High School - Gateway 1

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for materials attending to the full intent of the mathematical content contained in the high school standards for all students.

Each lesson opens with Engage and Explore, where students complete a Warm-Up, participate in a Launch Activity, and engage in a student-centered exploration during Investigate. Students then examine Key Concept examples and complete In-Class Practice and Practice exercises, and they apply mathematics to contextual situations in Connections to Real Life. Standards-Based Practice and Lesson Extra Practice provide additional opportunities for students to reinforce the knowledge and skills developed in each lesson. Across the program, the instructional materials address all aspects of all non-plus standards.

Three correlation resources document alignment to the standards and identify where students engage with the full intent of the standards across the course. The Standards Correlation (by Course) organizes standards by course component and identifies the standards addressed in each lesson. The Standards Correlation (by Standard) lists every lesson in which a specific standard is addressed. The Standards-Based Practice Correlation connects each Standards-Based Practice activity in the digital platform or Practice Workbook to a content standard and identifies the lessons in which that standard is reinforced.

Examples include:

• N-RN.3: Algebra 1, Chapter 9, Lesson 1, students apply properties of rational and irrational numbers. In Practice Exercise 39, students analyze statements about whether performing an operation on rational and irrational numbers results in a rational or irrational number. Students determine whether each statement is always, sometimes, or never true and explain their reasoning using an example, a counterexample, or proof. In Practice Exercise 40, students perform operations with radical expressions and determine whether each resulting expression represents a rational or irrational number.

• A-REI.6: Algebra 1, Chapter 5, Lessons 1-4, students solve systems of linear equations by graphing, substitution, and elimination. Lesson 1, Paul’s Notes–Closure states, “Write a sentence that summarizes what you have learned about solving systems of linear equations by graphing.” Lesson 2, In-Class Practice, Exercise 4 states, “Solve the system by substitution. 2x=4y-5, 2x-3y=1””. Lesson 3, In-Class Practice, Exercise 4 states, “Solve the system by elimination. x+4y=22, 4x+y=13”. In Lesson 4, Practice, Exercises 1-6, students solve a system of linear equations using any method and explain their choice of method.

• F-LE.4: Algebra 2, Chapter 6, Lesson 5, students manipulate a logarithmic formula, express the result as a logarithm, and evaluate the result using technology. In-Class Practice, Exercise 13 states, “In Example 6, the artist turns up the volume so that the intensity of the sound triples. By how many decibels does the loudness increase?” Chapter 6, Lesson 6, Standards-Based Practice, Exercises 1-3 require students to solve exponential equations and express the solutions as logarithms, which students evaluate using technology.

• G-CO.1: Geometry, Chapters 1, 2, 3, and 10, students use precise definitions based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Chapter 1, Lesson 1, In-Class Practice, Exercise 1 states, “Use the diagram in Example 1. Give two other names for line ST. Name a point that is not coplanar with points Q, S, and T.” Chapter 1, Lesson 2, Warm Up, Exercise 1, states, “Write the ordered pair that corresponds to point A.” Chapter 1, Lesson 5, In-Class Practice, Exercise 1, states, “Write three names for the angle (diagram is provided).” Chapter 3, Lesson 1, Practice, Exercise 7, states, “Identify all pairs of angles of the given type. Corresponding angles (students analyze a diagram with two intersecting lines and a transversal).” In Chapter 10, Lesson 1, In-Class Practice, Exercise 3, states, “Tell how many common tangents the circles have and draw them. State whether the tangents are external tangents or internal tangents.”

• S-CP.4: Geometry, Chapter 13, Mid-Chapter Practice Test, students construct a two-way table and answer questions using the table. Exercise 2 states, “You randomly survey students about whether they like hamburgers or hot dogs. Of 129 students who like hamburgers, 46 like hot dogs. Of 143 students who dislike hamburgers, 54 like hot dogs. Organize the results in a two-way table. Include the marginal frequencies.” Exercise 3 states, “A restaurant chain planning to expand its operations surveys people in three cities. The survey asks whether they would eat at the restaurant. The results, given as joint relative frequencies, are shown in the two-way table. What is the probability that a randomly selected person in City A will eat at the restaurant? Round your answer to the nearest tenth of a percent. The probability that a person in City A will eat at the restaurant is about ___ %.” In Chapter 13, Lesson 4, Practice, Exercise 7 gives the students the following task:  “Three different local hospitals in New York surveyed their patients. The survey asked whether the patient’s physician communicated efficiently. The results, given as joint relative frequencies, are shown in the two-way table. Determine whether being satisfied with the communication of the physician and living in Saratoga are independent events.”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. Materials intentionally develop the full intent of the modeling process throughout the series leading to culminating experiences that address all, or nearly all, of the modeling standards. 

