High School - Gateway 2

The instructional materials reviewed for Reveal Math Traditional meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The digital materials have exploration activities and applets to aid in the development of conceptual understanding, and they provide opportunities for students to demonstrate that understanding throughout the series.

Examples of where the materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding include, but are not limited to:

• In Algebra 1, Module 11, Lesson 3, students relate the factors of an equation to the solutions of a quadratic equation (A-APR.B), and in Algebra 2, Module 5, Lesson 5, “Explore”, students determine how the value of the discriminant impacts the type and number of zeros of a quadratic function. In “Learn”, students examine examples of the relationship between zeros, roots, and factors and demonstrate their conceptual understanding of the concept introduced in “Explore”. 
• In Algebra 1, Module 2, Lessons 2-4, students solve equations using the algebraic properties of equality (A-REI.1). Students use algebra tiles to develop their conceptual understanding and provide algebraic justifications for their steps in solving equations. In “Learn”, students provide the steps and properties justifying the steps when solving equations to demonstrate their understanding of solving equations. 
• In Algebra 1, Module 4, Lesson 1, students determine what it means for a point to be a solution to an algebraic equation (A-REI.10). “Math Background” states, “The coordinates of the points on the line are the solutions of the related linear equation.” The materials assess this idea within the “Inquiry” question, “How is the graph of a linear equation related to its solution?” In Algebra 2, Module 8, Lesson 1, students solve logarithmic equations and make the connection between the solutions of logarithmic functions and the points of intersection on the graphs of the function.
• In Algebra 1, Module 3, Lesson 2, students determine whether relations are functions (F-IF.A). In “Launch the Lesson”, students examine a website of 6.3 million selfies to determine if the relationship between the independent and dependent variables represents a function. Students demonstrate their understanding of what a function is and determine whether relations are functions in mappings, tables, graphs, etc. 
• In Geometry, Module 9, Lesson 1, students use similarity in right triangles to develop proportional relationships within right triangles (G-SRT.6). In Geometry, Module 9, Lesson 5, students complete an interactive activity which establishes the ratios between pairs of side lengths based on given angle measures.
• In Algebra 1, Module 5, Lesson 1, students make a generalization about how changing coordinates of points changes the slope of an equation. Within the lesson, students interpret the slope and y-intercept of an equation (S-ID.7), and in Algebra 1, Module 4, students interpret rates of change in real-world relationships.

An example of the materials not developing conceptual understanding and providing opportunities for students to independently demonstrate conceptual understanding is:

In Algebra 1, Module 8, Lesson 4, the materials include steps for writing equivalent expressions with rational exponents using the exponent of $$\frac{1}{2}$$ as an example (N-RN.1). The materials do not develop a connection between integer and rational exponents, and in “Explore”, practice problems, students do not independently demonstrate their understanding of this standard.

The instructional materials reviewed for Reveal Math Traditional meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Often, these procedural skills occur across multiple modules with varied practice. 

Examples of how the materials develop procedural skills across the series include, but are not limited to:

• A-SSE.2: In Algebra 1, Module 1, Lesson 4, students rewrite expressions using the distributive property and by factoring the greatest common factor out of an expression. In Algebra 1, Module 10, Lessons 4 - 7, students multiply and factor special products, and in Algebra 2, Module 5, Lesson 2, students learn more factoring techniques and use polynomial identities to rewrite expressions that are not easily factorable otherwise.
• A-APR.1: In Algebra 1, Module 10, Lessons 1 - 3, students practice the operations of addition, subtraction, and multiplication with polynomials. These skills are also developed in Algebra 2, Module 4, Lesson 3, along with showing closure under these operations. Through these lessons, students have multiple opportunities to practice operations with polynomials.
• F-BF.3: In Algebra 1, Module 4, Lessons 4 and 7, students examine transformations of linear and absolute value functions respectively. Transformations of functions are also developed in Algebra 2, Module 6 (square root functions); Algebra 2, Module 7 (exponential functions); and Algebra 2, Module 9 (reciprocal functions). In all cases, students use embedded technology to explore how the parameters change the graph of a function and identify and justify the effects of changing certain parameters.

