High School - Gateway 2

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Each Concept promotes student thinking with an introduction activity that is relevant and accessible to all students, preparing students for a more in-depth exploration of mathematical concepts in the investigations.

• Algebra I Concept 7.2 Intro asks students to estimate the area of a square with a given side length and the side length of a square with a given area, then explore both ideas using a dynamic geometry tool.

Tasks are designed so students develop understanding of the mathematics through an investigation.

• Algebra I Unit 3, Functions, has students engage with real-world phenomena involving dependent relationships to understand the “input/output” nature of mathematical relationships, classify data as representing functions or non-functions, and develop their own definitions of function. (F-IF.1, F-IF.2, F-IF.3, F-LE.1a, F-IF.1b)
• Geometry Concept 2.2 Investigation 2 has students make a conjecture and use a dynamic geometry tool to explore and then follow a scaffolded process to test their conjecture, write a general rule, and summarize their results. (G-CO.6, G-CO.7, G-CO.8)
• Algebra II Concept 5.2 Investigation 1 has students explore the possibilities for the intersection of some nonlinear equations and a line, then follows up with questions and tasks to formalize student understanding. (A-REI.7, A-REI.11)

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for providing intentional opportunities for students to develop procedural skills and fluencies.

Each Concept includes a Practice tab with:

• A Coach section that focuses on procedures and leads students through solving a problem. This section provides support and immediate feedback.
• A Play section that focuses on procedures without scaffolded support.
• A Check Your Understanding section focused primarily on procedural tasks.

The following are examples of items found in Practice:

• In Algebra II Concept 2.4 Investigation 1, students investigate rewriting logarithmic equations as exponential equations to solve logarithms (F-LE.4, F-LE.5). After exploring the inverse property of inverse exponents and logarithms to rewrite equations, students are given the opportunity to reach procedural fluency with the “Check Your Understanding. “
• In Geometry Concept 6.1, students work through right triangle trigonometry by solving problems around mapping out a race course and finding checkpoints along the way. The Coach section has the students practice the procedural skills of solving trigonometric ratios.

Procedural items are also embedded into Concepts and Investigations.

• In Algebra II Concept 7.1 students write rational numbers in different forms and connect this to writing rational expressions in different forms (A-APR.6). They apply their understanding of factoring to identify and generate equivalent rational expressions (A-SSE.2).
• Throughout the investigations in Algebra II Concept 6.1, students write polynomials in different forms, perform operations with polynomials, and interpret parts of the expressions helping them reach further procedural fluency with A-SSE.1b, A-SSE.2, and A-APR.1.

The instructional materials reviewed for the High School Discovery Traditional series meet expectations that they support the intentional development of students' ability to utilize mathematical concepts and skills in engaging applications, especially (but not limited to) where called for in specific content standards or clusters. The curriculum is written with an investigative, context-based approach that allows multiple opportunities for students to apply what they discover in each lesson.

Every Concept includes multiple Apply tasks that provide students meaningful opportunities to apply the mathematics they have learned.

• Algebra I Concept 6.2 Apply 1 and 2 provide opportunities for students to apply their abilities to represent and analyze data for population growth and building height. Students’ apply their knowledge of correlation and residuals to the contexts as well as model the data with a regression line and make predictions from it (S-ID.6a-c; S-ID.8). Apply 3 asks students to reason and to justify their understandings of correlation in the context of sports data. (S-ID.8)
• Geometry Concept 10.2 Apply 2 provides students with an opportunity to use volume formulas to solve problems (G-GMD.3) and apply geometric concepts in a modeling situation (G-MG.A).
• Algebra II Concept 11.2 Apply 3 has students determine a strategy for winning a contest involving cell phone data by making an estimate based on random sampling of the populations’ text message usage during a certain time period. (S-IC.1)

In addition, application tasks are typically included in Investigations.

• Algebra I Concept 6.1 Investigation 4 has students compare data about cereals’ nutrition. They apply their knowledge of measures of center and spread by determining which measure, center or spread, is most appropriate to use to make comparisons of the cereals’ nutritional data. Students then write an argument to justify the most healthy cereal based on these comparisons. (S-ID.2)
• Geometry Concept 7.2 Investigation 4 looks at circles inscribed in regular polygons, leading to informal justification of the formula for the circumference of a circle, and then has students apply what they’ve learned to a task about lights on a ferris wheel. (G-GMD.1)
• Algebra II Unit 8 has students interpret rational functions in a variety of applications. In the Introduction, students determine the function necessary to give a child of specific ages doses of medication. The application of medicine doses continues in Investigation 3 where students look at different medications and what different parts of the function mean in relationship to the medication. (F-IF.B)

The entire series is designed with application problems embedded within the lessons, as well as an Apply section at the conclusion of the lesson.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately and that the three aspects of rigor are balanced with respect to the standards being addressed.