There are multiple opportunities throughout the series for students to engage in one or more aspects of the modeling process and opportunities for students to engage in the full modeling process.

The Digital Teaching Experience, Teacher Toolkit: Course Essentials, Modeling Course Overview guides teachers in facilitating the full intent of the modeling process across course tasks and activities. The Modeling Course Overview includes prompts teachers use to engage students in the modeling process and examples from the series that illustrate the modeling process in action. The Digital Teaching Experience, Teacher Toolkit: Course Essentials, Modeling Standards Correlation identifies lessons and assessments across the series in which students engage with each modeling standard. Across the series, students engage with modeling standards in In-Class Practice Exercises, Practice Exercises, Performance Tasks, and Big Idea Tasks.

The lesson Instructional Guide includes Point-of-Use Teaching Notes, and the Supporting the Mathematical Practices: Facilitation Guide provides additional guidance for each lesson. Point-of-use notes in the Teacher Editions (Instructional Guide and Answer Guide) include prompts and questions that support student engagement in the modeling process. 

Examples where students engage in one or more aspects of the modeling process include:

• Algebra 1, Chapter 2, Lesson 5, Practice, Exercise 27 states, “You want to plan a road trip using a round-trip fuel budget of $200. You will use a car with a fuel range of 25–34 miles per gallon. Use current gasoline prices to determine the distances you can travel. Then plan the trip. Use inequalities to represent how much you will spend on gasoline, the speeds at which you can travel, and how long the trip will take.” The open-ended nature of the task prompts students to identify variables that affect fuel economy, conduct research on these variables, formulate models representing the cost of gasoline, speed of travel, and time spent traveling, and make computations with the models to plan the trip. (A-CED.1, A-REI.3)

• Algebra 2, Chapter 10, Lesson 3, In-Class Practice, Exercise 8 states, “You post an image on social media. Four of your friends repost the image, then four of each of their friends repost the image, and so on. Find the total number of people who reposted the image after the sixth round?” Students create an equation representing the relationship and perform a computation using the model. (A-SSE.4, F-IF.3,F-BF.2, F-LE.2)

Examples where students engage in the full modeling process include: 

• Algebra 1, Chapter 4, Performance Task, students write a proposal describing the design of a wind farm in their community. DESIGN A WIND FARM states, “Write a proposal to government officials for a new wind farm in your county. Include at least two advantages and two disadvantages of the wind farm. Choose the amount of land that will be occupied by your farm and the capacity of the turbines. Use functions and graphs to show the cost of the turbines and the total power generated. Relate the energy output of the wind farm to the amount of energy used in a typical household. Energy (kWh) = Power (kW) Time (h).” Students write a proposal describing the design of a wind farm for their community. Students determine the components of a government proposal, identify variables relevant to the situation, formulate algebraic and graphical models that represent these variables, and analyze the models to make decisions. Students compare multiple possible plans, interpret results, and revise models as needed. After interpreting and validating their models, students write and present their proposal for a new wind farm. (A-CED.2, F-BF.1, F-LE.2, N-Q.1)

• Geometry, Chapter 6, Performance Task, students are presented with fragments of a circular plate from an archaeological dig and determine the diameter of the original plate. RECONSTRUCTING THE PAST states, “You are an archaeologist learning about the daily lives of an ancient civilization. You discover the fragments of a circular plate shown above. Use one of the fragments and the mathematical relationships in this chapter to find the diameter of the original plate. Explain each step so that other archaeologists can replicate your method.” Students formulate a plan to determine the dimensions of the original plate using one or more of the provided fragments. Students compute the dimensions of the selected fragments and interpret the dimensions of the original plate based on information from those fragments. Students validate their results by checking whether the dimensions remain consistent when using a different fragment. Students report their findings in a form that other archaeologists can replicate. (G-MG.1)

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for, when used as designed, allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

The Digital Teaching Experience, Teacher Toolkit: Course Essentials includes a teacher-facing Pacing Guide for each course. The Pacing Guide organizes instructional time across units and lessons and prioritizes topics aligned to standards widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

Three correlation resources document alignment to the standards across the series and identify where widely applicable prerequisite (WAP) content is addressed: The Standards Correlation (by Course) identifies the standards addressed in each lesson by course component. The Standards Correlation (by Standard) lists every lesson in which a specific standard is addressed. The Standards-Based Practice Correlation connects each Standards-Based Practice activity in the digital platform or Practice Workbook to a content standard and identifies lessons where that standard is reinforced. Across courses, these resources show that the majority of lessons align to standards identified as widely applicable as prerequisites (WAPs) for college and careers.

