High School - Gateway 1

The instructional materials reviewed for HMH Into AGA meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials address all aspects for most of the non-plus standards across the courses of the series. Examples of non-plus standards addressed by the series include:

• A-SSE.1a: In Algebra 1, Lesson 2.1 defines terms, factors and coefficients. In the three tasks, students explore what variables, terms, and expressions represent in situations. For example, in Task 2, students explain the reason 44 - a is an expression for the weight of grapes. In Algebra 2, Lesson 4.4, students write an equation for volume and identify the real-world meaning of each factor in an example using the volume of ice sculptures.
• A-REI.4b: In Algebra 1, Lessons 17.2 and 17.3, students solve quadratic equations by factoring; in Lesson 18.1, students solve quadratic equations by inspection with square roots; in Lesson 18.2, students complete the square; and in Lesson 18.3, students use the quadratic formula. In Algebra 2, Lesson 2.3, students solve quadratic equations using completing the square and the quadratic formula and find the value of the discriminant to tell the number and type of solutions. They also solve quadratic equations with complex roots in this same lesson.
• F-BF.4: In Algebra 2, Lesson 7.1, students find the inverse of a function. In Task 3, students explore the relationship between the distance an image is sent from an orbiting spacecraft and the number of seconds since the message has been sent. Students see the inverse of the function D(s) is s(D).
• F-IF.2: Students use function notation throughout the materials. In Algebra 1, Lesson 4.1, students use multiple versions of function notation to become accustomed to the variety of forms it can take. Students use h(t), g(x), R(b), A(t) and S(p) in various tasks to meaningfully represent the problems they are addressing.
• F-IF.4: In Algebra 1, Lesson 19.2, students determine the axis of symmetry, vertex, maximum or minimum, and intercepts. In Lesson 19.3, students use quadratic functions situated in various real-world contexts to identify and interpret x-intercepts and vertices. In Lesson 20.4, students use a graphing calculator to find x-intercepts, end behavior, turning points, and intervals over which cubic functions are positive and negative. In Algebra 2, Lesson 16.3, students use the key features of periodicity and maximum and minimum to write trigonometric functions. 
• G-CO.12: Compass and straightedge constructions are included throughout the Geometry materials. In Lesson 1.1, students construct a midpoint. In Lesson 1.2, students copy and bisect an angle. In Lesson 3.2, students construct parallel lines through a point. In Lesson 3.3, students construct a perpendicular bisector and a perpendicular line through a point.
• G-SRT.6: In Geometry, Lesson 13.1, students analyze the relationships of opposite and adjacent side lengths of two right triangles that have the same acute angle. They also use tracing paper to investigate the relationships of opposite and adjacent side lengths of two right triangles that have the same acute angle, defining the tangent ratio. In Lesson 13.2, students use drawing tools to create right triangles, examine side ratios, and discover the sine and cosine ratios.
• S-ID.8: In Algebra 1, Lesson 6.2, students use a spreadsheet to create a table of data and residuals. They also use a graphing calculator to find the equation of the line of best fit and correlation coefficient, which is interpreted for linear models. 

The non-plus standards that are not addressed, or partially addressed, include:

• G-GPE.2: This standard was completely omitted from the materials.
• S-IC.4: The materials do not develop a margin of error through the use of simulation models for random sampling.

The instructional materials reviewed for HMH Into AGA 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:

• N-RN.1: In Algebra 1, Lesson 1.2, Task 3, students discover that the properties of integer exponents can be extended to rational exponents. In Algebra 2, Lesson 6.1, Task 1, students extend the definition of integer exponents to define an exponent in terms of a root. In Step it Out of the same lesson, students translate between rational exponents and radical expressions.
• A-SSE.A: In Algebra 1, Lesson 2.1, students write, interpret, and simplify linear expressions in one variable.
• A-SSE.B: In Algebra 1, Lessons 19.1-19.3, students write quadratic equations in different forms to find x-intercepts, maxima, minima, and vertices. In Algebra 2, Lesson 2.3, students complete the square to find complex solutions.
• A-CED.A: In Algebra 1, Lessons 2.2 and 2.4, students solve linear equations with grouping symbols and variables on both sides. In Algebra 2, Lesson 2.1, students write and solve quadratic equations.
• A-REI.7: In Algebra 1, Lesson 20.3, Task 1, students use a graphing calculator to translate y = 2x and y = x$$^2$$ to determine linear-quadratic systems with 2, 1, and 0 solutions. In the Check for Understanding of the same lesson, students use a graphing calculator to solve a system that includes a linear equation and a parabola. In Algebra 2, Lesson 2.4, students solve linear and quadratic systems using substitution, elimination, and graphs.
• G-CO.1:  In Geometry, definitions for lines, angles, and circles are introduced in the Build Understanding of Lessons 1.1, 1.2, 2.1, 3.3, and 15.1.
• G-SRT.B: In Geometry, students prove theorems including the triangle proportionality theorem in Lesson 12.3 and the triangle congruence theorem in Lesson 12.2.

