High School - Gateway 2

The instructional materials for the Discovering series meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, the instructional materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding.

The instructional materials develop conceptual understanding throughout the series. For example:

• N-RN.1: In Discovering Advanced Algebra, Lesson 4.3 Investigation, students “describe what it means to raise a number to a rational exponent.” In the Investigation, students create a table and a graph for y = x^(½) in order to state a conclusion about raising a number to the power of ½. Students explain how they would evaluate numerical expressions involving a rational exponent, and they conclude the Investigation by generalizing “a procedure for simplifying a^(m/n)."
• F-IF.A: Across the series, students develop an understanding of functions. In Discovering Geometry, Lesson 6.2, functions are developed through the algebraic nature of geometric transformations. In Discovering Advanced Algebra, functions are further developed in Chapter 3. Students learn about function notation and evaluate functions. They use real world situations to sketch and interpret graphs of functions. Students talk about reasonable domains and evaluate functions that are representing different situations. Students continue to develop their understanding of functions of different types in Chapters 4, 5, 6, and 7.
• A-APR.B: In Discovering Algebra, Lessons 8.4 and 8.6, the materials initially address the relationship between zeros and factors of polynomials as students find the zeros of quadratic equations by factoring and completing the square. In Discovering Advanced Algebra, Lessons 6.2, 6.3, and 6.4, students further develop their conceptual understanding of the relationship between zeros and factors of polynomials through polynomial equations of degree 3 and higher. In both courses, students determine factors of polynomial equations from graphs in addition to finding the zeros for given polynomial equations.

The instructional materials provide opportunities for students to independently demonstrate conceptual understanding throughout the series. For example:

• G-SRT.2: In Discovering Geometry, Lesson 7.1, students determine why two figures are similar. They are expected to answer that the angles are congruent and that the sides are proportional. On Chapter 7, Quiz 1 Form A, students explain why or why not two triangles are similar in Problems 3 and 4. On Chapter 7, Constructive Assessment Options, Problem 3, students extend the sides of a trapezoid to create similar triangles. Students explain why the triangles constructed are similar and determine the ratio of the corresponding sides. In Chapter 11, Constructive Assessment Options, Problem 7, students find the ratio between surface area and volume of two similar triangular pyramids and explain why their cross sections are similar.
• G-SRT.6: In Discovering Geometry, Lesson 12.1, students explore right triangles with acute angle measures of 20 and 70 degrees. As students draw similar triangles with these angle measures, students develop an understanding that side ratios in right triangles are properties of the angles in the triangle, which leads to definitions of trigonometric ratios for acute angles.
• S-ID.7: In Discovering Algebra, Lesson 3.3, students interpret the slope of a graph (speed) and starting location (intercept) in order to provide walking directions to another student. In Chapter 3, Quiz 1 Form A, students complete similar problems.

The instructional materials for the Discovering series meet expectations that the materials provide intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The materials routinely address procedural skills and start each chapter with “refreshing your skills.” The materials include an Exercise section so students can independently practice skills and concepts addressed in the lesson.

The instructional materials develop procedural skills throughout the series. For example:

• F-BF.3: In Discovering Algebra, Lesson 7.5, Problems 4 and 5, students describe the graph of an equation based on the graph of the parent function. Students also write an equation given the description of the transformation and the graph of the parent function. In Chapter 7, Quiz 3, students describe the transformation and write the equation given a parent function.
• A-APR.6: In Discovering Advanced Algebra, Lesson 6.8, students multiply and divide rational numbers without context to develop procedural skill. Students solve these stand-alone problems during the lessons in Example A and B as well as during the Exercise portion. In Quiz 3, students solve four problems involving rational equations without context to further develop procedural skill regarding rational expressions.
• G-GPE.7: In Discovering Geometry, Coordinate Geometry 9, students use the distance formula to determine the perimeters and areas of quadrilaterals and triangles.

The instructional materials provide opportunities to independently demonstrate procedural skills throughout the series in the following examples:

• A-SSE.2: In Discovering Advanced Algebra, Lesson 6.2, students complete guided examples of how a cubic expression for volume can be converted from one form to another (i.e., standard to factored). The materials include methods for using each form to find information about the behavior of the function (the graphed path of the volume expression). Students solve similar problems individually during Exercise 6.2.
• F-BF.3: In Discovering Algebra, Lesson 7.5, students individually practice transforming absolute value, quadratic, and exponential functions. In Discovering Advanced Algebra, Lesson 3.5, students individually practice transforming square root functions, in Lesson 3.7, circles, in Lesson 6.6, rational functions, and in Lesson 7.5, trigonometric functions.
• G-GPE.4: In Discovering Geometry, Coordinate Geometry 11, students individually use coordinates to prove the definitions of polygons, such as specific quadrilaterals and triangles, by completing the distance formula or slope.

