High School - Gateway 1

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Examples of standards that are attended to fully by the materials include:

• N-CN.1,2,7: The introduction of rational exponents is done in Math 1, Lesson 2.4 as students analyze conjectures about how rational numbers between whole number data points are approximated to develop a continuous exponential function from a discrete geometric sequence. In Math 2, Lesson 3.5, students are formally introduced to complex numbers and operations with complex numbers as they relate to solutions to quadratic equations. 

• A-SSE.3: In Math 2, Lesson 3.7, students verify different forms of a quadratic expression to solve a given equation. Students explain how the factored form helps to reveal the zeros and what that means in the context of the question. In Math 1, Lesson 2.6, students are guided through an exploration of how expressions with different rational exponents are equivalent yet highlight different mathematical properties.

• A-APR.1-3: These standards are addressed in Math 3, Lessons 3.3-3.8 and Lesson 3.10. In Lesson 3.3, students add and subtract polynomials algebraically and graphically while also making and testing conjectures about the sum and difference of polynomials. In Lesson 3.4, students multiply polynomials using area models and traditional algebraic methods. Students divide polynomials using long division in Lesson 3.5 and use the Remainder Theorem to determine if a divisor is a factor of a polynomial. In Lesson 3.7, students investigate the relationship between roots, zeros, and x-intercepts using cubic functions. Students write cubic functions in factored form in order to identify the roots. In Lesson 3.8, students find real and complex imaginary roots of polynomials and write the polynomials in factored form. 

• F-IF.3: In Math 1, Lesson 2.1, students work with arithmetic and geometric sequences including discrete and continuous linear and exponential situations. In Math 1, Lesson 2.2 students connect context with domain and use the domain to distinguish between discrete and continuous functions. In Math 1, Lesson 2.3, students name functions based on identifying the change over equal intervals to prove that the function is either linear or exponential.

• F-IF.7: In Math 1, Lesson 2.5, students apply their understanding of negative exponents to identify key features of the graphs of exponential functions. In Math 2, Lessons 4.1-4.4 students interpret and create graphs of piecewise functions then connect their understanding of piecewise functions to linear absolute value functions. In Math 3, Lesson 6.4, students make connections between an equation that models the height of a rider on a Ferris wheel to the amplitude, period, and midline of the graph of the function. 

• G-MG.1: In Math 3, Lesson 5.3, Retrieval Ready, Set, Go, Problems 10-11, students use a model to find the total surface area and volume of the Washington Monument. 

• S-CP.A: In Math 2, Lessons 10.1 and 10.2 students use samples to estimate probabilities. In Math 2, Lesson 10.5, students examine independence of events using two-way tables, and in Math 2, Lesson 10.6 students use data in various representations to determine independence.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The materials provide opportunities for students to engage in the modeling process. Additionally, all the modeling standards are addressed in the materials.

Examples where students engage in some, or all, aspects of the modeling process with prompts or scaffolding from the materials include, but are not limited to:

• Math 1, Lesson 1.2, “Growing Dots,” addresses standards F-BF.1 and F-LE.1, 2, and 5. The students describe a given pattern and predict how the pattern would change after 3 minutes, 100 minutes, and “t” minutes. The teacher notes prompt the teacher to ask students to share out specific strategies and solution paths. While the teacher notes are scripted and prompt the teacher to seek out specific strategies, the problem leaves students open to any approach they find logical. The teacher notes place equal value in any of the possible student strategies and encourage students to analyze and discuss the varying strategies.

• Math 1 Lesson 4.2 “Elvira’s Equation,” addresses standards A-SSE.1, A-CED.4, and A-REI.3. Students use notes from the manager of the cafeteria to fill out a chart related to a large number of aspects of the daily lunch process. Students must give each item measured a variable notation, describe the quantity being measured (pizza) and the unit to be used (a slice). When the table is complete students write equations that model questions that the manager needs answered. 

• Math 2, Lesson 1.3, “Scott’s Macho March,” addresses standards F-BF.1, F-LE.A, A-CED.1 and 2, and F-IF.4 and 5. Details about the number of push-ups Scott completes a day are provided, and students interpret the information, formulate a strategy, and compute their answers. Students extrapolate how the pattern will continue into the future as they are looking at the sum of the number of pushups that Scott has completed on a particular day. The teacher notes provide instructions for teachers to have students share out their answers, interpret what their answers mean in context, and evaluate each other’s answers and strategies. 