Examples of how the materials allow students to spend the majority of their time on widely applicable prerequisites (WAPs) include, but are not limited too:

• Algebra 1, Chapter 11, Lesson 6, Practice, Exercises 10-12, students describe how graphs are misleading or misinterpreted based on scale and origin in given data displays. In Exercises 10 and 11, students describe how graphs are misleading and how the graphs are misinterpreted. In Exercise 12, students redraw the graphs so the display is not misleading, using the given data, units, and scale. (N-Q.A)

• Algebra 1, Chapter 7, Lesson 6, Practice, students factor trinomials using multiple strategies, including algebra tiles and tables, to identify factor pairs of the first and last terms that produce the middle term. In Exercises 1–9, students factor trinomials with a leading coefficient not equal to one using strategies from the lesson. In Exercise 14, students apply factoring to a contextual problem: “The Parthenon in Athens, Greece, is an ancient structure with a rectangular base. The length of the base of the Parthenon is 8 meters more than twice its width. The area of the base is about 2,170 square meters. Find the length and width of the base.” (A-SS.A)

• Algebra 1, Chapter 6, and Algebra 2, Chapters 6 and 9, students graph exponential, logarithmic, and trigonometric functions and describe features of the graphs. In Algebra 1, Chapter 6, Lesson 3, Practice Exercise 13, students graph an exponential function and describe the asymptote, domain, and range. In Algebra 2, Chapter 6, Lesson 3, Practice Exercise 33, students graph a logarithmic function and describe the domain, range, and asymptote. In Algebra 2, Chapter 9, Lesson 4, Practice Exercises 1–6, students identify the amplitude and period of given sine and cosine functions and graph each function. In Algebra 2, Chapter 9, Lesson 4, In-Class Practice, Exercises 2–3, students graph one period of given tangent functions. (F-IF.C)

• Geometry, Chapter 5, Lesson 7, Investigate, students use triangle congruence criteria to solve problems and prove relationships in geometric figures. Exercise 1 states, “The figure shows how a surveyor can measure the width of a river by making measurements on only one side of the river. a. Explain how the surveyor can find the width of the river. b. Write a proof to verify that the method you described in part (a) is valid. c. Exchange proofs with another group and discuss the reasoning used.” Chapter 8, students use triangle similarity criteria to solve problems and prove relationships in geometric figures. Lesson 2, Example 3, students write a proportion using given shadow lengths of a person and a free-fall ride to find the height of the ride. Lesson 3, Warm-Up, Exercises 1 and 2, students determine whether two polygons are similar. Chapter Review with Calcchat, Exercises 1 and 2, students find the scale factor, identify pairs of congruent angles, and write ratios of corresponding side lengths for given polygons. (G-SRT.B)

• Algebra 2, Chapter 8, Lesson 2, In-Class Practice, Exercises 2 and 3, students distinguish between parameters and statistics. Exercise 2 states, “A survey found that the median salary of 1,068 statisticians is about $95,570. Is the median salary a parameter or a statistic?” Exercise 3 states, “The median age of all Senators at the start of the 118th Congress was 65.3 years. Is the median age a parameter or a statistic?” (S-IC.A)

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for materials, when used as designed, allow students to fully learn each standard.

Each lesson begins with Engage and Explore, which includes a Warm-Up, a Launch Activity, and Investigate. Lessons continue with Key Concept examples, In-Class Practice, and Practice, and conclude with Connections to Real Life tasks. Standards-Based Practice and Lesson Extra Practice provide additional tasks aligned to the lesson content. Across the program, students complete multiple tasks that require them to demonstrate understanding of the standards.

Three correlation resources document alignment to the standards across the course. The Standards Correlation (by Course) identifies the standards addressed in each lesson by course component. The Standards Correlation (by Standard) lists every lesson in which a specific standard is addressed. The Standards-Based Practice Correlation connects each Standards-Based Practice activity in the digital platform or Practice Workbook to a content standard and identifies lessons where that standard is reinforced.

