High School - Gateway 1

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

• N-CN.1,2,7: The introduction of rational exponents is done in Algebra 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 Algebra 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 Algebra 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 Algebra 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 Algebra 2, Units 3 and 4. In Lesson 3.1, students add and subtract polynomials algebraically and graphically while also making and testing conjectures about the sum and difference of polynomials. In Lesson 3.2, students multiply polynomials using area models and traditional algebraic methods. Students divide polynomials using long division in Lesson 3.3 and use the Remainder Theorem to determine if a divisor is a factor of a polynomial. In Lesson 4.3, 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 4.4, students find real and complex imaginary roots of polynomials and write the polynomials in factored form.

• F-IF.3: In Algebra 1, Lesson 2.1, students work with arithmetic and geometric sequences including discrete and continuous linear and exponential situations. In Algebra 1, Lesson 2.2, students connect context with domain and use the domain to distinguish between discrete and continuous functions. In Algebra 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.

• G-MG.1: In Geometry, Lesson 8.3, Retrieval, Ready, Set, Go, problems 10-11, students use a model to find the total surface area and volume of the Washington Monument. In Geometry, Lesson 8.4, students model how they would determine the volume of a nail.

• S-CP.A: In Geometry, Lesson 9.3 students use samples to estimate probabilities. In Geometry, Lesson 9.5, students examine independence of events using two-way tables; and in Geometry, Lesson 9.6 students use data in various representations to determine independence.

The materials reviewed for Open Up High School Mathematics Traditional 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. Tasks that involve modeling include a graphic of the modeling process in the teacher notes. Additionally, 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:

• In Algebra 1, Lesson 3.1, students sketch a graph given steps that Sylvia used to clean and refill her pool. Students complete the problem by responding to provided questions (F-IF.4). 

• In Algebra 1, Lesson 8.3, students interpret a graph detailing Michelle’s bike ride to and from a lake. Students create a function to model the situation and identify key characteristics of the function (F-IF.4, F-IF.7b). 

• In Geometry, Lessons 4.10 and 4.11, students solve real-world problems that include angles of elevation, angles of depression, and right angles (G-SRT.8).

• In Geometry, Lesson 6.8, students “deepen their understanding of volume formulas.” Students discuss why the formula for the volume of a cone is one-third the volume of a prism and compare the two volumes (G-GMD.3). 

• In Algebra 2, Lesson 1.2, students solidify their understanding of an inverse function. Students engage with mathematical models that represent relationships amongst car length, speed, and braking distance (F-BF.4). 

• In Algebra 2, Lesson 9.2, students analyze and determine a “good” score on the ACT given information about the mean and standard deviation. Students answer analysis questions that are provided (S-ID.A).

The materials reviewed for Open Up High School Mathematics Traditional 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 the ways the materials allow students to spend the majority of their time on the WAPs include:

• A-SSE: Throughout the AGA 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 Algebra 1, Lessons 7.3-7.6, students factor and rewrite expressions as directed in A-SSE.2. In Geometry, Lessons 7.5, students complete the square to reveal its center and radius (A-SSE.3b). In Algebra 2, 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.

• G-CO.9: In the Geometry course, students work with content related to standard G-CO.9. In Geometry Lesson 3.3, students develop proofs that show that points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment. In Geometry, Lesson 3.7, students prove that vertical angles are equal. In Geometry, Lessons 3.4-3.7, 5.1, and 7.3, students have additional opportunities to engage with the content of this standard.

• G-SRT: In Geometry, Lessons 4.1 and 4.2 students build upon their understanding of dilations developed in Grades 7 and 8 to solidify their understanding of dilations and how triangles are similar. In Geometry, Lesson 4.3 students use the AA, SSS, and SAS Similarity Theorems to prove that triangles are similar. In Geometry, Lesson 4.5 students practice applying theorems about lines, angles and proportional relationships. In Geometry, Lesson 4.7 students develop a new proof of the Pythagorean theorem based on similar triangles. In Geometry, Lesson 4.8 students apply prior understandings about similar triangles to develop the definitions of the trigonometric ratios. In Geometry, Lessons 4.10 and 4.11 students continue to work with right triangles, trigonometric relationships and methods for finding missing angles and sides in right triangles and applied problems. 

• S-ID.7: Students work with standard S-ID.7 in the Algebra I course. In Algebra I, Lesson 9.2, students use data sets to create scatterplots and determine the line of best fit. They examine different attributes of the graph of the line of best fit and determine if a linear model is appropriate. In Algebra I, Lesson 9.3, students interpret data based on the line of best fit, compare data sets, and make conjectures based on graphs of data sets and lines of best fit.

• S-IC.1: In Algebra 2, Lessons 9.5 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 9.6 and 9.8-12 as students work with confidence intervals and determine whether their results are statistically significant.

The materials reviewed for Open Up High School Mathematics Traditional 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 Algebra 1, Lesson 7.4, students find the square of a binomial expression, recognize perfect square trinomials, and create perfect squares from partial areas. In Algebra 1, Lesson 7.7, Problems 9-11, students engage with the difference of squares. In Geometry, Lesson 6.6, Retrieval, Ready, Set, Go, problems 1-5, students apply the Pythagorean Theorem to find the unknown lengths in figures. In Algebra 2, Lesson 4.5, 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 Algebra 1, Lesson 2.9, Retrieval, Ready, Set, Go, problem 7, students calculate the average rate of change given two tables. In Algebra 1, Lesson 2.9, 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: In Algebra 1, Lesson 8.7, students sketch graphs of piecewise-defined and absolute value functions. In Algebra 1, Lesson 8.7, Retrieval, Ready, Set, Go, problems 19-20, students graph cube root functions. The materials provide a limited number of problems for students to graph cube root functions.

