High School - Gateway 2

The instructional materials for enVisionMath A/G/A meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Overall, the instructional materials develop conceptual understanding throughout the series as well as provide opportunities for students to demonstrate conceptual understanding independently throughout the series.

Some examples across the series that develop conceptual understanding and present students with opportunities to independently demonstrate conceptual understanding include:

• A-REI.A: In Algebra 1 Lesson 1-2, students create and solve simple linear equations. Students evaluate various methods for solving linear equations, including using Algebra Tiles, and determine which operations are needed to solve a variety of problems. In Algebra 2 Topic 4, students solve rational equations. Students analyze and critique various methods and check for extraneous solutions. Further, in Lesson 5-4, students solve equations with exponents and radicals. Students look for relationships between square roots and squaring and solve algebraically and graphically.
• N-RN.1: In Algebra 1 Lesson 6-1, there is an explanation of why a power of 1/2 must be equivalent to the square root of the number. Then in Algebra 2 Lesson 5-1, a similar explanation is provided for a fourth root and for a rational exponent of 2/3. In Lesson 6-1, problems 20, 22 and 23, students explain concepts related to this standard. In Algebra 2 Lesson 5-1, problems 4, 6, 20 and 21, students also explain problems aligned to this standard.
• A-APR.2: Algebra 2 Lesson 3-4, example 3 provides an explanation of why the Remainder Theorem is true. In exercises in the lesson, students explain their reasoning involving the Factor Theorem.
• S-ID.7: In Algebra 1 Topic 2, students understand that linear equations can be written in three forms. Students also develop an understanding that choosing a form for writing a linear equation depends on given information, and equivalent forms can be obtained using the properties of equality. Students further their understanding of linear equations by interpreting the meaning of the slope and y-intercept of each form used in the context of the problem posed.
• A-APR.B: Students examine the relationship between factors and zeros in Algebra 1 Lesson 9-2. Students factor quadratic expressions to find the solutions of quadratic equations, and students further develop their understanding of the relationship between zeros and factors by finding factors when zeros of a quadratic function are given. In Algebra 2 Lesson 3-5, students extend their understanding of the relationship between factors and zeros to higher order polynomials.
• G-SRT.2: In Geometry Lesson 7-2, the lesson starts with examining examples of student work. The questions posed for the teacher in the teacher edition promote reasoning and problem solving, such as, “What is preserved with different types of transformations?” and “How might you use side length to help you determine whether there is a composition of transformations that maps one figure to the other?” Through these questions, students develop an understanding of a similarity transformation, which is the essential question for lesson 2. Also, in Lesson 7-2, there are questions for the teacher to help students develop an understanding of the connection between congruence and similarity.

The instructional materials for enVisionMath A/G/A meet expectations that the materials provide intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Opportunities for students to independently demonstrate procedural skills across the series are included in each lesson. An additional resource is available to provide problems for extra practice.

Some examples that show the development of procedural skills across the series include:

• A-APR.6: In Algebra 2 Lesson 3-4, students divide polynomials using long division and synthetic division. Students are also provided additional practice in Lessons 4-2 and 4-3.
• A-APR.1: In Algebra 1 Lesson 7-1, there are examples of simplifying, adding, and subtracting polynomials. The practice and problem solving set contains several practice problems for this skill. The next lesson provides examples of multiplying polynomials and has many problems for students to practice.
• A-REI.6: In Algebra 1 Lessons 4-1 and 4-2 and Algebra 2 Lessons 1-6 and 1-7, students work with systems of linear equations. Students solve systems of linear equations exactly and approximately.
• F-BF.3: Students transform different types of functions. For example, in Algebra 1 Lesson 3-3, students transform linear functions, then piecewise and absolute value functions in Lesson 5-4. In lesson 6-5, students transform exponential functions, and in Lessons 8-1 and 8-2, students work with transformations of quadratic functions. In Algebra 2 Lesson 3-7, students extend their understanding of transformations to higher degree polynomials, and in Lesson 5-4, they transform square root functions. In Lesson 6-4, students have the opportunity to use their understanding of transformations to work with logarithms, and in Lesson 7-6, students continue their work with transformations of trigonometric functions.
• G-GPE.4: In Geometry Lessons 9-2 and 9-3, students plan a coordinate geometry proof, prove theorems using coordinate geometry, derive the equation for a circle in the coordinate plane, and write equations for and graph circles.
• G-GPE.7: In Geometry Lesson 9-1, students use coordinate geometry to classify triangles and quadrilaterals. Students solve problems with polygons on the coordinate plane. Students use the distance formula, the midpoint formula, and the slope formulas to analyze polygons. Students are given multiple opportunities to compute perimeters and areas of polygons.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematics throughout the series. Additionally, the instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts.