Examples of how the instructional materials provide opportunities for students to independently demonstrate procedural skills include, but are not limited to:

• A-APR.6: In Algebra 2, Module 4, Lesson 4, students use long division and synthetic division to rewrite rational expressions, and there are many examples for students to practice each type of division.
• G-SRT.5: Within the Geometry course, there are many opportunities for students to justify congruence and similarity criteria for triangles. In Geometry, Module 5, Lessons 3 and 4, students use triangle congruence to solve problems. In Geometry, Module 8, Lessons 3 and 4, students demonstrate skill with triangle similarity by solving problems and demonstrate that two triangles are similar using the similarity criteria.
• G-GPE.5: In Algebra 1, Module 5, Lesson 2, students write equations of parallel and perpendicular lines through a given point. In Geometry, Module 3, Lesson 8, students justify the slope criteria for parallel and perpendicular lines and demonstrate finding parallel or perpendicular lines through a given point.

The instructional materials reviewed for Reveal Math Traditional 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 utilize different contexts across the series, such as population modeling, maximum profit, expenditures for a party, zip-lining, and numerous sports applications. Throughout the series, students apply conceptual understandings and procedural skills that have been developed.

Examples of applications across the series include, but are not limited to:

• A-REI.11: In Algebra 1, Module 7, Lesson 1, students predict the approximate year when the populations of China and India will be the same by approximating the average rate of change of both populations, writing a system of equations to represent the situation, and graphing the system to determine their solution. Students work in teams or small groups to contextualize and make sense of the problem.
• G-SRT.8: In Geometry, Module 9, Lessons 2 and 6, students use angles of elevation and trigonometry to solve problems in different outdoor settings. In Lesson 2, students find the length of a zipline given different measurements of length and angle measure, and in Lesson 6, students use trigonometric ratios to determine how far a drone is from a given location. 
• S-ID.2: In Algebra 1, Module 12, Lesson 6, Problem 30, students solve a problem to help Nickolas determine a new golfing partner. Nickolas asks for the stats from 2 of his friends. Students are asked to find the mean and standard deviation for each set of data. Students are then asked for the advantages/disadvantages for choosing each friend.
• S-IC.1: In Algebra 2, Module 10, Lesson 1, students determine if a raffle is fair or unfair. Students create and run a model or simulation of the raffle to determine fairness.

The instructional materials reviewed for Reveal Math Traditional meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Throughout the materials, students engage in each of the aspects of rigor. All three aspects of rigor are present independently throughout the materials, and there are instances where multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding.

Examples of where the instructional materials attend to conceptual understanding, procedural skills, and application independently throughout the grade level include, but are not limited to:

• Conceptual understanding is developed in the digital materials through “Launch the Lesson” in which students consider how the content relates to real-life scenarios. For example, in Algebra 2, Module 11, Lesson 2, students find the heights of clouds using trigonometry and using a clinometer as a mathematical tool.
• In Algebra 2, Module 3, Lesson 4, students apply factoring quadratic functions. Students “find the price range the store should charge to make the dress profitable” by factoring a quadratic equation, determining the zeros of the quadratic function, and interpreting when the selling price of the dress will be profitable.
• In some “Learn” sections, students develop procedural skills. For example, in Algebra 1, Module 7, Lesson 2, students find intersection points on a graph to find a solution. The materials demonstrate how substitution relates to finding intersection points, and students complete multiple practice problems to develop the procedural skill for themselves.

Examples of where two or more of the aspects of rigor are engaged simultaneously to develop students’ mathematical understanding include, but are not limited to:

• In Algebra 1, Module 5, Lesson 1, students use the web sketchpad to explore how changing points on a line impacts the slope of the line. Students practice procedural skills by writing equations in slope-intercept form given a slope and a point or two points. Students apply their understanding to create a linear equation that models the weight of a puppy tracked for 6 months. Students also interpret the y-intercept and slope in the context of the problem once they have created the equation.
• In Geometry, Module 7, Lesson 4, students write equations of four lines in the coordinate plane to form a rectangle. Students demonstrate their understanding of the definition of a rectangle. Students also determine how two lines can be perpendicular or parallel, find the distance of line segments, and find points of intersection for the equations of lines.
• In Algebra 2, Module 4, “Ignite”, students solve a problem involving a bridge by answering the questions, “What do you notice?” and “What can you ask?”. Students create a question about the scenario and develop a strategy to solve the question about the bridge. There are extensions in the materials for students to compare different spans for the bridge that would allow more practice with the procedure.

The instructional materials reviewed for Reveal Math Traditional meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards.