Examples of how the materials allow students to fully learn all of the non-plus standards include:

• Algebra 2, Chapter 3, Lessons 3 and 4, students solve quadratic equations with real coefficients that have complex solutions. In Lesson 3, In-Class Practice, Exercises 4–6, students solve quadratic equations with complex solutions. In Lesson 4, In-Class Practice, Exercises 3 and 4, students use the Quadratic Formula to solve quadratic equations with complex solutions. (N-CN.7)

• Algebra 1, Chapter 1, Lesson 1, Practice, Exercise 17, students use the fact that the angles of a quadrilateral sum to 360 degrees to write and solve a one-step equation to find a missing angle measure. Algebra 1, Chapter 1, Lesson 2, Practice, Exercise 8, students write and solve a two-step equation to find the number of hours spent repairing a car. In Algebra 1, Chapter 1, Lesson 4, In-Class Practice, Exercise 8, students write and solve a two-sided equation to determine the distance a boat traveled. Algebra 1, Chapter 1, Lesson 5, Practice, Exercise 22, students write and solve an absolute value equation to determine acceptable soccer ball masses. Algebra 1, Chapter 2, Lesson 2, Practice, Exercise 10, students write and solve a one-step inequality to determine possible sodium intake for the remainder of the day. Algebra 1, Chapter 2, Lesson 4, Practice, Exercise 17, students use information in a table to write and solve an inequality related to daily parking costs. Algebra 1, Chapter 2, Lesson 5, Practice, Exercise 20, students write compound inequalities to represent three elements in liquid form. Algebra 1, Chapter 2, Lesson 6, In-Class Practice, Exercise 10, students write an absolute value inequality to identify hotels within a per-player budget. Algebra 1, Chapter 6, Lesson 5, Practice, Exercise 30, students write an exponential function representing the amount of money in an account over time and determine when the account balance exceeds $200,000. Algebra 1, Chapter 9, Lesson 3, Practice, Exercise 11, students write and solve a quadratic equation to determine missing dimensions for a touch tank. Algebra 2, Chapter 7, Lesson 5, Practice, Exercise 26, students write and solve a rational equation to find the average speed of a kayaker. (A-CED.1)

• Algebra 1, Chapter 4, Lesson 6, Exercises 12–15, students write an equation for the nth term of an arithmetic sequence given the first four terms. Algebra 1, Chapter 6, Lesson 6, Practice, Exercises 10–13, students write an equation for the nth term of a geometric sequence given a list, graph, or table of the first four terms. Algebra 1, Chapter 6, Lesson 7, Practice, Exercise 23, students graph the first four terms of a sequence and write both recursive and explicit rules. Algebra 2, Chapter 10, Lesson 5, Investigate, Exercise 2, students write a recursive equation for a sequence in a ride-share context. Algebra 2, Chapter 10, Lesson 5, Practice, Exercises 12–15, students write a recursive rule given an explicit rule, and in Exercises 17–20, students write an explicit rule given a recursive rule. (F.BF.2)

• Geometry, Chapter 4, Lesson 6, Practice, Exercise 5, students prove two right triangles are similar by showing corresponding side lengths are proportional and corresponding angles are congruent. Geometry, Chapter 8, Lesson 1, Practice, Exercises 17–18, students determine whether pairs of polygons are similar by checking proportionality of corresponding sides and congruence of corresponding angles. (G.SRT.2)

• Algebra 1, Chapter 11, Lessons 2 and 3, students use statistics appropriate to the shape of a data distribution to compare the center and spread of two or more data sets. In-Class Practice, Exercise 3, students analyze a double box-and-whisker plot of surfboard prices to identify distribution shape and determine which data set shows greater spread. Practice, Exercise 12, students analyze double box-and-whisker plots of battery life data to identify distribution shape, determine the range of the upper 75 percent, compare interquartile ranges, and select a data set that meets a minimum battery life criterion. Lesson 3, Practice, Exercise 9, students compare histograms of daily high temperatures for two towns using distribution shape and appropriate measures of center and variation. (S-ID.2)

The instructional materials for Math & YOU: Concepts & Connections meet expectations for requiring students to engage in high school mathematics by focusing on problem contexts and attending to various types of real numbers.