• Math 2, Lesson 4.1 “Going to Pieces,” addresses standards F-IF.2 and F-IF.5-8. Students are given a piecewise function graph that purports to show the route of a pizza delivery car labeled only with ordered pairs at the end of each piece. Students must interpret the function related to the various activities of the driver, (traveling, sitting still, etc), then write a piecewise description of what they observed and a function to define it, including domains. Students also make observations about fuel efficiency related to the “traveling” sections of the graph.

• Math 3, Lesson 5.4, “You Nailed It,” addresses standards G-MG.1, 2, and 3. In this task, the students find the volume of an individual nail in order to estimate buying nails by the pound. Students use that information to calculate how many nails would be necessary for a particular building project and the cost of the nails. Students design a deck that will minimize the cost of materials, write an argument for the cost of materials, and defend their decisions to the whole class. 

• Math 3, Lesson 7.3, “Getting on the Right Wavelength” addresses standards F-TF.5 and F-BF.3. Based on a given picture of a ferris wheel along with a few details, students write equations to model the height of the rider at any given time and make predictions about how the wheel will behave in the future.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for, when used as designed, spending the majority of time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs and careers. Examples of how the materials allow students to spend a majority of their time on the WAPs include:

• A-SSE: Throughout the series, students engage with content related to the widely applicable prerequisites in A-SSE. Work related to the content standards found in this domain can be found in each course as both focus standards and supporting standards. For example, in Math 1, Lesson 2.10, students interpret the expression 2(n-1)as it relates to a given pattern. In Math 2, Lessons 2.3-2.6, students factor and rewrite expressions as directed in A-SSE.2. In Math 2, Lesson 9.2, students complete the square to reveal its center and radius (A-SSE.3b). In Math 3, Lesson 2.6, students interpret complicated expressions by viewing one or more of their parts as a single entity (A-SSE.1b). Students routinely factor polynomials of different powers in order to highlight different aspects of the function both algebraically and graphically.

• F-IF: Across the series, students engage with content related to the widely applicable prerequisites from this domain. For example, in Math 1, Lesson 1.7, students develop recursive and explicit functions and use them to find values of different terms in the sequence. In Math 1, Lesson 3.5, students match functions represented graphically, verbally, numerically, and analytically using cards containing important information about the functions. The information includes domain, range, values of functions at certain input values, comparisons of two functions, rate of change analysis, intercepts, and information about increasing, decreasing, and maximums. In Math 2, Lesson 2.9, students graph functions showing or describing key features such as the vertex, line of symmetry, intercepts, and transformations. In Math 3, Lesson 8.2, students reason and predict what graphs of functions have been built by combining functions of different types through addition and multiplication. Students graph the functions and identify key features of the graph.

• G-CO.9: In Math 2, students work with content related to standard G-CO.9. In Math 2 Lesson 5.5, students develop proofs that show that points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment. In Math 2, Lesson 5.7, students prove that vertical angles are equal. 

• G-SRT: In Math 2, Lesson 6.2 students build upon their understanding of dilations developed in 7th and 8th grade to solidify their understanding of dilations and how triangles are similar. In Math 2, Lesson 6.3 students use the AA, SSS, and SAS Similarity Theorems to prove that triangles are similar. In Math 2, Lesson 6.5 students practice applying theorems about lines, angles and proportional relationships. In Math 2, Lesson 6.7 students apply prior understandings about similar triangles to develop the definitions of the trigonometric ratios. In Math 2, Lessons 6.8 to 6.10 students continue to work with right triangles, trigonometric relationships and methods for finding missing angles and sides in right triangles and applied problems.

• S-IC.1: In Math 3, Lessons 9.6 and 9.8-9.12, students compare different sampling methods and types of studies and address what kinds of conclusions can be reached when using the different types of studies. Students form conclusions given specific sets of data. This standard serves as a supporting standard for Lessons 9.8-9.12 as students work with confidence intervals and determine whether their results are statistically significant.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for, when used as designed, letting students fully learn each non-plus standard. In general, students would fully learn most of the non-plus standards when using the materials as designed.

The non-plus standards that would not be fully learned by students across the series include:

• A-APR.4: In Math 2, Lesson 6.6, Retrieval Ready, Set, Go Problems 1-5, students apply the Pythagorean Theorem to find the unknown lengths in figures. In Math 2, Lesson 7.7, Problems 9-11, students engage with the difference of squares. In Math 3, Lesson 3.9, Retrieval Ready, Set, Go, Problems 9-12, students engage with the sums and differences of cubes. Students do not use proven polynomial identities to describe numerical relationships.

