High School - Gateway 2

The materials reviewed for enVisionMath A/G/A meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. Overall, the materials develop conceptual understanding throughout the series as well as provide opportunities for students to demonstrate conceptual understanding independently throughout the series. 

Examples across the series that develop conceptual understanding  include: 

• A-REI.A: In Algebra 1, Topic 1, Lesson 1-2, students create and solve simple linear equations using various methods, including Algebra Tiles, while determining which operations are needed in the process. In Algebra 2, Topic 4, students build on this understanding by solving rational equations, analyzing and critiquing various methods, and checking for extraneous solutions. In Lesson 5-4, students solve equations with exponents and radicals both algebraically and graphically, looking for relationships between square roots and squaring. 

• F-IF.4, F-BF.3, and F-TF.5: In Algebra 2, Topic 7, the materials provide multiple examples of the processes of graphing sine and cosine functions using the key features of the function—period/frequency, amplitude—including creating functions representative of real-world applications. In Lesson 7-4, students analyze shifts in sinusoidal graphs to develop functions based on the observed patterns of transformations. 

• G-SRT.2: In Geometry, Topic 7, Lesson 7-2, the lesson starts with examining examples of student work. Students use appropriate tools to find if there is a composition of transformations that will map one figure onto another figure and tell what it is for each student if there is one. A follow-up question requires students to describe the relationship between two figures that the students have drawn. In addition, there are questions posed for the teacher in the teacher edition that promote reasoning and problem solving, such as, “Name different types of transformations and what is preserved with each type.” 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. Teachers are provided with questions to help students develop an understanding of the connection between congruence and similarity.

• G-C.3: In Geometry, Topic 5, Lesson 5-2, students consider the relationship between the circumcenter of a triangle and a circumscribed circle of the triangle; they also consider the relationship between the incenter of a triangle and an inscribed circle of a triangle. In Practice & Problem Solving, students explain an error, explain reasoning, make a conjecture, justify their solution with a diagram, and explain their answer.

• S-ID.7: In Algebra 1, Topic 2, Lesson 2-1, students create a linear equation from two data points and interpret its meaning in the context of the application, demonstrating an understanding of key mathematical concepts such as slope and intercepts.

The materials reviewed for enVisionMath A/G/A meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, 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. The materials include a Practice and Problem Solving section, and Additional Skills Practices are available after each lesson.

Examples that show the development of procedural skills across the series include:

• A-APR.1: In Algebra 1, Topic 7, Lesson 7-1, students are engaged in examples of simplifying, adding, and subtracting polynomials. For Problem 12, students write an expression to represent the surface area of a figure, demonstrating their ability to perform this skill. They also write additional non-contextual problems for students to become proficient in this procedural skill. 

• A-APR.6: In Algebra 2, Topic 3, Lesson 3-4, students divide polynomial expressions using the process of long division and synthetic division, writing their answers in the form q(x) + r(x)/b(x) when applicable.

• F-BF.3: Students transform various types of functions throughout the series. In Algebra 1, students transform linear, piecewise, absolute value, exponential, and quadratic functions. This skill is extended to higher degree polynomials and square root functions in Algebra 2, as well as logarithmic and trigonometric functions. Students are given multiple opportunities to practice this skill throughout the materials. 

• G-CO.5: In Geometry, Topic 3, Lesson 3-2, students are given images or points before and after transformations on the coordinate plane and create the rule for the rigid motions that occurred. In Problem 11, students analyze the work of a hypothetical student to find the mistake when given a rule and their created image/preimage.

• G-GPE.7: In Geometry, Topic 9, 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 formula to find the characteristics of polygons. Students use these formulas to compute the perimeters and areas of polygons.

The materials reviewed for enVisionMath A/G/A meet expectations for supporting 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 materials include STEM Projects and Mathematical Modeling in 3 Acts that can be found in each topic. These provide multiple opportunities for students to engage in routine and non-routine applications of mathematics throughout the series, including opportunities to independently demonstrate mathematical flexibility in a variety of contexts. 

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

• A-REI.11: In Algebra 1, Topic 4, Lesson 4-1, students solve systems of linear equations by graphing. The materials provide a graphing utility for students to find solutions using technology. Students are asked to verify their answers using substitution, applying an understanding of systems of equations. For Problem 24, students determine the week when two students will have the same amount of money based on a description of their saving habits.

• F-IF.7e and F-IF.9: In Algebra 2, Topic 6, Lesson 6-1, students apply an understanding of exponential functions to problems involving population growth rates. For Problem 27, students create an exponential equation for a colony of bacteria and use their equation to predict the amount of bacteria in 5 days. 

• G-SRT.8: In Geometry, Topic 8, Lesson 8-2, Problem 44, students analyze a scenario in which they are given the dimensions of a boom lift at a particular angle. They use those dimensions and trigonometric ratios to find the lift's missing dimensions and determine if it will reach the height of a building in another scenario. 