Each Concept includes Discover, Practice, and Apply sections:

• Discover includes Intro, Investigation, Summary, and Extension sections that give students the opportunity to build conceptual understanding of the mathematics and practice procedural skills, typically in the context of a real-world example;
• Practice focuses on procedural skills with a Coach section that provides student support to develop fluency- for example, leading students through solving an algorithmic problem and giving immediate feedback- as well as a Play section where students demonstrate procedural fluency without support; and
• Apply includes extended tasks based on real-world applications.

In the Model Lesson section of the teacher materials, Progressions and Standards includes a diagram that identifies for teachers the balance of conceptual understandings, procedural fluencies, and applications that should emerge from each Concept in a Unit.

The following are examples of balancing the three aspects of rigor in the instructional materials:

• Algebra I Concept 8.2 asks student to examine data in a table to see what type of function it matches, to analyze the data in the table to identify key features, to write an equation to model the data, and to evaluate the equation for a given value. (A-SSE.3c, F-IF.7e, F-IF.8b, F-LE.1c, F-LE.2)
• Geometry Concept 6.1 gives students an opportunity to explore right triangle trigonometry first by developing the concepts within the context of racers finding distances on a map to determine shortest distances, as well as finding how to get back on course. Students then practice skills they have learned in the Coach and Play sections within the Practice tab. Finally, students apply what they have learned by designing a wheelchair ramp and hiking trails in the Apply tab. (G-SRT.7, G-SRT.8)
• Algebra II Concept 6.1 has students explore polynomials in different forms and develop an understanding of the meaning for writing the different forms of polynomials. Throughout the investigations, they use contextual situations, connections to prior knowledge about the different forms of quadratics, and peer interactions to conceptually develop their ability to write equivalent forms of polynomials, perform operations with them, identify zeros, and interpret different parts. Students practice these skills in the Coach and Play sections within the Practice tab. In the Apply tab, there are two application problems available for students to apply what they have learned using polynomial models that represent a lizard population and the design of a roller coaster. (A-SSE.1b, A-SSE.2, A-APR.1, A-APR.2,3 A-APR.4)

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for supporting the intentional development of making sense of problems and persevering in solving them as well as attending to precision (MP1 and MP6).

The course lists the Standards of Mathematical Practice in each section of the model lesson for the teacher. The teacher can see which Mathematical Practices are being addressed and in what way. Examples of the use of MP 1 and 6 to enrich the mathematical content include:

• Algebra I Concept 10.1 asks students to use precise language and clear definitions when discussing quadratic functions and their related equations; explain the correspondence among equations, graphs, and verbal descriptions to make sense of the given problems; use repeated reasoning as they persevere in creating the necessary adjustments to the model with algebra tiles representing values that will complete the square; and estimate solutions from graphs when calculating the exact roots of quadratic equations.
• Geometry Concept 1.2 Session 1 directs students to analyze given information and constraints while using a grid system to identify possible strategies for locating a statue.
• Algebra II Concept 9.2 Investigation 6 has students determine which trigonometric sine function gives the best prediction of temperature, which includes analyzing the preciseness of each model to find the most accurate model for predicting temperature.

In addition, each Concept includes multi-part, extended Apply tasks where students develop perseverance and precision in their work. For example, Geometry Concept 5.1 Apply 1 asks students to set up a projector to provide the best image; to do so, they must meet given constraints, experiment with the projector, and give a mathematical justification for their choice.

The materials clearly identify these mathematical practices, both in unit-level teacher notes and embedded in individual Investigations. For example, Algebra II Concept 3.2 Investigation 1 includes a teacher note to guide application of MP 1 and 6. Teachers are advised to have students discuss moving to the use of imaginary numbers. Key terminology, common mistakes and misconceptions are given as well.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for supporting the intentional development of reasoning and explaining (MP2 and MP3). MP2 and MP3 are used to enrich the mathematical content and are not treated separately from the mathematical content.