• F-IF.6: In Math 1, Lesson 2.9, Retrieval, Ready, Set, Go, Problems 3-7, students use both linear and exponential functions to calculate the average rate of change and, in Problems 10-17, students use a given exponential graph and student-generated equations to calculate the average rate of change. The materials have limited opportunities for students to estimate the average rate of change from a graph.

• F-IF.7b: Students are given limited opportunities to graph square root, cube root, and cubic functions by hand or using technology throughout the series. Work with cubic equations is limited to Math 3, Lesson 3.2 in which students graph cubics, check their graphs with technology, and compare cubic graphs to quadratic graphs.

• F-TF.8: In Math 2, Lesson 6.8, Problem 14, students reason about the Pythagorean identity. In Math 3, Lesson 7.5, Problem 3, students use a right triangle to show that the same Pythagorean identity is true for all acute angles. Students have a limited number of opportunities to learn how to find the trigonometric value of angles in all of the four quadrants using the Pythagorean identity.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. Students engage in investigations throughout each task that utilize real-world contexts appropriate for high school use. 

Examples of the materials using age-appropriate contexts include:

• In Math 1, Lesson 9.3, students analyze the Census Bureau’s income data to understand more about the differences between women’s and men’s salaries. Based on the data in this task and Lesson 9.2, “Making More $”, students make a case to support whether the difference in income may be explained by differences in education or discrimination and consider what other data would be useful. (S-ID.6-8)

• In Math 2, Lesson 10.3, students use data from a restaurant to predict how much food to prepare in order to avoid too much waste by computing conditional probabilities and applying the addition rule. (S-CP.1,7)

• In Math 3, Lesson 6.5, students continue to use the ideas, strategies, and representations discovered when completing the Ferris wheel tasks from the previous lessons. Students describe the periodic motion of the rider’s shadow on the Ferris wheel as the shadow moves back and forth across the ground when the sun is directly overhead. Students apply the cosine function to determine the distance horizontally from the center of the wheel and derive the function horizontal position of the shadow =25 cos (18x) . (F-IF.7e, F-TF.2,5)

Examples of the materials applying the key takeaways from Grades 6-8 include:

• In Math 1, Lesson 3.4, students use a given graph of two functions to answer questions regarding key features of the graph, and students interpret some of the key features. This is an application of a key takeaway from Grades 6-8 in applying basic function concepts to develop/solidify new understanding in this unit (A-APR.1, A-CED.3, A-REI.11, F-IF.7).

• In Math 2, Lesson 6.1, students consider a scenario where an employee at a copy center is enlarging a photo for a customer and makes a mistake. Students answer questions to determine what the mistake was and how the employee should have enlarged the photo. Students apply a key takeaway from Grades 6-8 regarding similar figures (G-SRT.1).

Examples of the materials using various types of real numbers include:

• In Math 1, Lesson 2.6, students verify that the properties of integer exponents also apply to rational exponents. Students use exponent rules to write equivalent forms of expressions involving rational exponents and rational bases. Expressions include rational numbers in the base as well as in exponents (N-RN.1,2, A-SSE.3).

• In Math 2, Lesson 1.6, students distinguish between relationships that are quadratic, linear, exponential or neither. The materials include relationships presented with tables, graphs, equations, visuals, and story context. Students also create a second representation for the relationships given. Graphing technology is recommended for this task. The contexts provided and the numbers used in the equations and graphs are all appropriate for high school students. (F-IF.4,5,9, F-LE.A) 

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. Overall, the materials include connections that are intentional and thoughtful as the tasks are reexamined so that familiar mathematical situations are viewed with a new level of sophistication. The sequence of the materials is designed to spiral concepts throughout the entire series.

Examples of the materials fostering coherence through meaningful mathematical connections in a single course include:

• In Math 1, Lessons 2.1-2.3, students analyze and build linear functions to model different scenarios. They represent the functions numerically, graphically, and analytically and focus on looking for the constant rate of change in linear functions. In Math 1 Lessons 9.1-9.5, students analyze bivariate data represented in tables and scatter plots. They apply what they learned about linear relationships earlier in Math 1 to identify which variables may have linear relationships and interpret the meaning of the slope and y-intercept of linear models. Students also change data to get the correlation coefficient to reach a given value by either making the data more or less linear. This allows them to use what they know about constant rates of change and linear functions. Additionally, students reference residual plots to see what characteristics may indicate linear or non-linear relationships. (F-BF.1, F-LE.1a-c,2, S-ID.6a-c,7,8)