Students work with course-level problems that reference prior mathematical knowledge. Examples, In-Class Practice, Practice, Interpreting Data, Performance Tasks, and Connecting Big Ideas include tasks situated in real-world contexts. Tasks introduce new concepts using simpler numerical values and later require students to perform operations and apply concepts using the full real number system. Across courses, tasks require students to apply concepts from Grades 6–8, including proportional relationships, systems of equations, and irrational numbers, within high school-level problems.

Examples of problems that allow students to engage in age-appropriate contexts include:

• Algebra 1, Chapter 11, Lesson 2, Example 2, students analyze a box-and-whisker plot representing song lengths in seconds from a concert. Students find and interpret the range and interquartile range and describe the distribution of the data. (S-IE.3)

• Geometry, Chapter 9, Lesson 5, In-Class Practice, Exercise 5, students use a right triangle representing a student skiing down a mountain, with the angle of depression and mountain height labeled, to find the distance from the student to the base of the mountain. (G-SRT.8)

• Algebra 2, Chapter 2, Lesson 2, Practice, Exercise 32, students write a quadratic equation to model the arch of the Gateshead Millennium Bridge, graph the function, and describe the domain and range in context. (F-IF.4, F-IF.7a)

Examples of problems that allow students to engage in the use of various types of real numbers include:

• Algebra 1, Chapter 9, STEM Performance Task, students calculate with rational and irrational values to examine the golden ratio in contextual problems. Exercise 1 states, “The golden ratio, also notated as \phi (pronounced “phi”), has long been used by artists and architects as a proportion that is visually appealing. The ratio of the width to the height of the Parthenon satisfies this ratio. Calculate the decimal approximation of \phi. Round your answer to the nearest thousandth.” Exercise 3 states, “The Golden Ratio is a solution of the quadratic equation \phi-\phi-1=0. a. Find the exact value of \phi by solving (-2\phi+1+\sqrt{5})(2\phi-1+\sqrt{5})=0, which is an equivalent form of \phi^2-\phi-1=0. b. What type of number is \phi? Why can it only be approximated by a decimal value?” (A-CED.1, A-REI.4)

• Geometry, Chapter 12, Lesson 4, Practice, Exercise 8, students determine the volume of a composite solid consisting of a rectangular prism with a cone removed. Students perform computations using rational and irrational numbers. (G-GMD.3)

• Algebra 2, Chapter 3, Lesson 4, Examples 2-3, students solve quadratic equations using the Quadratic Formula. Example 2 states, “Solve 25x^2-8x=12x-4 using the Quadratic Formula.” The equation has one real solution. Example 3 states, “Solve -x^2+4x=13 using the Quadratic Formula.” The equation has two complex solutions. (A-REI.4b, N-CN.7)

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

• Algebra 1, Chapter 9, Lesson 1, Practice, Exercise 12, students simplify a cube root expression. This exercise builds upon students’ knowledge of exponent rules from grade 8 (8.EE.1, 8.EE.2) in order to rewrite expressions involving radicals. (N-RN.2) 

• Geometry, Chapter 8, STEM Performance Task, students design a pool similar in shape to an Olympic-sized pool with a length of 30 meters. Students use proportional relationships to find the perimeter, calculate material needed for a cover, and determine the amount of chlorine needed. The task applies proportional relationships from grade 7 (7.RP.2) to problems involving similarity of figures. (G-MD.3)

• Algebra 2, Chapter 8, Connecting Big Ideas Task, students calculate probabilities using a normal distribution that represents the time it takes players to guess a word correctly in an online word game. The task applies probability concepts and measures of center and variation from grades 6 and 7 (6.SP.5, 7.SP.7) to determine probabilities in a normal distribution. (S-ID.4 )

The instructional materials for Math & YOU: Concepts & Connections meet expectations for making meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

The materials foster coherence by engaging students with related mathematical ideas across conceptual categories and across standards, clusters, and domains within a course and across multiple courses in the series. Students encounter connected concepts multiple times and apply them in different contexts, consistent with the progressions required by the Standards.