The materials provide multiple opportunities for students to engage in application of mathematics throughout the series. For example:

• G-SRT.8: In Geometry Lessons 8-1 and 8-2, students use the Pythagorean Theorem and trigonometric ratios to solve many different types of problems. In each lesson, there is one application problem intended to be solved by groups of students and several application problems intended to be solved by individual students. Lesson 8-5 primarily consists of application problems related to angles of elevation and depression in tandem with trigonometric ratios.
• F-IF.7a,c: Students solve application problems that involve the graphs of various types of functions. In Algebra 1 Lesson 2-1, students solve problems involving graphs of linear equations, and in lesson 9-1, students solve application problems involving graphs of quadratic equations.
• F-IF.7e,9: In Algebra 2 Lessons 6-1 and 6-4, students create and solve exponential and logarithmic application problems. In Lesson 6-1 example 3 and practice problems 12 and 24, students solve problems involving growth rates of populations. In Lesson 6-4, example 5 and practice problem 25, students solve problems using logarithmic functions to approximate the altitudes of airplanes.

Some examples that include opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts include:

• A-REI.4 In Algebra 2 Lesson 2-5, students solve a quadratic equation and identify the maximum and minimum values of the function. In example 4, students enclose a pasture in the shape of a rectangle with parameters on fencing materials and square footage available. In Practice problem 42, students find the length and width of a skate park given parameters on perimeter and area.
• A-REI.11: Algebra 1 Topic 9, students solve linear and quadratic systems of equations using methods of substitution, elimination, and graphing. Contexts for students to explain why the intersection of the two functions are solutions include comparing cell phone sales, costs of ropes course facilities, and the number of individuals who prefer rock climbing to zip-lining.

The instructional materials for enVisionMath A/G/A meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All three aspects of rigor are present independently throughout the program materials. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Each topic in this series includes a Topic Opener, STEM Project, and Mathematical Modeling in 3 Acts (which relates to the Topic Opener). Each lesson includes: Explore and Reason, Understand and Apply (which guides students through examples and problems to try on their own), Concept Summary, Practice and Problem Solving, and a formative quiz to determine understanding and mastery. This structure of the materials lends itself to balancing the three aspects of rigor.

The following are examples of balancing the three aspects of rigor in the instructional materials:

• Algebra 1 Topic 3 addresses Linear Functions. In the STEM Project, students explore how recycling can offset carbon dioxide production. Students use linear functions to determine recycling rates by planning a recycling drive at their school to increase the amount of trash that gets recycled. In the first three lessons, students develop procedural skill in using function notation, evaluating functions, graphing the lines described by functions, and graphing translations through applications, tables and graphs. In the Modeling in 3 Acts problem, students find a strategy for picking a checkout lane in the grocery store. In lesson 3-6, students apply linear functions through representing arithmetic sequences, determining a linear function from a scatter plot, and analyzing trend lines. Students use linear functions to solve real-world problems, such as the time to download a given file size and the number of hybrid cars sold in America over 16 years.
• Geometry Topic 7 addresses similarity transformations, similar triangles, and proportional relationships in triangles. In the STEM Project, students use similarity to find the dimensions of an engine part while given the dimensions of a model of the part. Students calculate key values related to the 3D printing of the part and describe steps for its production. Students extend their conceptual understanding of transformations to include dilations and develop the understanding that two figures are similar if a similarity transformation occurs. Students develop skill in identifying a series of transformations used in mappings, and application problems (such as comparing blueprints to actual measurements, working with a surveying device to determine the distance from the lens to the target, and constructing supports for a roof) integrate all aspects of rigor. In the Modeling in 3 Acts problem, students make scale models of a building project.
• Algebra 2 Topic 2 addresses extending understanding of quadratic functions. In the STEM Project, students explore how the design of a ballpark influences the number and frequency of home runs. Throughout the seven lessons, students develop the conceptual understanding that all quadratic functions are transformations of the parent function. Students develop procedural skill in factoring quadratic expressions and solving quadratic equations through factoring. There are many opportunities to apply the understanding of quadratic equations to real life as seen in the 3 Acts where students develop a conjecture to model kicking a soccer ball into a goal. Students also interpret key features of the graph of a quadratic function in terms of the context, which includes describing projectile motion, maximizing space of a rectangular patio, and determining maximum profits for a bike manufacturer.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. Overall, MP1 and MP6 are used to enrich the mathematical content and demonstrate the full intent of these mathematical practices across the series. The mathematical practices are identified in both the teacher and student editions.