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

• N-RN.B: In Algebra 2, Topic 5, Lesson 5-2, Problem 16, students discuss the advantages and disadvantages of first rewriting an expression. “Communicate Precisely Discuss the advantages and disadvantages of first rewriting \sqrt{\smash[b]{27}} + \sqrt{\smash[b]{48}} + \sqrt{\smash[b]{147}} in order to estimate its decimal value.

• G-MG.3: In Geometry, Topic 2, Mathematical Modeling in 3 Acts, students use their knowledge of parallel and perpendicular lines to decide what it means for roads to be “paved correctly” and identify examples meeting their criteria. 

• S-ID.2 and S-ID.3: In Algebra 1, Topic 11, Mathematical Modeling in 3 Acts, students interpret characteristics of multiple data sets presented as Histograms, Dot Plots, and Box Plots, using their interpretations to make inferences about how many text messages a student will receive “tomorrow.”

The materials for enVisionMath A/G/A meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. The three aspects are balanced with respect to the standards being addressed. 

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 materials:

• A-REI.2: In Algebra 2, Topic 4, Lesson 4-5, students integrate all aspects of rigor as they solve rational equations and investigate the presence of extraneous solutions. They connect the concept of the domain of rational functions to determining whether a solution is extraneous and critique the work of others who have already found such solutions. In the Practice & Problem Solving sections of the lesson, students develop procedural skills by solving both simple and more complex rational equations with and without extraneous solutions and further apply that understanding to real-world problems involving completion times when working together and distance, rate, and time with and without the help of a current. 

• F-IF.2, F-IF.4, and F-LE.2: In Algebra 1, Topic 3, students explore linear functions. In the STEM Project, students investigate 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 recycled trash. In Lessons 3-1-3-3, students develop procedural skills in using function notation, evaluating functions, graphing the lines described by functions, and graphing translations through applications, tables, and graphs. In Modeling in 3 Acts, students find a strategy for picking a checkout lane in the grocery store. In Lesson 3-6, students apply linear functions by 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. 

• G-CO.8: In Geometry, Topic 4, students demonstrate a conceptual understanding of congruence by using their knowledge of rigid motions from the previous topic to show that two triangles are congruent and prove theorems about triangle congruence.  They develop procedural fluency and skill through practice mapping one triangle onto another, recognizing congruent parts of congruent triangles, and identifying the appropriate congruence theorem based on the given information. Throughout the topic, students apply their knowledge and skills to solve real-world and mathematical problems, for example, Problem 26 in Lesson 4-6, where students find the width of a quadrilateral created by two overlapping triangles using an understanding of congruent triangles.

• G-SRT.5: In Geometry, Topic 7, STEM Project, students demonstrate conceptual understanding through application by using 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 the 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 skills 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), integrating all aspects of rigor.

• S-ID.2 and S-ID.3: In Algebra 1, Topic 11, Lesson 11-2, students demonstrate all three levels of rigor by examining data sets in multiple representations. Students demonstrate a conceptual understanding of descriptive statistics in the Critique & Explain as they analyze the reasoning for given prices of various paintings in a gallery. Throughout the lesson and in the Practice & Problem Solving sections, students demonstrate procedural skills in obtaining a 5-number summary of data distribution and the effect of any outliers. Finally, students apply their knowledge by determining characteristics and constructing a display to compare and analyze data of two smart phone batteries in Problem 24.

The materials reviewed for enVisionMath A/G/A meet expectations that they support the intentional development of overarching mathematical practices (MPs 1 and 6) in connection to the high school content standards. Overall, MP1 and MP6 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. 

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, Topic 5, Lesson 5-3, students use step functions to understand why the functions appear differently in two scenarios. They then determine which graph is correct and why. 

• In Geometry, Topic 10, STEM Project, students are asked to use an aerial view of two cities and their knowledge of trigonometry and tangents to calculate unknown dimensions. They then do the same thing as they independently design a trio of space cities. Students make sense of the calculations required to describe the “space cities accurately.”

• In Algebra 2, Topic 7, Lesson 7-5, Practice & Problem Solving, Problem 22, students determine a function that models a triangle's height to construct a treehouse's hexagonal floor. Students make sense of the triangle's shape to fit the floor pattern. 

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

• In Algebra 1, Topic 3, Lesson 3-6, Example 5, Try It!, students are given two scenarios regarding the length of time that running shoes last, along with another statistic (the number of miles run and a person's age) and have to determine if the other statistic affects the longevity of the shoes. Students must understand that other variables play a role, and correlation does not mean causation.

• In Geometry, Topic 9, Lesson 9-2, students justify their reasoning within proofs and are precise in calculating needed values. 

• In Algebra 2, Topic 11, Lesson 11-6, students communicate precisely using data and statistical measures to support or reject a hypothesis.

The 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. 

Most of the time, MP2 and MP3 are used to enrich the mathematical content. There is an intentional development of MP3 and MP3 that reaches the full intent of the MPs. There are many examples in the 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 answers. 