Examples of MP1 include, but are not limited to:

• In Geometry, Module 1, Lesson 4, “Think About It”, students answer, “Why do you think the Distance Formula uses absolute value?” Students make sense of the idea that line segments cannot have negative values for their lengths.
• In Geometry, Module, 5, Lesson 1, “Teaching the Mathematical Practices” states, “In Example 1, guide students through the use of the 4-step plan to identify the meaning of the problem and look for entry points to its solution.” The materials provide explicit instructions for how to make sense of a mathematical problem.
• In Algebra 2, Module 9, Lesson 2, teachers remind students to check their answers by selecting values for variables in their simplified expression.

Examples of MP6 include, but are not limited to:

• In Algebra 1, Module 5, Lesson 5, the materials define the linear regression correlation coefficient and explain its meaning. Students attend to precision when explaining the graph, paying specific attention to labels and units in the problem.
• In Geometry, Module 5, Lesson 6, “Communicate Precisely” states, “Encourage students to routinely write or explain their solution methods. Point out that they should use clear definitions when they discuss their solutions with others.” This occurs in multiple modules and lessons throughout the series.
• In Algebra 2, Module 11, Lesson 2, students calculate with precision to determine the trigonometric ratios for an angle whose terminal side passes through a given point.

The instructional materials reviewed for Reveal Math Traditional meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards.

Examples of MP2 include, but are not limited to:

• In Algebra 1, Module 6, Lesson 2, students model multi-step inequalities with algebra tiles and consider their representation of the inequality algebraically. Students answer the question, “How can you model and solve a multi-step inequality?” to reason abstractly about the process.
• In Geometry, Module 3, Lesson 9, using the relationships between pairs of angles formed when two parallel lines are cut by a transversal, students write an equation that abstractly represents the relationship between the angle measures for a pair of angles and solves the equation to find the missing angle measures.
• In Algebra 2, Module 7, Lesson 2, “Teaching the Mathematical Practices” states, “Encourage students to consider how they could write a system of equations to represent the relationships among the number of visitors that visited the three parks.” Throughout the lesson, students write systems of equations that represent relationships between quantities in real-world scenarios, and in Problem 26, students explain how to recognize a system with infinitely many solutions.

Examples of MP 3 include, but are not limited to:

• In Algebra 1, Module 2, Lesson 1, “Think About It”, students construct an argument about if it is possible for a function to be discrete and continuous at the same time.
• In Geometry, Module 2, Lesson 2, students construct arguments and communicate them to others. In Example 1, students answer the following question, “Adrian claims that if two complementary angles are both acute, then a pair of supplementary angles must both be obtuse. Do you agree? Explain why or why not.”
• In Geometry, Module 5, Lesson 1, students use dynamic geometry software to make conjectures about the interior angles of triangles. Through their exploration, students construct an argument for the triangle angle sum theorem.
• In Algebra 2, Module 6, Lesson 6, students solve an equation that includes an extraneous solution. Students critique their own reasoning by answering the question, “In the example above, could you tell that four was an extraneous solution before checking the result? Explain your reasoning.”

The instructional materials reviewed for Reveal Math Traditional meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. In the materials, students find and use patterns and generalize findings from regularity in repeated reasoning. Within the print materials, there are Higher Order Thinking problems that provide students with opportunities to describe patterns and make connections from their repeated reasoning.

Examples of MP 7 include, but are not limited to:

• In Algebra 1, Module 9, Lesson 6, “Explore and Develop”, students use the structure of a geometric sequence to write a recursive formula for the sequence. In Algebra 2, Module 7, Lesson 4, students use the structure of geometric sequences to convert between recursive and explicit forms of the sequences.
• In Geometry, Module 9, Lesson 5, students explore the structure of right triangles to discover trigonometric ratios. Students complete an inquiry process that ends with the question, “If two right triangles have the same angle measure, what do you know about the trigonometric ratios of the angle?”
• In Algebra 2, Module 3, Lesson 6, students use the discriminant to make connections between different values of the discriminant and the number and types of roots of the quadratic equation.

Examples of MP 8 include, but are not limited to:

• In Algebra 1, Module 7, Lesson 1, students examine the slopes and y-intercepts of equations in a system for patterns to determine the number of solutions of the system.
• In Algebra 1, Module 8, Lesson 1, students explore provided examples to look for a pattern to determine the product of two expressions with exponents. In “Explore”, students express regularity in repeated reasoning to write the general product of two expressions with exponents.
• In Algebra 2, Module 5, Lesson 3, students find patterns in expanding and simplifying polynomial expressions to provide justification for polynomial identities.