High School - Gateway 1

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials omit the full intent of the modeling process for more than a few modeling standards across the courses of the series.

Each topic of the enVisionMath A/G/A series contains “Mathematical Modeling in 3 Acts” and STEM projects. In each lesson, students are posed a problem, usually by watching a video. Students develop questions of their own, formulate a conjecture, and explain how they arrived at the conjecture. In most of the tasks, the needed information is not given, and students determine what information is essential. Students compute a solution for the problem and interpret their results. Students are guided through validating their conjecture and considering reasons why their answers might differ. Students engage in the full modeling process within the “Mathematical Modeling in 3 Acts” and STEM projects. However, several modeling standards are not addressed within these 3 Acts and STEM projects.

Some of the modeling standards for which the full intent of the modeling process has been omitted include:

• N-Q.2: In Algebra 1 Lesson 1-3, there are several application problems, but none allow the student to complete the full modeling cycle. For example, on page 23 problem 50, a performance task is presented in which two individuals paint a wall. Although students determine when the painters have painted the same amount, students are not given the opportunity to create a conjecture and defend it as the rates are provided.
• A-SSE.1a: In Algebra 1 Lesson 7-5, problem 37, students use a quadratic expression to represent the area of a swimming area. Students factor the expression to determine possible dimensions of the swimming area. They interpret the factors as expressions that give the length and width of the swimming area. However, this problem does not include the full modeling cycle because students do not formulate a model, the formula is given to them. Students are not provided an opportunity to validate their results.
• A-SSE.1b: In Algebra 1 Lesson 6-3, students are shown the relationship between the growth rate and the growth factor in exponential growth. In problem 27, students determine when a particular plant will become invasive. Students are given all the necessary information and told to write an exponential growth formula. Students do not have any choice of which type of model to use, nor to identify the variables for this particular model. Students are not given the opportunity to validate their results. Other examples of parts of the modeling cycle being omitted with this modeling standard can be found in Algebra 1 Lesson 7-5, problems 37 and 41. These problems require students to compute and interpret but not formulate or validate.
• A-SSE.3a: In Algebra 1 Lesson 9-2, students solve quadratic equations by factoring. Students are given three problems (35, 36 and 39) that contain components of the modeling cycle. However in each problem, students are told to use a quadratic equation to describe the situation. In problem 35, students are given a graphical model and asked to write a quadratic equation. In problems 36 and 39, students are given a context with a diagram that is labeled with variables. Students write a quadratic equation using those variables to answer the questions. All three problems missed opportunities to have students formulate needed information and validate their findings. In Algebra 2 lesson 2-3, this topic is revisited. In problem 41, students use an equation that models the height of a drone in terms of time. This problem does not allow students to formulate a solution method because the equation is given, and students are directed how to solve the problem using the equation. Students do not validate their answer.
• A-SSE.3c: In Algebra 2 Lesson 6-2, example 1, students are shown how to use the properties of exponents to transform expressions for exponential functions; however, there are no problems where students do this as part of a modeling cycle.
• A-SSE.4: In Algebra 2 Lesson 6-7, students are shown how to derive the formula for the sum of a finite geometric series. Students use that formula to calculate a monthly payment on a loan. Students are given the formula to use and all the necessary information to solve the problem. The students compute and interpret within the problem. In problem 39 where students are not directed to use the formula for the sum of a geometric series and must formulate the correct model, then compute and interpret their result. However, the opportunity for students to validate their results is missing.
• G-GPE.7: In Geometry Lesson 9-1, students use coordinates to compute the perimeter of different polygons. In Homework problems 26-28, students use parts of the modeling process while formulating an equation and computing the perimeter of the polygon, but they are not provided the opportunity to validate their model nor interpret their results within the context of the problem.
• G-MG.2: In Geometry Lesson 11-2, problem 27, students are given information for making candles and are prompted to determine the weight of a specific order of candles. Students do not identify the needed variables as the dimensions of each candle is presented. The problem is specific and does not allow for interpreting individual findings.
• F-IF.5: In Algebra 1 Lesson 8-1, students identify key features of a quadratic function. There are several application problems for students to practice determining the average rate of change over a specific interval. Problems 26-28 provide students with aspects of the modeling process, such as evaluating a reasonable rate of change and defending that conclusion. However, students do not interpret or evaluation the solution. The questions do not provide multiple access points or various solutions.
• F-IF.6: In Algebra 2 Lesson 1-1, problem 29, students explain the meaning of the rate of change in the context of students jumping. There is only one possible solution, missing the opportunity for students to defend their solution and/or validate their conjecture.
• S-ID.6b: In Algebra 1 Lesson 3-6, students graph residuals from a linear model of data, and in lesson 8-4, students graph residuals from a quadratic model of data. Students do not complete the entire modeling process with this standard. In both lessons, students work with residuals in context, but students do not validate models or analyze results.

