High School - Gateway 2

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Every unit attends to the learning cycle, interweaving aspects of mathematical proficiency. “Some lessons are devoted to developing a concept, others to solidifying understanding, yet others to practicing mathematics.” For example, in Algebra 1, Unit 4, each lesson builds upon the previous to reinforce concepts:

• In Algebra 1, Lesson 4.1, students develop conceptual understanding by explaining each step in the process of solving an equation (A-REI.1).

• In Algebra 1, Lesson 4.2, students solidify their understanding by rearranging formulas to solve for a variable (N-Q.1, N-Q.2, A-REI.3, A-CED.4).

• In Algebra 1, Lesson 4.3, students practice solving literal equations (A-REI.1, A-REI.3, A-CED.4). 

The materials systematically support the development of conceptual understanding as students build on the previous day’s learning with each new lesson.

• N-RN.3: In Algebra 2, Lessons 3.5 and 3.6 develop the concept of irrational numbers. The students begin by plotting rational numbers on a number line and then move on to plotting irrational numbers on the number line. This activity helps students understand the similarities and differences between irrational and rational numbers. Once students establish this understanding in Lesson 3.5, the students develop their understanding of the properties of irrational numbers in Lesson 3.6. The materials address irrational numbers and their properties over two lessons, so students have the opportunity to develop a more thorough understanding of the mathematical concept.

• F-IF.7: In Algebra 1, Lesson 7.1, students develop an understanding of transformations on a graph and how a graph 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 lesson, students identify the key features of the graph and how changes to a corresponding equation will change the graph. This development is continued in Algebra 1, Lesson 7.2, where “students write and graph quadratics with multiple transformations.”

• G-CO.10: In Geometry, Lessons 3.1, as students explore ways of knowing the triangle interior angles sum theorem - one based on experiments with specific triangles and the other based on a transformational argument - they consider the difference between making a conjecture based on experimentation versus reasoning about the validity of the conjecture using diagrams. In Lesson 3.2, students generate proofs for specific cases showing that the base angles of isosceles triangles are congruent; and in Lesson 3.3, students prove that the base angles of isosceles triangles are congruent for all cases.

• G-GPE.B: In Geometry, Lessons 7.1-7.3, students engage in a complete learning cycle. In Lesson 7.1, students find the distance between two points; and in Lesson 7.2, students prove that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals. In Lesson 7.3, students apply coordinate geometry to quadrilaterals to prove that a given quadrilateral is a parallelogram, a rectangle, a rhombus, or a square.

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Examples in the materials where students independently demonstrate procedural skills include, but are not limited to:

• N-CN.7: In Algebra 2, Lesson 3.5, Retrieval, Ready, Set, Go, students solve six quadratic equations, three of which have complex solutions (problems 28, 31, and 32). In Algebra 2, Lessons 3.6-3.8 and Lesson 4.5, students solve additional problems that have complex solutions.

• A-SSE.2: In Algebra 2, Lesson 2.3, students derive the product, quotient, and power rules for logarithms by graphing various logarithmic functions and generalizing the patterns found in corresponding equations. In Retrieval, Ready, Set, Go, problems 1-12, students practice rewriting radical expressions with fractional exponents and logarithmic expressions with radical exponents. In problems 15-16, students practice applying the rules of logarithms by identifying which of the given expressions are, in fact, equivalent, and showing why. In problems 17-22, students practice converting exponential equations into their logarithmic equivalents, and vice versa. In Algebra 2, Lesson 4.6, students use the given features of a polynomial to find other features, such as writing the function in factored form when given the graph of the function or its roots. Because only some information about the function is provided, students write the equation of the polynomial from scratch or write the equation in a different form than is given in order to find the other missing information. Students continue practicing these types of problems on their own in Retrieval, Ready, Set, Go, problems 10-15. In problems 1-8, students also practice rewriting rational expressions by dividing out common factors from the numerator and denominator. 

• F-BF.3: In Algebra 1, Lesson 7.1, students complete scaffolded questions about the effect on the graph of f(x) by replacing it with f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative). In Retrieval, Ready, Set, Go, students complete problems to develop and demonstrate their procedural skill. In Lessons 7.2 and 8.3-8.4, students write equations and graph functions that have multiple transformations for quadratic and absolute value functions, respectively. Students continue to write equations and graph functions that have multiple transformations for cubic, rational, logarithmic, and trigonometric functions in Algebra 2, Units 2-8.

• G-GPE.7: In Geometry, Lesson 7.1, students determine the distance between two points on the coordinate plane. At times they are only required to calculate the distance vertically or horizontally. Other times, students are asked to determine the distance between two points on a diagonal. Students use the Pythagorean Theorem to determine the distance and then formalize that understanding into the distance formula. Students practice with the procedure and formalize their thinking (strategies) conceptually. The procedural use of the distance formula is reinforced in Geometry, Lesson 7.3, where students prove that figures are parallelograms by comparing the lengths of opposite sides.

