High School - Gateway 2

The instructional materials reviewed for Mathematics Vision Project Traditional series meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Every three tasks in each module follow the develop/solidify/practice sequence. This allows students to develop conceptual understanding across many tasks. For example, in Algebra I, Module 4, each task builds upon the previous to reinforce concepts:

• Task 1 Develop - Explaining each step in the process of solving an equation (A-REI.1).
• Task 2 Solidify - Rearranging formulas to solve for a variable (N-Q.1,2; A-REI.3; A-CED.4).
• Task 3 Practice - Solving literal equations (A-REI.1,3; A-CED.4).

Within this progression, students develop their conceptual understanding of what it means to solve an equation in each task. The materials develop the understandings through each task, so students can build on the previous days’ learnings.

• F-IF.7: In Algebra I, Module 7, Task 1, students develop an understanding of transformations on a graph and how it relates to a corresponding equation. Students explore the changes of a graph in relationship to the area of a square. By the end of this task, students identify the key features of the graph and how changes to a corresponding equation will change the graph. This development is continued in Task 2.
• G-CO.10: In Geometry, Module 3, Task 1, students explore why the interior angles of a triangle add up to 180 degrees. Their understandings of the sum of angles along a straight line and transformations are expanded as they prove relationships about triangles. Students also use the key mathematical concepts of transformations and congruence to prove other theorems about triangles.
• N-RN.3: In Algebra II, Module 3, Tasks 5 and 6 develop the concept of irrational numbers. The tasks begins with plotting real numbers on a number line and moves to plotting irrational numbers on the number line. This activity helps students understand how irrational numbers behave in similar and different ways to the rational numbers. Once students establish this understanding in Task 5, the students develop their understanding of the properties of irrational numbers in Task 6. The materials address irrational numbers and their properties over two tasks, so students can develop a more thorough understanding of the mathematical concept.

The instructional materials for Mathematics Vision Project Traditional series 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. Every task begins with an engaging scenario that is either a direct, real-world application of the content for that day or provides a unique, novel problem for the students to solve. The applications within the series enable students to develop their conceptual understanding of the mathematics and the abstract notation or procedural skills once students have built that understanding. There are also scenarios that recur throughout the series. As a result, students contextualize many different mathematical ideas to the same scenario.

The series includes numerous applications across the series, and examples of select standard(s) that specifically relate to applications include, but are not limited to:

• A-REI.11: In Algebra I, Module 3, Tasks 4-6 use a Water Park scenario where students, in task 4, determine when both pools are the same height using intersection points. In task 5, they compare the graphs of the pools to the sum of both pools. In task 6, students set the two function rules equal to each other to determine the intersection point. The scenario spans the three tasks, so students develop their understanding about the intersection points of two graphs and the different properties of functions.
• G-SRT.8: In Geometry, Module 4, Task 10, students determine the height of a tree using angle of elevation and shadows. Students work within the same scenario to determine unknown angles of depression and elevation. In Ready, students work multiple real-world problems using trigonometric ratios to determine missing lengths and angles.
• F-IF.7: In Algebra II, Module 5, Task 1, students write, graph, and solve rational equations in the context of winning the lottery. Students compare different points on the equation and graph based on different ways of splitting the prize money. In Set, students interpret an equation and graph to determine different ways of paying for a gift among friends.

The instructional materials for Mathematics Vision Project Traditional series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The materials engage students in each of the aspects of rigor in a cycle throughout the materials. Each module contains a Developing Understanding task to build conceptual understanding in students, a Solidifying Understanding task to build on that conceptual knowledge, and a Practicing Understanding task. Within each task there are Ready, Set, Go! activities that spiral procedural skills for students. 

For example, in Geometry, Module 6, Task 4, students develop conceptual understanding of the equation of a circle centered at the origin. Students practice procedural skills with the equations of circles in Ready, Set, Go!, Questions 10-15. In Task 5, students solidify their understanding with a sprinkler application problem and determine the equation of a circle when it is not centered at the origin. Students practice this concept in Ready, Set, Go!, Questions 11-19. In Task 6, students continue working with equations of circles with different challenges in the task and the Ready, Set, Go! questions.

The instructional materials for Mathematics Vision Project Traditional series meet expectations that the materials support the intentional development of making sense of problems and persevere in solving them, as well as attending to precision (MP1 and MP6) in connection to the high school content standards.

Examples of MP1 include, but are not limited to:

• In Geometry, Module 2, Task 1, students construct two shapes, a rhombus and a square. Within the materials, teachers are prompted for students to get enough time to explore the constructions fully. Within this task, students make sense of the constructions and persevere through the process of making the construction.
• In Algebra II, Module 1, Task 1, students recall previous information about functions and their graphs to make sense of inverse functions. Through the task, students notice how a function and its inverse are related and make sense of the relationship. Students determine what makes two functions inverses from their observations about the Pet Sitter situation.  
• In Algebra II, Module 5, Task 5, students identify and record errors in the Rational Expression and Functions activity. Students provide strategies to help others avoid these errors in the future. Through this, students make sense of the problems they are completing by determining where the errors might occur.