Some examples of where and how the materials use MP1 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

• In Algebra 1 Lesson 4-1, problem 35 poses a question about maximizing volume. Students relate the height to the radius and write a formula to meet the needs of the manufacturer.
• In Algebra 1 Lesson 5-3, students use step functions to make sense of why the functions appear to be different given two scenarios. Students determine which graph is correct and why.
• In Algebra 1 Lesson 6-5, students describe two ways to identify how an exponential function is transformed.
• In Algebra 2 Lesson 7-5, students determine a function that models the height of a triangle to construct a hexagonal floor of a treehouse. Students make sense of the shape of the triangle to fit the floor pattern.

In some places, identification of MP1 does not reflect the intent of the mathematical practice as seen in the following examples:

• In Geometry Lesson 4-5, problem 21, the materials indicate students are making sense and persevering in identifying congruent triangles; however, the problem states that students use the Hypotenuse-Leg Theorem.
• In Geometry Lesson 5-1, problem 22, the materials indicate students are making sense and persevering in finding the perimeter of a garden using perpendicular bisectors; however, students are given a diagram including dimensions which reduces students' making sense of how to determine the amount of fencing needed.

Some examples of where and how the materials use MP6 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• Many of the problem sets contain a problem titled “Communicate Precisely,” which asks students to write a clear explanation. For example, in Algebra 1 lesson 5-1, problem 2, students compare the domain and range of g(x) = a|x| and f(x) = |x|. In Algebra 2 lesson 6-1, problem 4, students compare and contrast exponential growth and exponential decay. Both of these examples require students to be precise in their words.
• In Algebra 1 Lesson 3-4, students explore arithmetic sequences. In problem 48, students are given the number of rows at an outdoor concert and and that two more chairs in each additional row. Students precisely write a recursive formula, an explicit formula, and graph the sequence for setting up rows of chairs at the concert.
• In Geometry Lesson 9-2, students justify reasoning within proofs and are precise in the calculation of needed values.
• In Algebra 2 Lesson 11-6, students communicate precisely by using data and statistical measures to support or reject a hypothesis.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards.

The majority of the time MP2 and MP3 are used to enrich the mathematical content. There is an intentional development of MP2 and MP3 that reaches the full intent of the MPs. There are many examples in the instructional materials of MPs 2 and 3 where students are asked to reason abstractly and quantitatively and to critique a solution to determine if it is correct or to find the mistake. Every lesson has at least one error analysis problem, and there are many occasions throughout the topics where students are asked to construct an argument to support their answer.

Some examples of where and how the materials use MP2 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1 Topic 5, Mathematical Modeling in 3 Acts, students are presented with a video showing a person running on uneven terrain. Students generate a graph that matches the presented situation. Students rely on their understanding of concepts related to nonlinear functions to develop a representative model. Students engage in abstract and quantitative reasoning to identify constraints which will affect the graph of the scenario. Students compare the speed of a runner going uphill, downhill, and on a flat surface.
• In Geometry Lesson 1-3, students reason abstractly to represent real-world scenarios using points on the coordinate plan and calculate the distance and midpoint between those points.
• In Geometry Lesson 3-4, students use symbolic rules for rigid motion. Students also explain the relationships between the rules and the figures which involve application of the rules.
• In Geometry Lesson 5-5, students reason abstractly by using triangles to represent real-world situations. Students also interpret their solutions in terms of the original context.
• In Algebra 2 Lesson 2-7, problem 26, students are given an equation to model the height of a football and an equation to model the height of a blocker’s hands. Students determine if it is possible for the blocker to knock down the ball and what other information is needed. Students must decontextualize the situation in order to think about the function needed to represent the situation and solve the equations as a system. Students must also contextualize the situation in order to interpret both the equations and the solution in context.