Teacher-facing materials identify where these connections occur within and across courses, including Mathematics of the Chapter, Chapter Insight Videos, the COHERENCE Through the Grades chart located at the start of each chapter, and the Standards Correlation (by Course).  They are all available in the Teacher Toolkit: Course Essentials. These resources identify how chapter-level mathematics connects to prior and future learning across lessons, chapters, and courses. In Mathematics of the Chapter, teachers review how the mathematics of the chapter fits within a broader learning progression across the course and the series. Each overview includes “What We’re Doing,” “Why We’re Doing It,” and “Essential Background,” which reference how students’ work in the chapter connects to prior and subsequent course-level mathematics. The COHERENCE Through the Grades chart explicitly identifies how chapter learning builds from previous learning and extends to future learning within and across courses. The Standards Correlation (by Course) identifies primary and secondary standards aligned to each lesson, illustrating connections among standards and conceptual categories across the series.

The materials foster coherence and make meaningful mathematical connections within a single course. For example:

• Algebra 1, Chapter 8, Lesson 4, Practice, Exercise 21, students write and graph a quadratic function that models the path of a flare shot from a boat using the vertex and an additional point, connecting algebraic modeling (A-CED.A) with interpreting and graphing functions (F-IF.B).

• Geometry, Chapter 9, Connecting Big Ideas, students use a city map to identify streets that intersect at specified angles, justify whether streets are parallel (G-GPE.5), identify similar triangles (G-SRT.B), and calculate distances using similarity (G-SRT.C), requiring students to integrate multiple geometric concepts within a single task.

• Algebra 2, Chapter 10, Lesson 3, In-Class Practice Exercise 5, students use two given terms of a geometric sequence to write a rule for the n^{th} term of the sequence (A-SSE.B, F-BF.2) and then graph the first six terms of the sequence (F-LE.A), connecting sequence structure, function rules, and graphical representations.

The materials also foster coherence and make meaningful mathematical connections across courses in the series. For example:

• Algebra 1, Chapter 4, Lesson 4, Practice, Interpreting Data, Exercise 17, students describe the relationship between two quantitative variables shown in a scatterplot (S-ID.6). In Algebra 2, Chapter 6, Lesson 1, Practice, Interpreting Data, Exercise 26, students describe relationships between quantitative variables in a different context using scatterplots (S-ID.6), applying the same type of reasoning across courses.

• Algebra 1, Chapter 4, Lesson 3, Practice, Exercises 3-6, students write equations of lines that pass through a given point and are parallel to a given line. In Practice, Exercises 9–12, students write equations of lines that are perpendicular to a given line. In Geometry, Chapter 3, Lesson 5, Practice, Exercises 7–10, students write equations of lines that are parallel and perpendicular to given lines while connecting algebraic equations to geometric relationships in the coordinate plane (G-GPE.5).

• Algebra 1, Chapter 3, Lesson 7, Practice, students transform linear functions. Exercise 17 states, “How does the value of p in the equation y=g(x)+p affect the graph of y=g(x)? How does the value of p in the equation y=g(x+p) affect the graph of y=g(x)?” Algebra 1, Chapter 3, Lesson 8, Practice, students transform absolute value functions. Exercise 24 states, “Explain how the graph of each function compares to the graph of y=x for positive and negative values of k, h, and a. a. y=\left|x\right|+k b. y=\left|x-h\right| c. y=a\left|x\right| d. y=\left|ax\right|.” Algebra 1, Chapter 10, Lesson 1, Practice, students transform square root functions. Exercise 20 states, “Explain whether the graph of g(x)=1.25\sqrt{x} is a vertical stretch or a vertical shrink of the graph of f(x)=\sqrt{x}” Algebra 1, Chapter 10, Lesson 2, In-Class Practice, students transform cubic functions. Exercises 3-5 states, “Match the function with its graph. 3. m(x)=\sqrt[3]{x-2}; 4. g(x)=\sqrt[3]{x}+2; 5. m(x)=\sqrt[3]{x}-2. (Students are given three graphs of cubic functions).” Algebra 2, Chapter 2, Lesson 1, Investigate, students transform quadratic functions. Exercises 1-6, students match six quadratic functions with their graphs and then use technology to verify. Exercise 7 states, “How do the constants a, h, and k affect the graph of the quadratic function g(x)=a(x-h)^2+k?” Algebra 2, Chapter 4, Lesson 7, In-Class Practice, students transform polynomial functions. Exercise 5 states, “Let the graph of g be a horizontal stretch by a factor of 2, followed by a translation 3 units right of the graph of f(x)=8x^3+3. Write a rule for g.” Algebra 2, Chapter 6, Lesson 4, In-Class Practice, students transform exponential and logarithmic functions. Exercise 4 states, “Describe the transformation of f represented by g. Then graph each function. 4. f(x)=e^{-x}-5.” Example 4 states,“Describe the transformation of f(x)=log_{\frac{1}{2}}x represented by g(x)=2log_{\frac{1}{2}}(x+4). Then graph each function.” (F-BF.3)

The instructional materials for Math & YOU: Concepts & Connections meet expectations for the materials to explicitly identify and build on knowledge from Grades 6-8 to the high school standards.

