High School - Gateway 1

The materials reviewed for Big Ideas Learning AGA meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The materials include a few instances where all aspects of the non-plus standards are not addressed across the courses of the series. 

The following are examples from the Student Edition (unless otherwise noted) for which the materials attend to the full intent of the standard:

• N-RN.3: In Algebra 1, Chapter 9, Section 1, Explore It, students experiment with the sums and products of irrational and rational numbers. Students use their findings to explain why the sum or product of two rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

• A-CED.2: In Algebra 1, Chapter 3, Section 5, Explore It, students create equations in two variables to represent the number of child tickets and the number of adult tickets sold for a charity. Then, students graph their equation on a coordinate grid and interpret the intercepts. 

• A-REI.4a: In Algebra 1, Chapter 9, Section 4, students use completing the square to transform quadratic equations into an equation of the form $${{(x-p)}^{2}}=q$$. In Algebra 2, Chapter 3, Section 3, students derive the quadratic formula from the form $${{(x-p)}^{2}}=q$$.

• F-IF.7c: In Algebra 2, Chapter 4, Section 1, students graph polynomials and describe end behaviors. In Algebra 2, Chapter 4, Section 5, students identify zeros when suitable factorizations are available. 

• F-BF.3: In Algebra 1, Chapter 3, Section 7, students identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k. In Algebra 1, Chapter 4, Section 8, students recognize even and odd functions from their graphs. In Algebra 2, Chapter 4, Section 8, students recognize even and odd functions from their algebraic expressions.

• G-SRT.4: In Geometry, Chapter 8, Section 4, students prove a line parallel to one side of a triangle divides the other two proportionally. Within the same lesson, students prove the converse of the Triangle Proportionality Theorem. In Geometry, Chapter 9, Section 1, students prove the Pythagorean Theorem using triangle similarity. 

• G-GPE.1: In Geometry, Chapter 10, Section 7, Explore It, students derive the equation of a circle using the Pythagorean Theorem. Within the same lesson, students complete the square to find the center and radius of a circle given an equation. 

• S-ID.6a: In Algebra 1, Chapter 4, Section 4, students create linear models to fit a given data set and use the models to solve problems. In Algebra 1, Chapter 6, Section 3, students use technology to create an exponential function to fit a data set about the consumption of bottled water. Students then use the function to estimate the amount of bottled water consumed in 2022. In Algebra 1, Chapter 9, Section 2, students use technology to create a quadratic function representing the number of students with the flu after a break from school. Students then use the function to solve problems. 

The materials attend to some aspects, but not all, of the following standards:

• F-LE.1a: In Algebra 1, Chapter 3, Section 6, Explore It, students create tables to determine if the rate of change is constant, and students explain how a constant rate of change represents a linear equation. In Algebra 1, Chapter 6, Section 3, the materials state, “As the independent variable x changes by a constant amount, the dependent variable y is multiplied by a constant factor, which means consecutive y-values form equivalent ratios.” The materials do not provide opportunities to prove linear functions grow by equal differences over equal intervals or prove exponential functions grow by equal factors over equal intervals. 

• F-TF.8: In Algebra 2, Chapter 10, Section 7, the materials state the Pythagorean identities but the identities are not proven within the materials. Within the same section, students use the Pythagorean identity to find $$sin{\theta }$$, $$cos{\theta }$$, or $$tan{\theta }$$. 

• G-CO.2: In Geometry, Teacher Edition, Chapter 4, Section 1, teachers describe transformations as functions that take points in the plane as inputs and give other points as outputs. In Geometry, Chapter 4, Section 1, Explore It, students translate figures using technology. In Geometry, Chapter 4, Section 3, Explore It, students use technology to perform a 90° rotation, a 180° rotation, and a 270° rotation. The materials do not provide opportunities to compare transformations that preserve distance and angle to those that do not. 

• G-CO.6: In Geometry, Chapter 4, Section 4, students use the definition of congruence in terms of rigid motions to decide if figures are congruent. In Geometry, Chapter 5, College and Career Readiness, students perform a composition of transformations to map one figure onto a second figure. The materials do not provide opportunities to predict the effect of a given rigid motion on a given figure. 

• G-SRT.2: In Geometry, Chapter 4, Section 6, Practice, students use the definition of similarity in terms of similarity transformations to decide if two figures are similar. In Geometry, Chapter 8, Section 1, the materials state, “Because the ratio of corresponding lengths of similar polygons equals the scale factor, $${\frac {AB'} {AB}}={\frac {DE} {AB}}$$.” There is no evidence of explaining using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding sides. 

• G-C.5: In Geometry, Chapter 11, Section 1, the materials define the radian measure of the angle as the constant of proportionality. In Geometry, Chapter 11, Section 2, Explore It, students derive the formula for the area of a sector. In Geometry, Chapter 11, Section 1, the materials state, “In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.” There is no evidence of using similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the radius.

The materials reviewed for Big Ideas Learning 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 how the materials allow students to spend the majority of their time on the WAPs include: 

• N-RN.1: In Algebra 1, Chapter 7, Section 2, students use the meaning of rational exponents to describe and correct an error in rewriting the expression $${{({\sqrt[{3}] {2}})}^{4}}$$ in rational exponent form. Within the same section, students, “Explain how extending the properties of integer exponents to rational exponents allows you to express radicals in terms of rational expressions.” In Algebra 2, Chapter 5, Section 1, students apply understanding rational exponents by identifying which of four expressions written in radical notation or rational exponent notation does not belong. 