The instructional materials reviewed for HMH Into AGA meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from grades 6-8.

Examples where the materials use age-appropriate contexts include:

• In Algebra 1, Lesson 1.3, Problem 38, students determine the average rate of speed when going down a zipline, given the vertical and horizontal distances.
• In Algebra 1, Lesson 18.3, Journal and Practice Workbook, Problem 9, students create and solve a quadratic function to determine how long after a football player kicks a football to a returner does the returner catch the ball. 
• In Geometry, Lesson 15.4, Problem 31, students find the equation of a ferris wheel given the diameter. Students also explain if a given point value can be possible for one of the cars of the ferris wheel to be attached to the wheel.
• In Geometry, Lesson 13.1, Journal and Practice Workbook, Problem 8, students use right triangle trigonometry to find the measure of an angle located at a shed formed by the ground and the line to the top of a cell phone tower. 
• In Algebra 2, Lesson 7.3, On Your Own, Problem 29, students describe the overall rate of change in the context given a square root function that models the number of likes received on a video after d days.
• In Algebra 2, Lesson 14.1, On Your Own, Problem 27, students write a recursive and explicit formula that gives the loan balance after n months and the amount of the monthly payment for a loan at a car dealership.

Examples where the instructional materials use various types of real numbers include:

• In Algebra 1, Module 18, students solve many types of quadratic equations that result in whole, rational, and irrational solutions. 
• In Algebra 1, Lesson 4.3, students identify the zeros and extreme values of functions. Problems 18, 19, 23, 24, and 29 have fractional values for these features.
• In Geometry, Module 14, students solve missing sides and angles using Law of Sines and Cosines that result in integer and rational numbers.
• In Geometry, Lesson 4.2, students write the equation of a line perpendicular to a line passing through a given point. Problems 8-10 have slopes and y-intercepts that are fractions. 
• In Algebra 2, Lesson 5.2, students solve polynomial equations with integer, rational, irrational, and imaginary roots.

Examples where students apply key takeaways from Grades 6-8 include:

• In Algebra 1, Lesson 4.3, students answer questions regarding key features of graphs from graphs of functions, which applies knowledge from Grade 8 Functions (8.F).
• In Algebra 1, Lessons 22.1-22.2, students create dot plots, box plots, and histograms. Students apply concepts and skills of basic statistics and probability from 6.SP.4.
• In Geometry, Lesson 6.2, students apply understanding of congruence and similarity through translations, rotations, reflections, and dilations (8.G) to learn about compositions with rigid motions.
• In Geometry, Lesson 12.3, students find the distance between two streets using proportional relationships, which applies key takeaways from 7.RP.2.
• In Algebra 2, Lesson 15.1, students find the relationship between arc length and radius by applying ratio and proportional relationships, applying learning from Ratios and Proportional Relationships (RP) in Grades 6 and 7.
• In Algebra 2, Unit 5, Project Epoxy Proxy, Problem 2, students write rational expressions for the total weight of various mixtures. Students apply Ratios and Proportional Relationships (RP) from Grades 6 and 7.

The instructional materials reviewed for HMH Into AGA meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials foster coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the standards. Each Module in the Teacher Edition has Teaching For Success that identifies Mathematical Progressions with bulleted lists of prior learning, current development, and future connections. Each lesson has Mathematical Progressions for teachers, which is more specific and targeted. In Prior Learning, grade levels and lessons are identified, without standards. In Future Connections, specific lessons within the course and/or series are identified.

Examples where the materials foster coherence within courses include:

• In Algebra 1, Lesson 2.2, students create and solve linear equations in real-world context (A-CED.1). In Lesson 3.1, students create and graph linear equations in two variables (A-CED.2). In Lessons 9.1-9.4, students solve systems of linear equations through graphing, substitution and elimination (A-REI.5).
• In Algebra 1, Lesson 4.2, students write and graph linear functions (F-IF.7a). In Lesson 9.1, students write and graphically solve systems of linear equations (A-REI.6).
• In Geometry, Unit 3, Lessons 5.1-5.3, students use descriptions of rigid motions to transform figures (translate, rotate, and reflect) on a coordinate plane. In Unit 4, Lessons 7.1-7.3, students continue to transform functions on a coordinate plane and use the definition of congruence in terms of rigid motions (G-CO.5,7).
• In Geometry, Module 12, students use the definition of similarity to decide if two figures are similar (G-SRT.2). In Lessons 13.1 and 13.2, students use the information they have learned about similar triangles to discover the tangent, sine, and cosine ratios in right triangles (G-SRT.6). Students also use trigonometric ratios in Lesson 19.2 to develop a formula for the volume of a cone (G-GMD.1).
• In Algebra 2, Lessons 7.3 and 7.4, students graph square root and cube root functions and analyze the graphs in real-world contexts (F-IF.7b). In Lesson 7.5, students solve radical equations and graph the equations using graphing calculators (A-REI.2).
• In Algebra 2, Lesson 2.3, students use completing the square and the quadratic formula to find complex solutions to quadratic equations (N-CN.7). The materials connect this to previous learning in Algebra 2, Lesson 1.3, by graphing and transforming quadratic functions (F-BF.3). Students further build on their knowledge of complex solutions by finding real and complex solutions to higher order polynomials in Algebra 2, Lesson 5.1 (A-APR.3).