• S-CP.3: In Geometry, Lessons 9.4-9.6, students connect conditional probability and general probability and come to understand the meaning of independence, using tests to determine if two events are independent, and using the Multiplication Rule to find probabilities for independent events. In Lesson 9.4, Retrieval, Ready, Set, Go, problems 9-14, students find the conditional probability in contextual situations. In Lesson 9.5, Retrieval, Ready, Set, Go, problems 7-10, students use formulas for P(A and B) and P(A|B) to determine independence; in problems 11-22, students find general and conditional probabilities. In Lesson 9.5, Retrieval, Ready, Set, Go, problems 5-15, students find general and conditional probabilities and determine independence within a range of contexts.

The materials reviewed for Open Up High School Mathematics Traditional series 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. 

Lessons often begin with an engaging scenario that is either a direct, real-world application of the content for that day or a unique, novel problem for the students to solve. The applications within the series support students as they develop their conceptual understanding of the mathematics and subsequent acquisition of abstract notation and procedural skills. 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-CED.3: In Algebra 1, Lessons 5.1, 5.2, 5.4, and 5.5, and within a pet-sitting context, students use systems of equations and inequalities to build a business model, minimize cost, and maximize profit. 

• A.REI.3: In Algebra 1, Lesson 4.1, students “develop insights into how to extend the process of solving equations to multistep equations” associated with the actions of Elvira, a cafeteria manager. This storyline continues in Lesson 4.2, where students “apply the equation solving process developed in the previous task to solving literal equations and formulas” as Elvira continues to work to improve the efficiency of the cafeteria.

• F-IF.4,5: In Algebra 1, Lesson 3.2, students use tables and graphs to interpret key features of functions (domain and range, increasing and decreasing intervals, intercepts, rates of change, discrete vs. continuous) while analyzing the characteristics of a float moving down a river. Students interpret water depth, river speed, and distance traveled.

• F-IF.7: In Algebra 2, Lesson 5.1, 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 Retrieval, Ready, Set, Go, problems 7-10, students interpret an equation and a graph to determine different ways of paying for a gift among friends.

• F-TF.5: In Algebra 2, Lesson 6.1, students will use right triangle trigonometry to develop a function for finding the height of a rider at any position on a stationary Ferris wheel with specified midline (center) and radius (amplitude). In Algebra 2, Lesson 6.2, students model the circular motion of a rider as the angle measures the amount of rotation around a moving Ferris wheel, creating a function that models the amplitude, period, and average height (or midline) of the rider.

• G-SRT.8: In Geometry, Lesson 4.10, 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 the Retrieval, Ready, Set, Go problem set, students work multiple real-world problems using trigonometric ratios to determine missing lengths and angles.

• S-ID.2: In Algebra 1, Lesson 9.7, students select the bridge design that will support the most weight upon calculating and comparing the mean and standard deviation of their weight-bearing data. In Algebra 1, Lesson 9.8, students calculate measures of center (mean and median) and spread (interquartile range and standard deviation) of test results of six classes; by analyzing these results, students determine which class performed the best on the test and wins free pizza. 

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Overall, the three aspects are balanced with respect to the standards being addressed.

Throughout the materials students engage in each of the aspects of rigor in a cycle: each unit contains at least one Developing Understanding lesson to build conceptual understanding in students, at least one Solidifying Understanding lesson to build on that conceptual knowledge, and at least one Practicing Understanding lesson. Real-world applications are often incorporated in these cycles. Within each lesson there are Retrieval, Ready, Set, Go problems that spiral procedural skills for students. Examples include:

• In Algebra 1, Lesson 2.1 (a Developing Understanding lesson), students build upon their experiences with arithmetic and geometric sequences and extend to the broader class of linear and exponential functions with continuous domains. Students compare these types of functions using multiple representations. In Algebra 1, Lesson 2.2 (a Solidifying Understanding lesson), students discern when it is appropriate to represent a situation with a discrete or continuous model, thus deepening their conceptual understanding. Within this lesson, students also practice modeling with mathematics by connecting the type of change (linear or exponential) with the nature of that change (discrete or continuous), thus supporting student development of procedural skill and fluency. Throughout both of these lessons, students learn the concepts and procedures through their application to real-world contexts (e.g., medicine metabolized within a dog’s bloodstream and pool filling/draining). (F-IF.3, F-IF.5, F-BF.1, F.LE.1, F.LE.2)