The instructional materials for the Discovering 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. Overall, the instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematics and independently demonstrate the use of mathematics flexibly in a variety of contexts.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematics throughout the series. For example:

• N-Q.2: In Discovering Algebra, Lesson 2.3, students use conversion rates within contextualized problems. In Exercise 11, students find the conversion factor from a table and use it to solve multiple problems. This problem is multi-step and non-routine.
• S-IC.1: In Discovering Advanced Algebra, Lesson 9.1, students use statistics as a process for making inferences about population parameters based on a random sample from that population within contextualized problems.
• Chapter 6 of Discovering Geometry addresses applications of transformations. Students explore transformations through activities such as "Finding A Minimal Path" and "Exploring Tessellations."

The instructional materials include opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. For example:

• F-IF.4: In Discovering Algebra, Lesson 4.3, Graphs of Real World Situation, students are provided with four problem contexts and need to match them to their corresponding graphs (six are given).
• A-SSE.3: In Discovering Advanced Algebra, Lesson 5.1, students construct a function from a table and answer questions using their function related to the problem context.
• G-SRT.8: In Discovering Geometry, Lesson 12.2, students use trigonometric functions to solve single and multi-step contextualized problems.

The instructional materials for the Discovering series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The materials represent each aspect of rigor both independently and together.

The materials frequently prompt students to explain their reasoning to demonstrate conceptual understanding, and students complete at least one investigation in each lesson to better understand the concept in the lesson. The lessons also provide opportunities for students to practice problems to increase procedural skills with certain topics. The materials frequently use application/contextualized problems to relate concepts to real-world scenarios. Problems oftentimes address conceptual understanding with procedural skill or conceptual understanding with application.

All three aspects of rigor are present independently throughout the program materials in the following examples:

• Functions and transformations are addressed in Discovering Algebra and Discovering Geometry. The “rules” related to different transformations, presented in Discovering Algebra, represent the procedural skills for this topic. The use of transformations with functions and combinations of transformations in Discovering Geometry represent conceptual understanding of this topic.
• In Discovering Algebra, Lesson 5.1, students solve systems of linear equations. Most of the problems in this lesson do not have contexts, so students develop procedural skills in relation to A-REI.6.
• In Discovering Geometry, Lesson 4.6, students complete several proofs. Students develop conceptual understanding of the triangle congruence criteria in regards to G-CO.8 and G-SRT.5 through the completion of the proofs.
• In Discovering Advanced Algebra, Lesson 7.6, Exercise 14, students apply trigonometric functions to model a person’s distance from the ground at different places on a double ferris wheel and at different points of the ferris wheel's rotation.

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a topic throughout the materials in the following examples:

• In Discovering Advanced Algebra, Lesson 6.1, students write a function to model the height of a falling object due to gravity (application). Students graph the function and write a description of the graph in the context of the problem (conceptual understanding) for S-ID.6a.
• In Discovering Algebra, Lesson 6.1, students compare adding the same amount to an account each year with earning interest on the amount each year (application). From this, students compare linear growth to exponential growth (conceptual understanding). In Example C, students divide consecutive terms to find the constant multiplier of a sequence of numbers (procedural skill).

The instructional materials for the Discovering series meet the expectation that the materials support the intentional development of reasoning and explaining (MP2 and MP3), in connection to the High School Content Standards. The majority of the time, MP2 and MP3 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others.

Some examples where MP2 (Reason Abstractly and Quantitatively) is used to enrich the mathematical content include:

• In Discovering Algebra, Lesson 4.2, Investigation Step 1, students examine a series of tables, determine which relations are functions (MP2), and explain their reasons for their answers.
• In Discovering Geometry, Lesson 8.1, Investigation 1, students transform a parallelogram and a triangle labeled with dimensions expressed as variables to derive the formulas for the area of a parallelogram and triangle, respectively. Students reason abstractly by transforming general figures and manipulating variable dimensions, and they can reason quantitatively by contextualizing the general figures and calculating numerical areas to verify the derived formulas are valid (MP2).
• In Discovering Advanced Algebra, Lesson 3.1, students examine a graph of the speed and time two cars traveled. Students determine which car will be in the lead after 1 minute (MP2) and explain their reasoning.
• In Discovering Advanced Algebra, Lesson 2.7, students reason abstractly and quantitatively (MP2) by adding and subtracting rational numbers. Students add fractions using fraction bars, and they add and subtract rational expressions in an abstract manner.