Examples where the materials foster coherence across courses include:

• In Algebra 1, Lesson 5.2, students transform linear functions, including translations, reflections, and stretches or compressions (F-BF.3). In Geometry, Module 5, students define and apply translations, rotations, reflections, and symmetry. In Geometry, Lesson 6.1, students define and apply dilations, stretches, and compressions (G-CO.2, G-SRT.1). In Algebra 2, Lesson 1.3, students explore transformations of functions, including translations, stretches, compressions, reflections, and combined transformations (F-BF.3). The Build Understanding section of the lesson reminds students that in previous courses they performed transformations of figures and function graphs.
• In Algebra 1, Lesson 2.2, students solve multi-step equations. In Geometry, Lesson 3.2, students solve multi-step equations from diagrams to find the value of x that makes two lines parallel (A-REI.3).
• In Geometry, Lessons 13.1 and 13.2, students define trigonometric functions in terms of the ratio of sides in a right triangle (G-SRT.8). In Algebra 2, Lesson 15.1, students understand the radian measure of an angle as the length of the arc on the unit circle subtended by the angle (F-TF.1).

The instructional materials reviewed for HMH Into AGA meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials explicitly identify content from Grades 6-8 and support the progressions of the high school standards.

The materials explicitly identify content from Grades 6-8 by using language directly from prior, grade-level standards in Mathematics Progressions Prior Learning. The materials identify the grade level of the prior learning and the unit where the prior learning is found. The prior grade level and unit are noted in parentheses, and examples include:

• In Algebra 1, Lesson 4.2, Mathematical Progressions Prior Learning, “Students: Identified linear relationships graphically.” (Gr8, 6.3 and 6.4) 
• In Algebra 1, Lesson 6.1, Mathematical Progressions Prior Learning, “Students: Have created scatter plots with bivariate data and examined association between variables. (Gr8) Have identified linear association in bivariate data, informally fit a line to data, and informally assessed the fit of the line of the data. (Gr8) Have used a linear model of bivariate data to solve problems, interpreting the slope and intercepts of the linear equation.” (Gr8) 
• In Geometry, Lesson 4.1, Mathematical Progressions Prior Learning, “Students: Explained why the slope is the same between any two distinct points on a nonvertical line in the coordinate plane.” (Gr8, 5.1)
• In Geometry, Lesson 17.1, Mathematical Progressions Prior Learning, “Students: Knew the formulas for circumference and area of a circle and used them to solve problems.” (Gr7, 10.1 and 10.2) 
• In Algebra 2, Lesson 2.1, Mathematical Progressions Prior Learning, “Students: Used square root symbols to represent solutions of equations of the form x$$^2$$ = p. “ (Gr8, 3.1)
• In Algebra 2, Lesson 11.1, Mathematical Progressions Prior Learning, “Students: studied linear proportional relationships.” (Gr8, 5.1)

Examples where the materials make connections between Grades 6-8 and high school concepts and allow students to extend their previous knowledge include:

• In Algebra 1, Lesson 9.3, (A-REI.C) students extend 8.EE.8 as they analyze and solve systems of linear equations using the elimination method and estimate or solve pairs of simultaneous linear equations by inspection. 
• In Algebra 1, Lesson 10.22, students extend their prior learning from 6.SP.A, understanding a set of data can be described by its center, spread, and overall shape, to S.ID.3, interpreting differences in shape, center, and spread in the context of the data sets.
• In Geometry, Lesson 1.4, students use knowledge of the pythagorean theorem (8.G.8) to justify the distance formula. They use this to find the perimeter/area of polygons (G-GPE.7) on the coordinate plane.
• In Geometry, Lesson 3.1, students prove theorems about parallel lines cut by a transversal (G-CO.9). This extends from 8.G.5 where students use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal.
• In Algebra 2, Lesson 1.1, students discover there are constraints on domain and range in function models (F-IF.5). This is an extension of 8.F.1 where students understand that a function is a rule that assigns each input to exactly one output.
• In Algebra 2, Lesson 2.3, students find the value of the discriminant and solve quadratic equations using the quadratic formula (A-REI.4). This is an extension of 8.EE.2 where students learn to solve equations of the form x$$^2$$ = p.