Examples where the materials intentionally develop the full intent of the modeling process across the series to address modeling standards include:

• In Algebra 1 Topic 2, students are presented with a situation in which height is measured in unconventional ways. Students watch a video that shows the height of a basketball player in terms of various objects. Students see a stack of cups being built next to him. As students attempt to figure out the basketball player’s height in foam cups, they have to formulate what information they would need. Data is given about smaller stacks of cups. Students utilize the information previously taught about linear functions. When students develop a plan, they complete computations. After they compute, students validate their findings when they view the final video which shows all the cups falling into place. Students report their findings compared to the final solution. This task addresses A-CED.1,3,4.
• In Algebra 2 Topic 6, students explore and apply concepts related to exponential equations and functions. Students watch a video of an athlete performing a running drill. They extrapolate both the time and distance for a certain round of the drill. Students formulate how to determine how far the athlete runs in the twentieth round and how long it will take. Students determine what information they need and consider how their ideas might relate to exponential functions. Students interpret their findings, validate them with each other, and view the final video which reveals the total time and distance. Students report their findings compared to the final solution. This task addresses F-LE.5, S-ID.6a.
• In Geometry Topic 11, students explore and apply concepts related to surface area and volume. Students are presented with different packaging options for candles. The problem is to determine the packaging option with the least surface area for a constant volume. Students watch a video which shows 24 individually-boxed candles. The smaller boxes are then packed inside one cardboard box. Students determine the dimensions of the package that has the least surface area. Students formulate a solution as they speculate how they could analyze the differences in surface area among the packages to find the one with the least surface area. Students compute a solution and think strategically to make sure they have found every possible set of dimensions for the packaging. They validate their results with each other to include ones not seen in the video. The final video shows the dimensions and surface area of each box. Students approach this solution in a variety of methods and report to each other. This task addresses G-GMD.3,4.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for, when used as designed, letting students fully learn each non-plus standard. Overall, the series addresses many, yet not all, of the standards in a way that would allow students to fully learn the standards.

The non-plus standards that would not be fully learned by students across the series include:

• N-Q.3: In Algebra 1 Lesson 6-3, students find the population in one example. The “Common Error” box explains rounding to whole numbers when referring to people. In lesson 9-6, students round to the nearest hundredth but are given no context as to why to round to this particular number. Students are not given sufficient opportunities to practice choosing a level of accuracy when reporting quantities throughout the series.
• A-SSE.1: In Algebra 1 Lesson 7-5, students factor polynomials and interpret the factors in terms of the context; however, no opportunity exists to practice interpreting terms and coefficients in terms of the context.
• A-SSE.3c: In Algebra 1 Lesson 6-2, problems 25 and 28, students use properties of exponents to transform exponential functions. No other problems were given to help support students' learning of this standard.
• A-APR.4: In Algebra 2 Lesson 3-3, students prove the difference of cubes' identity. There is not an opportunity to prove any other identities nor use them to describe numerical relationships.
• A-REI.1: In Algebra 1 Lesson 1-2, students explain each step of solving a simple equation in an example, but there are no other opportunities for students to practice constructing a viable argument with other simple equations. In Algebra 2 lesson 5-4, students do not practice constructing arguments to justify solution methods.
• A-REI.5: In Algebra 1 Lesson 4-3, students are provided opportunities to practice solving systems of equations by elimination; however, students are not provided an opportunity to prove, given a system of two equations in two variables, replacing one equation by the sum of that equation, and a multiple of the other produces a system with the same solutions.
• F-IF.3: In Algebra 1 Lesson 1-4, example 1 shows a sequence being a function. However, there are no other opportunities found for students to recognize that sequences are functions. In addition, sequences are not defined using function notation. In Algebra 2 lesson 3-4, students determine whether each sequence is arithmetic, but students do not have the opportunity to develop a connection between sequences and functions.
• F-LE.3: In Algebra 1 Lesson 8-5, students compare linear and exponential graphs to determine which function will exceed the other in one problem. There are no other opportunities for students to address this standard.
• G-SRT.2: Geometry Lesson 7-2 addresses similarity transformations, but students do not explain similarity for triangles in terms of proportionality of all corresponding pairs of sides.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School CCSSM Standards. The instructional materials do not explicitly identify content from Grades 6-8.

Some examples where the materials do not explicitly identify content from Grades 6-8 include:

• At the beginning of each topic in the teacher editions, the Topic Overview Math Background Coherence standards from middle school are identified as being from Grades 6-8, but specific standards are not listed. Concepts from middle school are explicitly identified but not connected to any standard.
• The Algebra 1 Teacher Edition, page 2B states how students will use their understanding of rational and irrational numbers from Grade 8 to compare and order rational and irrational numbers in Topic 1 of Algebra 1. The materials state that students will use their understanding of solving inequalities in the form $$px + q \gt r$$ or $$px + q \lt r$$ from Grade 7 to solve compound inequalities.
• In the Geometry Teacher Edition, page 2B, informal geometric arguments from Grade 8 are referenced and connected to building formal proofs in high school, but no specific standards are identified.

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

• Algebra 1 Teacher Edition, page 86B describes how using functions to model relationships is extended to understanding domain and range. The materials state, “In Grade 8, students began to explore linear and nonlinear functions. Students learned about the key features of linear functions, including slope and rate of change.”
• Algebra 1 Topic 8 Quadratic Functions, page 312B references, “In Grade 8, students compared linear and nonlinear functions, learned about increasing and decreasing intervals, and sketched functions from a verbal description. Students explored key features of linear functions including slope and rate of change” as a precursor to the learning of quadratics.
• Geometry Teacher Edition, page 342B references ratios and proportions and connects that to understanding trigonometric ratios (G-SRT.C). Teachers use this resource to make the connection as it is not present in the student edition.
• Geometry Teacher Edition, page 462B refers to formulas for volume of cones, cylinders, and spheres from middle school, and these are used to solve problems aligned to G-GMD involving those shapes in applications of prisms, cylinders, pyramids, cones, and spheres.