• In Algebra 1, Lessons 8.5-8.7, students work with inverse functions and engage in all three aspects of rigor. The emphasis of this learning cycle, students work with inverse functions. In Lesson 8.5 (a Developing Understanding lesson), students compare the two different ways of modeling the same real-world context (two friends who ride bikes for exercise) to see characteristics of inverse functions, such as: the graphs are reflections over y = x, the inputs and outputs in the tables are reversed, and the equations have reciprocal slopes with the variables reversed. In Lesson 8.6 (a Solidifying Understanding lesson), students find inverse functions from equations and graphs. In the culminating lesson of the unit, Lesson 8.7 (a Practice Understanding lesson), students combine the work of piecewise functions, inverse functions, and absolute value functions using tables, graphs, and equations. (F-LE.2, F-LE.3, F-LE.5, F.BF.2, F-IF.5, F-IF.7, F-IF.9, A-REI.10)

• In Geometry, Lesson 6.4 (a Develop Understanding lesson), students use the formulas for arc length and area of a sector, developed in Lesson 6.3, and proportional reasoning to calculate the ratio of arc length to radius. Students use this ratio to define a constant of proportionality for any given central angle that intercepts arcs of concentric circles and develop radians as a way to measure an angle. Students conduct this work as they help Madison design her circular garden. In the Retrieval, Ready, Set, Go problem set, students practice the procedures by answering problems related to area, circumference, and the arc length of circles. (G.C.5)

• In Geometry, Lesson 7.2 (a Solidifying Understanding lesson), students use the slope formula and determine the relationship between parallel and perpendicular lines. Students engage with procedures by finding the slope between two points. Students then think conceptually when comparing the slopes of two lines to determine if the lines are parallel, perpendicular, or neither. Students also use rotations and transformations to hypothesize about the relationships between the slopes of lines that have undergone a transformation. (G.GPE.B, G.GPE.5)

• In Algebra 2, Lesson 5.2 (a Solidifying Understanding lesson), students engage with the same function f(x) = 1/x, but with a primary focus on the procedures of graph transformations and writing equations from graphs, without incorporating application problems. The lesson begins with practice problems that activate students’ prior knowledge about transformations of functions addressed in earlier lessons in the course. The lesson then revisits a graph of the function f(x) = 1/x and students name the asymptotes and anchor points. Later, given a set of problems containing either a graph or a description of a function that is a transformation of f(x) = 1/x, students write the equation for each. In yet another problem set in the lesson, students match a given equation to one of the given phrases that describes a transformation from y = 1/x. (F-IF.7d+, A-CED.2) 

The materials reviewed for Open Up High School Mathematics Traditional  series 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 where students make sense of problems and persevere in solving them include, but are not limited to:

• In Algebra 1, Lesson 3.1, students make sense of a story context to make a graph of a function without any knowledge of shape or scale. Working within the context of a pool’s water level over time, students persevere in making sense of increasing and decreasing intervals, minimum and maximum values, and domain and range. 

• In Geometry, Lesson 2.1, students construct two shapes: a rhombus and a square. Within the materials, teachers are prompted to provide students with enough time to explore the constructions fully. Within this lesson, students make sense of the constructions and persevere through the process of making the construction.

• In Algebra 2, Lesson 1.1, students make sense of two different views of a problem situation, each using a different dependent and independent variable. Because students who attempt to model the problem using the same variables in both cases will likely encounter confusion, students have an opportunity to persevere and in doing so see the importance of defining and making consistent use of variables as well as highlighting features of a relationship between a function and its inverse.

Examples of where students attend to precision include, but are not limited to:

• In Algebra 1, Lesson 2.2, students identify the domains of two sequences. As students “connect the type of change---either linear with a constant rate of change or exponential with a constant change factor, with the nature of the change, either discrete or continuous”---they further conceptualize that an arithmetic sequence is a linear function with a domain that is limited to the natural numbers and that a geometric sequence is an exponential function with a domain of the natural numbers. In Lesson 2.2, problems 6-11, students discuss a situation that is not clearly discrete or continuous and attend to precision as they make mathematical arguments about modeling choices.

• In Geometry, Lesson 1.1, students attend to 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, Lesson 8.3, “students naturally attend to precision as they attempt to refine the parameters of the equations they write to model the data given in graphs and tables.” Students become aware of how small changes in some parameters of certain equations result in large differences in results, while other parameters may be less sensitive to small changes. 

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for supporting 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 where students reason abstractly and quantitatively include, but are not limited to:

• In Algebra 1, Lesson 4.4, students are given a statement and determine which of the two expressions represent a larger value. In addition, students reason abstractly about an expression, compare it to another expression, and explain their reasoning.

• In Algebra 1, Lesson 7.1, students reason abstractly by relating the numeric results in a table to the graphs and explaining how 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, Lesson 6.2, students reason about how an infinite process might converge on a unique value and 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.