Lessons intentionally build from knowledge students learned in Grades 6-8 and require students to extend that prior understanding to develop new grade-level concepts aligned to high school standards. The materials explicitly identify these connections in teacher-facing resources that support instructional coherence across grade bands.

Teachers access explicit connections to Grades 6-8 standards in the COHERENCE Through the Grades resource available for each chapter in the Digital Teaching Experience and the Teacher Edition. This resource presents a learning progression that identifies how the chapter content builds from prior learning, including middle school standards, and connects to future learning within the course and across subsequent courses. Teachers use additional chapter-level resources, including the Chapter Overview and Chapter Insight Video, to support instruction that builds on students’ prior knowledge. At the lesson level, teachers reference the Instructional Guide, which identifies prerequisite skills, including those from Grades 6-8.

The materials provide multiple examples where students explicitly build on Grades 6–8 knowledge to meet high school expectations. 

Examples include: 

• Algebra 1, Chapter 4, Lesson 1, Practice, Exercises 4-6, students extend their understanding of equations written in the form y=mx+b as defining linear functions (8.F.3) to write equations of linear functions in slope-intercept form from a graph (F-BF.1, F-LE.2).

• Geometry, Chapter 12, Lesson 6, Practice, Exercises 4-5, students apply their understanding of volume and surface area formulas (7.G.6, 8.G.9) to model a real-world scenario and solve problems involving density (G-MG.2).

• Algebra 2, Chapter 8, Lesson 1, Practice, Exercise 8, students build on their understanding of describing data distributions using measures of center and variability (6.SP.5) to analyze a normal distribution and determine probabilities based on a given mean and standard deviation (S-ID.4).

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for having assessment information included in the materials to indicate which standards are assessed.

CCSS standards are identified on the digital assessments for each item on the following formal assessments: Chapter Performance Task, Big Idea Task, Mid-Chapter Tests, Chapter Test, Multi-Chapter Test, and End-of-Course Test. The print materials do not always identify the CCSS or Practices; however, the problems on the print assessments are identical to the problems in the digital assessments. Two correlation resources demonstrate assessment items’ alignment to the standards. The Assessment Correlation (by Course) is organized by assessment and identifies the standards addressed on each assessment. The Assessment Correlation (by Standard) lists every assessment item in which a specific standard is addressed. 

The Digital Teaching Experience and the Teacher Toolkit: Course Essentials identify the Standards for Mathematical Practice (SMPs) for Big Idea Tasks and Chapter Performance Tasks; however, the materials do not identify the SMPs consistently for each item on other formal assessments.

Examples include:

• Algebra 1, Chapter 7, Chapter Test, Exercise 7 states, “Factor the polynomial. x^2-12x+36= ____.” (A-SSE.2, A-SSE.3)

• Geometry, Chapter 4, Big Ideas Task, Exercise 5 states, “Use the diagram. (There is a line that contains points A, B, and C (from left to right). Point Z lies above the line.) a. Dilate the line by a scale factor using the center of dilation Z. Locate the images of points A, B, and C. b. What appears to be the relationship between the lengths A’B’ and AB? B’C’ and BC? Check your prediction by measuring the segments. c. Repeat parts (a) and (b) using different scale factor. What do you notice? Make a conclusion about how the scale factor affects the lengths of the segments in the diagram.” (G-SRT.1, MP1, MP8)

• Algebra 2, End of Course Test, Exercise 52 states, “A junior class consisting of 326 students voted on whether they want the winter dance to have a glow-in-the-dark theme or a disco theme. In a random sample of 64 juniors, 77% are in favor of a disco theme. Which of the following statements is true? a. The sample size is large enough to produce valid results, so it is likely that the winter dance will have a disco theme. b. The sample size is not large enough to produce valid results, so it is likely that the winter dance will have a glow-in-the-dark theme. c. The sample size is not large enough to produce valid results, so you cannot draw a valid conclusion about the theme of the winter dance.” (S-IC.3, S-IC.6, S.ID.4)