• In Math 2, Lesson 3.1, Students solve quadratic equations by applying knowledge of quadratic function behavior developed in Math 2, Units 1 and 2. Students discover that factoring a quadratic expression and completing the square are ways to not only find zeros and the vertex of a function, but to find solutions to a quadratic equation as well. In Math 2, Lesson 3.2, students build on the knowledge from Lesson 3.1 to derive the quadratic formula. In Math 2, Lesson 3.3, students synthesize learning from the first two lessons to solve a system consisting of a quadratic and linear equation. Students apply algebraic methods to solve the system and then show the solution graphically. In the remaining lessons of Math 2, Unit 3, students solve quadratic equations that result in both irrational and complex solutions. (N-CN.7, A-SSE.3a,b, A-CED.1,4, A-REI.4,7,10)

Examples of the materials fostering coherence through meaningful mathematical connections between courses include:

• In Math 1, Lesson 1.4, “Scott’s Push-Ups'' students analyze the pattern of push-ups Scott will include in his workout. Students examine tables, graphs, and recursive and explicit formulas that focus on how the constant difference is represented in different ways and define the function as an arithmetic sequence. In Math 2, Lesson 1.3, “Scott’s Muscle March” students revisit Scott’s workout, but this time his push-up pattern creates a quadratic model. Again, students use algebraic, numeric, and graphical representation to represent a story with a visual model. In Math 3, Lesson 3.1, “Scott’s March Motivation” students develop an understanding of how the degree of a polynomial determines the overall rate of change. (A-CED.1,2, F-IF.4,5, F-BF.1, F-LE.1,2,3,5)

• In Math 3, Lesson 6.6, “Diggin’ It” students discover alternative ways of measuring a central angle of a circle: in degrees, as a fraction of a complete rotation, or in radians. Students use right triangle trigonometry to find the coordinates of points on a circle and use the relationship between arc length measurements and radian angle measurements all within the context of an archeological dig. This task builds upon what students learned in Math 2, Lesson 8.4 where students encountered the idea that the length of an arc intercepted by an angle is proportional to the radius and defined the radian measure of the angle as the constant of proportionality. In Math 3, Lesson 6.7, “Staking It,” students’ previous understanding of radians as the ratio of the length of an intercepted arc to the radius of the circle on which that arc lies and uses radian measurement as a proportionality constant in computations. (F-TF.1,2, G-C.5)

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The materials explicitly identify the standards from Grades 6-8 in the Progression of Learning section of the teacher materials. This information appears routinely in the design of the teacher materials but not in the student materials.

Examples where the teacher materials explicitly identify standards from Grades 6-8 and build on them include, but are not limited to:

• In Math 1, Lesson 1.1, the Progression of Learning states, “Beginning in grade 7 (7.EE.A.2), students have used variables to describe a changing quantity.” Students create algebraic expressions to model patterns and identify different parts of the expression in terms of what those parts represent in the problem (A-SSE.1) which builds on work students did in Grade 7 rewriting expressions to shed light on problems and how quantities are related (7.EE.2).

• In Math 1, Lesson 3.1, students make connections to the key features of graphs of functions listed in F-IF.4 by connecting the features to a situation using the water level of a pool over a period of time. The Progression of Learning references that this lesson builds on students’ experience with functions in Grade 8 (8.F.1-5) by expanding the concepts to different functions and developing the key features to use as tools for analysis in future lessons. 

• In Math 1, Lesson 4.1, the Progression of Learning states, “Students were introduced to solving one- or two-step equations related to the context of a word problem in grade 7 mathematics (7.EE.B.4). Students have also learned how to interpret the order of operations when evaluating an expression in grade 6 (6.EE.A.2), and will need to draw upon that understanding in today’s lesson." Students build on that knowledge by developing strategies to solve multi-step equations. 

• In Math 2, Lesson 6.6, the Progression of Learning identifies 8.G.6 and 8.G.7 as students’ first encounter the Pythagorean theorem. It also mentions students’ prior work with solving proportional statements in Grade 7 (7.RP.2c, 7.RP.3). Students build on this prior knowledge in this lesson as they develop right triangle trigonometric ratios. The application of the previous understandings of the Pythagorean theorem continues into Math 3, Lesson 7.5 as students derive and justify the Pythagorean identity.

• In Math 2, Lesson 10.1, the Progression of Learning states, “In grade 7, students learned to develop probability models based on data and to find the probability of compound events with tables, tree diagrams and simulations (7.SP.6, 7.SP.7a,b, 7.SP.8a, b, c). This lesson builds on students’ experience with using tree diagrams to find probabilities to introduce conditional probability.” Students write and interpret conditional probability statements in the context of medical testing.