• In Algebra 2, Lesson 4.2, students identify the characteristics of and graph the basic cubic function. Students come to understand that the same transformations they used to graph quadratic functions can be applied to cubic functions. Students reason abstractly and quantitatively as they compare the rates of change and end behavior of quadratic and cubic functions. In the Retrieval, Ready, Set, Go problem set, students reason quantitatively by substituting in values to compare different power functions. Students reason abstractly by making generalizations based on their knowledge of exponents.

Examples where students construct viable arguments and critique the reasoning of others include, but are not limited to:

• In Algebra 1, Lesson 3.2, students interpret two representations (a table and a graph) and determine if Sierra’s statements are correct. Within the lesson, students analyze the situations, justify their reasoning, and communicate their conclusions to others. Students listen to the reasoning of others and decide if they make sense.

• In Geometry, Lesson 4.8, students consider examples of right triangles and explain why their specified ratios are the same, or nearly the same. Students identify and justify that having one acute angle congruent in all of the triangles creates a set of similar right triangles. Students use this observation to 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.

• In Algebra 2, Lesson 6.4, students work in partners to apply their understanding about sinusoidal functions to the construction of their own Ferris wheel in terms of radius, height, period, and direction of rotation, along with its graph and equation. Students then share their responses with another pair and decide if they agree with the connections they each have made between the three representations of each of their Ferris wheels: the description, the graph, and the function equation. Students are given sentence frames to provide support for constructing viable arguments and critique the reasoning of others; the sentence frames allow students to explain the connections in the representations of their Ferris wheel and provide supporting evidence for their claims.

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. 

Examples where students engage in modeling include, but are not limited to:

• In Algebra 1, Lesson 5.3, students manipulate a system of equations to model the constraints of setting up a cat and dog boarding 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, Lesson 6.1, students analyze a plan to build a regular, hexagonal gazebo. In the plan, there are several statements with which students must agree or disagree and then design their own gazebo.

• In Algebra 2, Lesson 8.3, students model two real-world behaviors: a dampened oscillation (the up and down motion of the bungee jumper) and Newton’s Law of Heating. To do so, students draw on mathematics they know well, such as how to model periodic behavior and exponential growth and decay, along with their emerging understanding of combining functions to model more complex behavior. Students use the graphing calculator to test and refine their assumptions until they have an accurate model that fits the data given in the tables and graphs provided in the lesson.

Examples where students choose appropriate tools strategically include, but are not limited to:

• In Algebra 1, Lesson 2.8, 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, Lesson 2.2, 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 2, Lesson 2.5, students use various tools, such as tables, graphs, and technology, to find missing values for exponential functions. Because some problems require the use of a calculator whereas others do not, students make appropriate decisions about using technology, like finding exact values for log expressions without relying on a calculator when they can.

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards.

Examples where students look for and make use of structure include, but are not limited to:

• In Algebra 1, Lesson 8.3, students recognize that the operation of absolute value leads to a graph with two distinct sections. They relate the two sections of the graph to their experiences with piecewise functions and write equations in both piecewise and absolute value forms. Students use the structure of the graph to write piecewise functions and relate the structure of piecewise function rules to equivalent absolute value functions. 

• In Geometry, Lesson 1.6, students uncover the structure of regular polygons through experimenting with rotational symmetry and line symmetry. They notice that different types of line segments serve as lines of symmetry in regular polygons with an even number of sides versus those with an odd number of sides. 

• In Algebra 2, Lesson 3.4, students gain insight into the different types of roots for quadratic functions by making connections across multiple representations. Students connect information included in a table with the vertex and factored forms of a quadratic equation. Students use graphs to explain why roots of a quadratic function might be integers, non-integer rational numbers, irrational numbers, or non-real numbers.

Examples where students look for and make use of repeated reasoning include, but are not limited to:

• In Algebra 1, Lesson 8.6, students use the reasoning of how a function and its inverse are related to observe how they can write the inverse function by paying attention to the operations in the original function and the order in which they are performed. Students become familiar with characteristics of an inverse function and how to restrict are prompted to see that when finding an inverse you can sometimes just “undo” the operations in the opposite order of the original function.

• In Geometry, Lesson 1.2, students make a conjecture about the relationship between the slopes of perpendicular lines based on the collection of data obtained in four experiments. They also provide an informal justification as to why the conjecture is true by examining the rotation of the legs of a right triangle and relating this to the rise and run that determines slopes of the hypotenuse of a right triangle before and after rotation.

• In Algebra 2, Lesson 4.1, students apply previous experiences with functions and their representations to reason throughout this lesson. Students reason about the different rates of change and create recursive forms of equations as they seek generalizations. For example, students understand that a cubic function has a first difference that is quadratic, a second difference that is linear, and a third difference that is constant. Students generalize this pattern for all polynomial functions.