Some examples of where and how the materials use MP3 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• Algebra 1 Lesson 2-2 shows two different people writing a linear equation based on a point and a slope, and each person uses a different method. Students determine if the two equations represent the same line and provide a mathematical argument to support their answer. In this problem, students critique the reasoning of someone else by examining the approach each person used. Students construct a viable argument to explain their conclusion about whether or not the two equations represent the same line. In lesson 3-3, problem 12, students examine work with a mistake in it and describe how to correct the mistake.
• In Geometry Lesson 6-4, problem 11, students find the error in an argument of the proof that a quadrilateral with one pair of congruent sides and one pair of parallel sides is a parallelogram.
• In Geometry Lesson 7-4, problem 14, students construct a viable argument by proving the altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
• In Algebra 2 Lesson 2-4, problem 10, students agree or disagree with a statement about the result of raising the number i to an integer power and explain their thinking. In Lesson 2-7, problem 7 presents two possible methods to solve a linear-quadratic system. Students decide which method is correct. In Lesson 2-7, problem 10, students construct a viable argument for why a linear-quadratic system cannot have more than two solutions.
• In Algebra 2 Lesson 9-2, example 5, students examine a system of equations and answer the following question from the teacher edition: “Which of the three methods for solving a system of equations would you be least likely to use? Explain.” Students construct an argument for which method is not possible.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards. The majority of the time, MP7 and MP8 are used to enrich the mathematical content and to reach the full intent of the MPs.

Some examples of where and how the materials use MP7 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1 Topic 2, students learn that linear equations in different forms are composed of several parts that can be manipulated to reveal the slope and intercepts of the equations. Students apply their previous understandings of parallel and perpendicular lines to identify methods for writing the equation of a line parallel or perpendicular to a given line.
• In Algebra 1 Topic 8, students see the standard form of a quadratic function as composed of several objects, including values of a, b, and c that can be used to graph the intercepts, axis of symmetry, and vertex of the parabola that represents the function. Students look at the structure of the vertical motion model and relate it to the standard form of a quadratic equation.
• In Geometry Topic 3, Lesson 3-5, students use the structure of a symmetric figure to determine that it can be mapped onto itself after a reflection, rotation, or series of rotations.
• In Geometry Topic 7, Lesson 7-1, students use the structure of similar triangles to understand relationships in triangles.
• In Algebra 2 Topic 1, Lesson 1-4 students use patterns in arithmetic sequences and series to find the common difference of a series and write both recursive and explicit definitions for each sequence.
• In Algebra 2 Topic 2, Lesson 2-1, students use the structure of a quadratic equation to rewrite an equation from standard form to vertex form.

Some examples of where and how the materials use MP8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1 Topic 3, students graph transformations of linear functions by multiplying or adding specific values of k to the input or output of a function. Students make generalizations to recognize that multiplying the output of a linear function by k scales its graph vertically.
• In Algebra 1 Topic 8, Lesson 8-1, students use repeated reasoning to make generalizations about the effects of how changing the values of a function affects the graph of that function.
• In Geometry Topic 1, Lesson 1-4, students look for regularity in repeated reasoning when they identify patterns and rules for sequences and look for general rules to define the sequences.
• In Geometry Topic 5, Lesson 5-3, problem 6, students express regularity in repeated reasoning when investigating where the orthocenter is located for any right triangle.
• In Algebra 2 Topic 3, Lesson 3-1, students generalize how the sign of the leading coefficient and the degree of a polynomial affect the end behavior of a graph of a polynomial function. In lesson 2, students generalize the procedures used to add, subtract, and multiply polynomials and recognize whether or not the operations are closed for polynomials. In lesson 5, students generalize the relationship between the multiplicity of zeros and the appearance of the graph of a polynomial function.
• In Algebra 2 Topic 7, Lesson 7-1, students use regularity in repeated reasoning to determine that in isosceles right triangles, trigonometric co-functions are equivalent.