• A-SSE.1b: In Algebra 1, Chapter 1, Section 2, the materials provide two methods to solve multi-step equations. One method uses the distributive property to solve the equation while the other interprets the expression in parenthesis as a single quantity. In Algebra 1, Preparing for Chapter 7, students consider a launched rocket modeled by the expression -16t(t-13). Students complete a table of values using -16t and t-13 as separate quantities. Students reflect on the helpfulness of viewing an expression as a product of individual factors versus a single object. In Algebra 2, Chapter 6, Section 1, students consider the compound interest formula $${{A=P(1+r)}^{t}}$$. Students interpret quantities P and $${{(1+r)}^{t}}$$ before determining whether one quantity depends on the other. 

• F-IF.6: In Algebra 1, Chapter 8, Section 6, students calculate and interpret the average rate of change of a function over a specified interval for quadratic and exponential functions. In Algebra 1, Chapter 10, Section 1, students calculate and interpret the average rate of change of square root functions. In Algebra 1, Chapter 10, Section 2, students calculate and interpret the average rate of change of cube root functions. In Algebra 2, Chapter 10, Section 4 and Section 5, students calculate and interpret the average rate of change of trigonometric functions. 

• G-CO.10: In Geometry, Chapter 5, Section 1, students prove the sum of the measures of the interior angles is 180 degrees, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles, and the acute angles of a right triangle are complementary. In Geometry, Chapter 6, Section 4, students prove base angles of isosceles triangles are congruent and triangles are equiangular if and only if triangles are equilateral. In Geometry, Chapter 6, Section 4, students prove the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length. 

• S-ID.7: In Algebra 1, Chapter 4, Section 5, students use technology to find an equation for a line of best fit modeling the grade point averages and number of hours spent watching television each week for several students. Students interpret the slope and y-intercept of the equation for the line of best fit. In Algebra 2, Chapter 1, Section 3, students interpret the slope and y-intercept of the equation for a line of best fit modeling the number of active users on a social media site since 2009.

The materials reviewed for Big Ideas Learning AGA meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards. The materials foster coherence through meaningful mathematical connections in a single course and throughout the series.

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

• In Algebra 1, Chapter 3, Section 1 and Section 4, students develop a formal definition for functions (F-IF.1). In Algebra 1, Chapter 8, Section 1, students build on their understanding of functions to interpret key features of graphs and tables representing functions (F-IF.4).

• In Geometry, Chapter 8, Section 2, students develop the Angle-Angle criterion to prove triangles are similar (G-SRT.3). In Geometry, Chapter 9, Section 4 and Section 5, students build upon similar triangles by establishing that side ratios in right triangles are properties of the angles in the triangle leading to the definitions of trigonometric ratios (G-SRT.6).

• In Algebra 2, Chapter 4, Section 4, students use the structure of polynomials to rewrite the polynomial in factored form (A-SSE.2). In Algebra 2, Chapter 6, Section 5, students use the structure of logarithmic functions to rewrite the functions (A-SSE.2). In Algebra 2, Chapter 11, Section 3 and Section 4, students build on their understanding of rewriting expressions to develop the formula for the sum of a finite geometric series (A-SSE.4). 

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

• In Algebra 1, Chapter 7, Lesson 4, students calculate the solutions of polynomials. Within the same lesson students calculate roots of quadratic equations and explain the meaning of the solutions. In Algebra 1, Chapter 9, Lesson 5, students write quadratic equations with no real solutions (A-REI.4b). In Algebra 2, Chapter 3, Lesson 3, students solve quadratic equations by using square roots, completing the square, or factoring (N-CN.7). In Algebra 2, Chapter 3, Lesson 3, students’ build upon understanding of solving quadratic equations with real and complex roots by writing functions in vertex form to identify characteristics of the function (F-IF.8).

• In Algebra 1, Chapter 1, Section 1, students explain steps for solving one-step linear equations. In Algebra 1, Chapter 1, Section 2, students explain steps for solving multi-step equations. In Algebra 1, Chapter 1, Section 5, students explain steps for solving linear equations with variables on two sides. In Geometry, Chapter 2, Section 4, students explain steps for solving linear equations. In Algebra 2, Chapter 5, Section 4, students explain steps for solving radical equations (A-REI.1).

• In Algebra 1, Chapter 3, Section 6, and in Chapter 8, Section 1, students interpret key features of functions (F-IF.4). In Algebra 2, Chapter 2, Section 1 through Section 3, students build on functions by graphing polynomial functions (F-IF.7c). Finally, in Algebra 2, Chapter 2, Section 4, students calculate the average rate of change in polynomial functions (F-IF.6).

The materials reviewed for Big Ideas Learning AGA meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. Throughout the series, the materials provide a Chapter Progression chart located at the beginning of each chapter in the Teaching Edition. The “Coherence Through the Grades” chart provides identification of prior learning from middle school, current learning within the chapter, and connections to future learning. The online Teacher Edition materials contain the standards for prior learning, current learning, and future learning while the physical Teacher Edition materials do not contain the standard name. 

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

• In middle school, students analyze and solve pairs of simultaneous linear equations (8.EE.8). In Algebra 1, Chapter 5, Section 1 through Section 5, students solve systems of linear equations both graphically and algebraically (A-REI.6). In Algebra 1, Chapter 5, Section 5, students use properties of equality to prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other equation produces a system with the same solution (A-REI.5).

• Students build upon their knowledge of properties of integer (8.EE.1) exponents when writing equivalent expressions involving integer exponents. In Algebra 1, Chapter 6, Section 1, students use properties of integer exponents and begin writing expressions with rational exponents as radical expression (N-RN.2). In Algebra 2, Chapter 5, Section 1 and Section 2, students extend their knowledge of properties of integer exponents to include rational exponents (N-RN.1).

• In Geometry, Chapter 3, Section 2, students prove relationships between pairs of angles formed when parallel lines are cut by a transversal (G-CO.9) which extends their knowledge of the angles created when parallel lines are cut by a transversal (8.G.5).