeated reasoning. Suggested prompts: ‘In Exercises 4 and 7, how do changes in the angle between and/or the length of the diagonals affect the shape? What characteristics of the shape do not change? Can you express the patterns you are observing in mathematical terms? Are there any conditions that result in a special case?’ ‘How did your repeated calculations lead you to any conjectures?’”

• Algebra 2, Chapter 4, Lesson 2, Student Edition, students analyze patterns in consecutive square numbers, represent these patterns using an algebraic identity, and generalize their reasoning to prove the relationship. Practice Exercise 38 states, “The first four square numbers are represented below. (Image of 1\times1, 2\times2, 3\times3, and 4\times4 tiles provided.) a. Find the differences between consecutive square numbers. What do you notice? b. Show how the polynomial identify (n+1)^2-n^2=2n+1 models the differences between consecutive square numbers. Then prove the identity.” The Answer Guide states, “Ask probing questions to uncover students’ repeated reasoning. Students will find that the right side of the identity in part (b) models the pattern of differences from part (a). Probe to discover how they connected the left side of the identity with the figure.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students look for repeated reasoning in the results of adding, subtracting, and multiplying polynomials and expanding binomials (SMP.8). Encourage students to focus on the reason behind the patterns they are noticing to draw generalizations. For example, the sum, difference, and product of two polynomials is still a polynomial, addition and multiplication of polynomials are commutative, and the number of terms in the expansion (a+b)^n is always n+1. Practice Exercise 38 provides another opportunity for students to make sense of patterns. Help them apply repeated reasoning with prompts. ‘What patterns do you see in the differences between consecutive square numbers? Does that pattern continue? How do you know? What does the left-hand side of the identity represent? The right-hand side? How does this identity help generalize the patterns you saw?’”