• F-TF.8: In Geometry, Lesson 4.9, Problem 14, students reason about the Pythagorean identity. In Algebra 2, 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 Traditional series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age appropriate contexts, apply key takeaways from Grades 6-8, and vary the types of real numbers being used. 

Examples of the materials using age-appropriate contexts include:

• In Geometry, Lesson 8.3, students find the volume of a frustum (created by rotating a trapezoid around the y-axis) and approximate the volume of a vase by replacing the curved edges of the vase diagram with segments. Teachers have students share several different strategies for approximating the volume (G-GMD.1,4).

• In Algebra 2, Lesson 7.1, students use the information from Ferris Wheel tasks in the previous unit to develop strategies for transforming the functions to represent different initial starting positions for the rider. Students focus on horizontal translations and may recognize that either sine or cosine functions can be used with an appropriate horizontal shift (F-BF.3, F-TF.5).

• In Algebra 2, Lesson 8.2, students sketch the shape of given graphs and justify their sketches. These functions combine linear, quadratic, absolute value, and trigonometric functions. While doing this, students design plans for a new amusement park ride (F-BF.1b).

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

• In Algebra 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 Geometry, Lesson 4.1, 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 Algebra 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 Algebra 2, Lesson 3.4, students write the equation of given graphs of parabolas in vertex, standard, and factored forms. Students use irrational numbers and the radical form of i to write the factored form of equations (A-REI.4a,b). In Algebra 2, Lesson 3.5, students learn about i and calculate solutions for given quadratic equations that have both real and imaginary roots (N-CN.1,7).

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. Overall, the materials provide tasks in similar contexts throughout the series, so students can make connections to previous and future learning. The practice problems in Retrieval, Ready, Set, Go revisit topics in a spiral manner for students to maintain skills throughout the series. 

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

• In Algebra 1, Lesson 6.1, students describe a growing pattern that represents a quadratic function. They build upon interpreting expressions and writing recursive and explicit equations from Unit 1 (A-SSE.1, F-BF.1) to develop the idea that quadratic functions show linear rates of change. In Unit 6, students write equations to represent quadratic relationships (A-CED.2). 

• In Geometry, Lesson 4.9, students develop relationships between the sine and cosine of complementary angles and justify whether given conjectures are true or false; and, in Problems 8-16, students use right triangles to justify if certain conjectures related to trigonometric identities are true (F-TF.8). In Geometry, Lesson 4.10, students solve for unknown angles and side measurements in a right triangle. In Geometry, Lesson 4.11, students solve applied and mathematical problems using all the concepts and skills developed by the previous tasks. 

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

• In Algebra 1, Lessons 1.4 and 6.3, students analyze the pattern of push-ups that Scott will include in his workout. Students examine tables, graphs, and recursive and explicit formulas that show how the constant difference is represented in different ways, define the function as an arithmetic sequence, and recognize that a quadratic function is a model for the sum of the linear function. In Algebra 2, Lesson 4.1, students revisit Scott’s workout and develop understanding related to the degree of a polynomial function and the overall rate of change. Students use multiple representations to arrive at this understanding (A-CED.1,2, F-IF.4,5, F-BF.1, F-LE.1,2,3,5).

• In Geometry, Lessons 7.7 and 7.8, students define a parabola geometrically using the focus and directrix. In Lesson 7.8, students connect this to quadratic functions and parabolas, which were addressed in Algebra 1, Units 6 and 7 (Functions and Algebra conceptual categories) (G-GPE.2). The concepts are further connected in Algebra 2, Lessons 3.4 and 3.5, where students discover a need for complex solutions to quadratic equations and define the imaginary unit (A-REI.4, N-RN.3, N-CN.A).

The materials reviewed for Open Up High School Mathematics Traditional 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 Algebra 1, Lesson 2.6, students build on 8.EE.1, where they applied the properties of integer exponents, and their beginning work with fractional exponents from Lesson 2.4. Students rewrite expressions using the properties of exponents.

• In Algebra 1, Lesson 9.1, students extend their prior learning from 8.SP.1 and 8.SP.2, constructing scatter plots and informally fitting lines to data. Students plot data sets, calculate the correlation coefficient, and learn to use it as a measure of the strength and direction of a linear relationship.

• In Geometry, Lesson 4.3, students build on their experience with 8.G.4, where they applied a sequence of transformations to determine the similarity of two figures, and with 7.G.1, where they solved problems involving scale drawings of geometric figures. Students extend this understanding to develop a new definition of similarity for polygons. 

• In Geometry, Lesson 4.7, students extend their prior learning from 8.G.6 and 8.G.7, where they engaged with the Pythagorean Theorem and used it to determine unknown side lengths of right triangles, and from 7.RP.2c and 7.RP.3, where they wrote proportionality statements based on similar triangles. In Lesson 4.7, students prove the Pythagorean Theorem in two different ways algebraically and use similar right triangles and the Pythagorean Theorem to develop right triangle trigonometric ratios.

• In Algebra 2, Lesson 3.3, students build on their experience with factoring and dividing whole numbers (6.NS.1) as they learn the long division process for polynomials and use the quotient and remainder to write multiplication statements that are equivalent to the dividend.

• In Algebra 2, Lesson 3.5, students extend their prior learning from 8.NS.1 and 8.NS.2, where they were introduced to the irrational numbers and approximated their values. Students revisit irrational numbers and the entire set of real numbers to contrast with imaginary numbers.