Some examples where MP3 (Construct Viable Arguments and Critique the Reasoning of Others) is used to enrich the mathematical content include:

• In Discovering Algebra, Lesson 2.1, students examine the work of three different students and answer the following questions: “There are many ways to solve proportions. Here are three student papers, each answering the question ‘13 is 65% of what number?’ What steps did each student follow? What other methods can you use to solve proportions?” Students analyze the different solutions to determine what steps were taken.
• In Discovering Algebra, Lesson 5.2, "The Slopes Investigation," students plot separate rectangles and find the slopes of the four sides to conclude that opposite sides have equal slopes and adjacent sides have slopes that are opposite reciprocals. They move from the concrete shape to the abstract slopes and construct an argument to support their findings (MP2 and MP3).
• In Discovering Geometry, Chapter 4, Constructive Assessment Options, Problem 2, students agree or disagree with "Chloe" as to whether her triangle on her quiz has enough information to solve the problem. Students provide reasoning to support their answer.
• Discovering Advanced Algebra, Lesson 2.2, Exercise 12 contains a system of equations. Three fictitious students in the problem recommend different ways to solve the system by substitution, elimination, or graphing. Students determine which method works the best for this particular problem and why.

An example of the misidentifications for MPs 2 and 3 is in Discovering Geometry, Lesson 7.2. The teacher’s notes for summarizing the lesson include: “Return to the list of all potential shortcuts: AA, SSS, SAS, SAA, ASA, and SSA. 'You’ve considered the first three as similarity shortcuts in this lesson; what about the last three?' Ask whether it would help to consider cases in which SSA failed as a congruence shortcut [SMP 1,3,6].” Students do not construct a viable argument or critique the reasoning of others (MP3).

The instructional materials for the Discovering 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. The majority of the time, MP7 and MP8 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, support is present for the intentional development of seeing structure and generalizing.

Some examples where MP7 (Look for and Make Use of Structure) is used to enrich the mathematical content include:

• In Discovering Algebra, Lesson 3.7, students examine the graph of two lines and use the structure of the graphed lines to determine how the lines and their equations are similar.
• In Discovering Algebra, Lesson 6.1, students look for and describe patterns in the data they have collected. They look for structure when they analyze the pattern to see if it is linear. By examining data and determining that linear data grows at equal amounts over equal intervals, students look for and make use of structure.
• In Discovering Geometry, Lesson 6.4, Congruence Shortcuts, students complete a series of compositions to see if certain compositions of transformations can be combined into a single transformation.
• In Discovering Advanced Algebra, Lesson 5.2, Investigation, students expand a binomial raised to the second power resulting in a perfect-square trinomial. Students use the structure of the perfect-square trinomial to rewrite other expressions as perfect-square trinomials (completing the square) which develops the vertex form of a quadratic equation. Students also use the structure of completing the square to determine how to find the x-coordinate of a vertex given the general form of a quadratic equation. MP7 is not identified in the teacher materials for this lesson, though students use structure to proceed through the investigation.

Some examples where MP8 (Look for and Express Regularity in Repeated Reasoning) is used to enrich the mathematical content include:

• In Discovering Advanced Algebra, Lesson 6.7 Investigate Structure, students graph rational functions. Students determine the slant asymptote equation for a general rational function in terms of variables by recognizing patterns from the graphs of the provided functions.
• In Discovering Algebra, Lesson 3.1 The Toothpick Patterns Investigation, students complete a series of repeated steps to determine a recursive formula for finding the number of toothpicks in subsequent terms.
• In Discovering Geometry, Lesson 5.1, students examine different angle sums in polygons and look for a pattern to determine the polygon interior angle sum formula.

An example of the misidentifications for MPs 7 and 8 is in Discovering Geometry, Lesson 2.3. Students draw a model of handshakes using points and line segments and, after completing a table, are told, “Notice that the pattern does not have a constant difference. That is, the rule is not a linear function. So we need to look for a different kind of rule.” The teacher’s note states: “Step 3. Students may be drawing points rather randomly. “How could you arrange the points to be sure that every pair is connected by a line segment?” [Using vertices of a convex polygon is a good arrangement.] [SMP 1,2,4,7,8]" Duplicating a representation given in the investigation does not engage students in MP4. Furthermore, students do not use MP7 or MP8 as students are told that the pattern does not have a constant difference. Students do not engage in MP7 or MP8 on their own due to the steps that are provided for the investigation.