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, relying on their understanding of concepts related to nonlinear functions to develop a representative model. Students engage in abstract and quantitative reasoning to identify constraints that will affect the graph of the scenario, comparing the speed of a runner going uphill, downhill, and on a flat surface. 

• In Geometry, Topic 10, Lesson 10-1, Practice & Problem Solving, Performance Task, and Problem 32, use their knowledge about arcs and sectors to reason abstractly and quantitatively as they answer questions about building a stage for a concert.

• In Algebra 2, Topic 9, STEM project, students make sense of the quantities in the parts of conic sections by answering questions pertaining to the characteristics of whispering galleries and the relationship between the location of a foci and the ratio of the length to width of an ellipse.

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

• In Algebra 1, Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 12, students are given an error analysis question where they need to describe and correct the error a student made when finding the function rule for the date in the table.

• In Geometry, Topic 7, Lesson 7-4, Practice & Problem Solving, and Problem 14, students write proofs of a theorem and its corollaries by constructing arguments based on various "Given" and "Prove" scenarios. 

• In Algebra 2, Topic 7, Lesson 7-3, Practice & Problem Solving, Problem 10, students analyze the work of a hypothetical student to find and correct the error made in solving for the period of a trigonometric function.

The materials for enVisionMath A/G/A partially meet expectations that the materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. There is intentional development of MP4 and MP5 that reach the full intent of the MP throughout the series. 

Examples of where and how the materials use MP4 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1, Topic 8, Mathematical Modeling in 3 Acts, students develop a model to represent the best path taken by a basketball to make a basket. They use their understanding of quadratic functions to test and validate their model. 

• In Geometry, Topic 1, Mathematical Modeling in 3 Acts, students create a model to perform calculations involving measurements of an object from an image (a bike wheel). They identify variables and the relationship between them in the shape being applied to the object and apply skills that will be revisited in the context of volume and surface area of composite shapes that represent objects from the real world in Topic 11.

• In Algebra 2, Topic 1, Lesson 1-3, Model & Discuss, students graph the income each store would receive selling their different guitar string packages. “A music teacher needs to buy guitar strings for her class. At Store A, she can buy a single pack of strings. At Store B, she can buy a bundle of 4 packs of strings. A. Make graphs that show the income each store receives if the teacher needs 1-20 packs of guitar strings. B. Describe the shape of the graph for store A. Describe the shape of the graph for store B. Why are the graphs different? C. Compare the graphs for stores A and B. For what number of guitar strings is it cheaper to buy from store B? Explain how you know.” 

Examples of where and how the materials use MP5 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Algebra 1, Topic 3, Lesson 3-5, Practice & Problem Solving, Problem 28, students are asked to compute and interpret the correlation coefficient for linear data using a graphing calculator tool. “A store records the price of kites in dollars, x, and the number of kites, y, sold at each price.” Students make a scatter plot and then calculate and interpret the slope in the context of the problem. 

• In Geometry, Topic 1, Lesson 1-2, students learn to use a straightedge and compass to perform basic constructions. In Practice & Problem Solving, Problem 10, students explain how to “use a compass to determine if two segments are the same length”. 

In Algebra 2, Topic 4, Lesson 4-3, Practice & Problem Solving, Problem 14, students use tools and their knowledge of domain restrictions to show two rational expressions, \frac{(-6x^2)+21x}{3x} and -2x + 7, are equivalent. They use their understanding of x intercepts to extend their explanation. 

The 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. 

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 10, Lesson 10-3, students analyze the key features of function graphs. This includes the domain, range, maximum and minimum values, axis of symmetry, and end behaviors. By discovering that the leading coefficient and exponent of a polynomial affect the end behavior of functions, students can determine a polynomial’s structure.

• In Geometry, Topic 7, Lesson 7-1, students use the structure of similar triangles to understand relationships in triangles after undergoing a dilation. In Practice & Problem Solving, Problem 25, students determine where a light should be placed to display a shadow of a particular size on the wall 2 ft away.  

• In Algebra 2, Topic 9, Lesson 9-1, students use the structure of the equation of a parabola to identify key features of the parabola graph. In the Practice & Problem Solving sections of the lesson, students write the equation of a parabola shown in a graph in Problem 10 and describe the shape of a parabola “whose focus is very near the directrix” in Problem 13. 

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 9, Lesson 9-6, Example 2, Try It!, students form a system of equations to determine the solutions of the single quadratic equation x^2 + 1 = x + 3. Students generalize that the solution to a system of equations is when all expressions/equations are equal, or where the two equations intersect.

• In Geometry, Topic 5, Lesson 5-3, students express regularity in repeated reasoning when investigating where the orthocenter is located for any right triangle. In Practice & Problem Solving, Problem 6, students extend that generalization to explain, “for any right triangle, where is the orthocenter located”? 

• In Algebra 2, Topic 7, Lesson 7-1, students use regularity in repeated reasoning to determine that the trigonometric values of coterminal angles are equal. In Practice & Problem Solving, Problem 19, students explain “the relationship between a positive and a negative angle that share a common terminal side.”