Examples of MP6 include, but are not limited to:

• In Algebra I, Module 2, Task 2, students identify the domains of two sequences in the Please Be Discrete task. Students determine that one is arithmetic and the other geometric. Students discuss how discrete and continuous functions are related, specifically around their domains. In order to discuss the difference between these two functions, students must be precise to show the differences between the two functions.
• In Geometry, Module 1, Task 1, students use precision in their language for transformations. Students use precise definitions for each of the transformations so the final image is a “unique figure, rather than an ill-defined sketch”. The materials prompt students to see how precision is needed when defining geometric relationships to make sure that images are well defined. 
• In Algebra 2, Module 8, Task 3, students attend to precision as they determine different parameters of their equations. The Bungee Simulator is a sophisticated graph that combines a sinusoid and exponential decay. In order to match a function to the graph provided, students utilize precision with their parameters to create a function that models the situation.

The instructional materials for Mathematics Vision Project Traditional series 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. Throughout the materials, there are many opportunities for students to critique the reasoning of others and to reason abstractly.

Examples of MP2 include, but are not limited to:

• In Algebra I, Module 4, Task 4, students are given a statement and determine which of the two expressions represent a larger value. In the Which is Greater task, students reason abstractly about an expression, compare it to another expression, and explain their reasoning.
• In Algebra I, Module 7, Task 1, students reason abstractly by relating the numeric results in a table to the graphs and explain the way the graph is transformed. Students examine the abstract relationships between the different representations (table, graph, and function) and how a change in one form impacts a change in the other.
• In Geometry, Module 5, Task 5, students engage in reasoning that considers how an infinite process might converge on a unique value. In Polygons to Circles, students examine the case of how an inscribed regular polygon with more and more sides converges on the shape of a circle. This limiting process provides an informal proof for the circumference and area of a circle.

Examples of MP3 include, but are not limited to:

• In Algebra 1, Module 3, Task 2, students interpret two representations (a table and a graph) and determine if Sierra’s statements are correct. During the task, students analyze the situations, justify their reasoning, and communicate their conclusions to others.
• In Geometry, Module 4, Task 8, the whole-class discussion begins by sharing several examples of triangles, so equivalent ratios can be observed. From this, students hypothesize the trigonometric relationships for all right triangles. Students determine how the ratios are related and construct an argument for what they believe about the trigonometric ratios. The materials promote a discussion for why these ratios are equivalent in all right triangles.
• In Algebra II, Module 4, Task 5, students use prior knowledge about polynomials and function behavior to construct an argument for the end behavior of various polynomial and exponential functions. They compare the functions and defend their conclusions.

The instructional materials for Mathematics Vision Project Traditional series 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. 

Examples of MP4 include, but are not limited to:

• In Algebra 1, Module 5, Task 3, students manipulate a system of equations to model the constraints of setting up a pet-sitting business. They determine the best use of space to provide maximum profit. Students also have to understand what terms in their expressions are related to the different constraints. From this, they derive various forms of the equations to determine maximum profit.
• In Geometry Module 5, Task 4, students analyze a plan to build a regular, hexagonal gazebo. In the plan, there are several statements students have to agree or disagree with and then design their own gazebo.
• In Algebra II, Module 8, Task 3, students model a bungee jump simulation and use calculator technology to create their model.

Examples of MP5 include, but are not limited to:

• In Algebra I, Module 2, Task 8, students compare linear and exponential growth related to two small companies. They are encouraged to use a calculator or spreadsheet to determine if this growth is continuous or discrete.
• In Geometry, Module 2, Task 2, students use the circle as a tool to create congruent line segments. Students also consider transformations as tools to think about congruence when creating mappings.
• In Algebra II, Module 2, Task 5 notes that students should use various tools, such as tables, graphs, and technology, to compare functions and their end behavior.

The instructional materials for Mathematics Vision Project Traditional series meet expectations that the materials support the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards.

Examples of MP7 include, but are not limited to: 

• In Algebra I, Module 8, Task 3, students use their understanding of absolute value to help solidify their understanding of piecewise functions. They develop the graph for an absolute value function using their understanding of piecewise functions. 
• In Geometry, Module 1, Task 6, students uncover the structure of regular polygons through the ideas of rotational and line symmetry. They notice the relationship between the number of sides in a regular polygon and the shape’s rotational and line symmetry. 
• In Algebra II, Module 3, Task 5, students use their knowledge of the quadratic formula to predict the nature of the roots of a parabola. Students relate their understanding of the different forms of the quadratic equation to the graph of a parabola to make predictions about roots. 

Examples of MP8 include, but are not limited to:

• In Algebra I, Module 8, Task 6, students are prompted to see when finding an inverse you can sometimes just “undo” the operations in the opposite order of the original function. Students also build an understanding of how to restrict the domain of the inverse based on this process.
• In Geometry, Module 5, Task 10, students determine the relationship between the area and perimeter of similar figures. Through the task, students develop the pattern for the relationship of properties between these scaled figures.
• In Algebra II, Module 4, Task 1, students explore how to determine the degree of a polynomial function. Students look at different rates of change to determine the type of function. For example, students understand that a cubic has a “first difference that is quadratic, a second difference that is linear, and a third difference that is constant”. Students are prompted to understand this pattern in all polynomial functions.