Examples of the use of MP 2 and 3 to enrich the mathematical content include:

• Algebra I Concept 2.1 Investigation 5 has students decontextualize a given real-world situation by representing it symbolically with equations and inequalities and then re-contextualize the algebraic solution sets of the inequalities in terms of the original real-world situation.
• Geometry Unit 1 Concept 1.2 Investigation 2 asks students to reason abstractly and quantitatively to derive an expression for the midpoint of a segment and to develop a formula for partitioning a segment in the coordinate plane in a given ratio.
• Algebra II Concept 9.2 Investigation 6 has students work to determine which trigonometric function gives the best prediction of temperature, justifying the reasonableness of their algebraic results to their original conjectures.
• Geometry Concept 5.1 Investigation 2 has students reason abstractly and quantitatively (MP 2) as they describe dilations as functions. In Investigations 1 and 3, MP3 is embedded as students make conjectures about dilations and then explore and justify their conclusions. They culminate their explorations in Investigation 3 by constructing arguments for the properties of similarity transformations.
• The Algebra II course demonstrates MP2 in Concept 5.1 Investigation 2 where students analyze how inequalities and graphs can be used to solve a real-world problem. MP3 is evident in Concept 11.2 Investigation 2 when students evaluate their model to determine whether it creates a realistic representation of the game and justify their reasoning to others.

The materials clearly identify these mathematical practices, both in unit-level teacher notes and embedded in notes to teachers for individual Investigations. For example, Algebra I Concept 4.1 Investigation 1 includes a teacher note to guide application of MP 2 and 3. The Investigation gives students tables of data about the density of two candies and asks them to graph, analyze, and discuss the condition under which each floats. The teacher note describes key points to address in the discussion with students to identify key features of the graphs.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for supporting the intentional development of modeling and using tools (MP4 and MP5). MP4 and MP5 are used to enrich the mathematical content and are not treated separately from the mathematical content.

Examples of the use of MP 4 and 5 to enrich the mathematical content include:

• In Algebra I Concept 4.1 Investigation 4 students are asked to use a graphing tool to draw parallel and perpendicular lines by changing slope and y-intercept in the context of locating fences and then analyze how changes in equations’ features affect their graphs using a graphing calculator tool.
• In Geometry Concept 4.2 Investigation 1 students examine art designs and try to construct their own version. Students choose and use geometric tools in order to recreate the design and write detailed descriptions of how to do it.
• In Algebra II Concept 4.1 students explore and make sense of the equations for circles and ellipses. Interactive tools such as Conic Section Interactive, Circle Interactive, Ellipse Interactive, Dynamic Geometry Tool, and Graphing Calculator Tool are embedded in the investigations and available for student use. In addition, students build a model of conic sections through these investigations and apply the graphing form of an ellipse to the orbital path of Pluto.

The materials clearly identify these mathematical practices, both in unit-level teacher notes and at the beginning of individual Investigations. For example, Geometry Concept 8.1 Intro includes a teacher note to guide application of MP 4 and 5 by reminding teachers of the possible tools students could use to explore locations within a triangle and different geometric models that could be used to represent a real-world problem. These teacher notes at the Concept level provide general guidance for teachers to emphasize MP 4 and 5, but there is limited support at the Investigation level for these math practices.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for supporting the intentional development of seeing structure and generalizing (MP7 and MP8). MP7 and MP8 are used to enrich the mathematical content and are not treated separately from the mathematical content.

Examples of the use of MP 7 and 8 to enrich the mathematical content include:

• Algebra I Concept 10.2 Investigation 1 asks student to use completing the square to develop the quadratic formula and look at how values in the quadratic formula provide information about the roots of the equation.
• In Geometry Concept 6.2 Investigation 3 students use geometric software to explore the relationship between the Pythagorean Theorem and the law of cosines. They analyze the structure of the triangle and understand the role of an auxiliary line in the development of the formula, determine which calculations are repeated, and look for other methods.
• In Algebra II Concept 1.1 students explore recursive functions. In Investigation 3, students use a hands-on paper tearing activity to record the outcomes of tearing their paper in half repeatedly. They observe the structure and pattern of the results (a geometric sequence) and use their observations to generalize their results into both recursive and explicit formulas. Repeated reasoning appears throughout other Investigations in Concept 1.1 as students use similar processes to develop understanding of geometric sequences, as they did to develop arithmetic sequences. In addition, students use the regularity in the reasoning associated with the arithmetic and geometric patterns to generalize to formulas, including Sierpinski’s Triangle.

The materials clearly identify these mathematical practices, both in unit-level teacher notes and embedded in individual Investigations. For example, Algebra II Concept 9.1 Investigation 5 (Angle Measure) includes a reminder that MP 8 (generalizing) is met when students see a proportional relationship among arc lengths with the same central angle and gives a recommendation for discussing this relationship and relating it to radians.