High School - Gateway 2

The materials reviewed for Open Up High School Mathematics Integrated 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. Most of the lessons across the series are exploratory in nature and encourage students to develop understanding through questioning and various activities. Concepts build over many lessons within and between courses in the series. Examples include:

• N-RN.A: In Math 1, Unit 2, a contextual situation offers students the opportunity to understand how values of a dependent variable can exist on the intervals between the whole number values of the independent variable for a continuously increasing exponential function. Next, students examine the role of positive and negative integer exponents and begin to understand the need for rational exponents. Students further develop their conceptual understanding by verifying that the properties of integer exponents remain true for rational exponents.

• A-APR.B: In Math 3, Unit 3, students develop an understanding of multiplicity and a deeper understanding of the relationship between the degree and the number of roots of a polynomial. Then, students use their background knowledge of quadratic functions and end behavior to extend their understanding to higher-order polynomials. The polynomials in this unit are factorable and allow students opportunities to solidify their understanding of end behavior, the Fundamental Theorem of Algebra, the multiplicity of a given root, and what the multiplicity would look like graphically. Finally, students extend their understanding of the Fundamental Theorem of Algebra and the nature of roots by applying the Remainder Theorem.

• A-REI.A,B: Math 1, Unit 4 builds students’ conceptual knowledge by first introducing multivariable linear equations and then having students express given relationships in equivalent forms. Students engage with inequalities as they encounter the contextual need for inequalities. Students consider the differences and similarities between solving inequalities and solving equations, including that inequalities produce a range of solutions, the inequality symbol must be changed when multiplying or dividing by a negative number, and the reflexive property is true only for equations.

• G-GPE.1: Math 2, Lesson 9.1, students cut out triangles and pin them to a coordinate plane to build a unit circle effectively developing their understanding of the relationship between the Pythagorean Theorem and the equation of a circle at the origin. Students connect their geometric understanding of circles as the set of all points equidistant from a center to the equation of a circle. This task focuses on a circle (constructed of right triangles) with a radius of 6 inches in order to focus on the Pythagorean theorem and use it to generate the equation of a circle centered at the origin. After constructing a circle at the origin, students consider how the equation would change if the center of the circle is translated. 

• G-GPE.5: In Math 1, Lesson 8.3, students prove that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals. The proofs use the ideas of slope triangles, rotations, and translations and are preceded by a specific case that demonstrates the idea before students are asked to follow the logic using variables and thinking more generally.

• G-GPE.6: In Math 1, Lesson 8.2, students use similar triangles and proportionality to find the point on a line segment that partitions the segment in a given ratio. Students first find the midpoint of a segment using two possible strategies and use similar triangles to find segments in ratios other than 1:1. The formula for finding the midpoint of a segment is formalized during the discussion. The discussion can also be extended to derive a formula for finding the point that partitions a segment in any given ratio.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills and students independently demonstrate procedural skills throughout the series. Examples include: 

• N-CN.7: In Math 2, Lesson 3.4, students write 10 equations in vertex form, standard form and factored form. Five of these equations have complex roots. In Math 2, Lesson 3.5, students find complex roots of quadratic equations in problems 11, 13, and 14. In Math 2, Lesson 3.5 Retrieval Problem 2 has a complex solution and Go Problems 28, 31, and 32 have complex solutions. Additional practice is provided in the RRSG problem sets for Lessons 3.6-3.8. 

• A.APR.2: In Math 3, Lesson 3.5 students explore the relationship between the remainder and factor in problems 3-8 and make conjectures based on their work. In Math 3, Lesson 3.5 RRSG, students practice this concept in the Set Problems 10-20.

• A-CED.4: In Math 1, Lesson 4.2 students apply the equation solving process to solve literal equations and formulas. There are multiple opportunities to solve a formula for a given variable throughout the lesson and several practice examples in the RRSG exercises. Students are presented with multiple opportunities to practice this standard in the RRSG throughout the remainder of Math 1, Unit 4. Additionally, practice for this standard is also found in the RRSG exercises of Math 1, Lessons 5.5, 8.3, and 9.5. The standard is revisited in Math 2, Lessons 4.4 and 4.6.

• G-GPE.7: In Math 1, Lesson 8.1, students calculate the length of ribbon needed to create a specific pattern. Students then find the perimeter of a hexagon in a different pattern. In RRSG Set Problems 9-12 students find the perimeter of two triangles and two quadrilaterals. In Math 2, Lesson 9.3 RRSG Retrieval Problem 1 and Ready Problems 1-4 students also find the perimeter of polygons using ordered pairs in the coordinate plane.

• S-CP.3: In Math 2, Unit 10, students investigate conditional probabilities in a wide variety of different contexts and using different models such as tree diagrams, Venn diagrams, two-way tables, and formulas. Students then use what they know about conditional probabilities to determine whether events are independent. There are many opportunities for students to practice in the student tasks and Retrieval Ready, Set, Go (RRSG) problem sets.

The materials reviewed for Open Up High School Mathematics Integrated 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. The materials use real-world situations in which students can apply mathematical concepts. In situations where a real-world context is not immediately appropriate, the materials begin with abstract situations (graphs, dot models, etc.) and build to the application of the concept in a real-world situation in a later task. Every lesson involves a task, and every task is a real-world situation or a mathematical model that builds to a real-world situation.

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

• A-CED.3: In Math 1, Lessons 5.1 and 5.2, the pet-sitting problem uses systems of equations and inequalities to build a business model, minimize costs, and maximize profit.

• F-IF.4,5: In Math 1, Lesson 3.2, students use tables and graphs to interpret key features of functions (domain and range, where function is increasing/decreasing, x and y 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 using the function skills they are developing.

• F-BF.1: In Math 2, Lesson 1.2, students develop a mathematical model for the number of squares in the logo for size n. Students are encouraged to use as many representations as possible for their mathematical model.

• F-TF.5: In Math 3, Lesson 6.2, students use the Ferris wheel to determine how high someone will be after 2 seconds, after observing that the Ferris wheel makes one complete rotation counterclockwise every 20 seconds. Students are continuing the work from a previous task in Math 3, Lesson 6.1. Students then determine elapsed time since passing a specific position. Students generate a general formula for finding the height of a rider during a specific time interval and are then asked how they might find the height of the rider for other time intervals.

The materials reviewed for Open Up High School Mathematics Integrated 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. The instructional materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations by giving appropriate attention to developing students’ conceptual understanding, developing procedural skill and fluency, and providing engaging applications.

The materials engage students in each of the aspects of rigor in a pattern that repeats itself throughout the materials. Each unit contains Developing Understanding (conceptual understanding), Solidifying Understanding, Practicing Understanding tasks, in addition to the Retrieval Ready, Set, Go (procedural skill) problem sets.

For example, Math 1, Lesson 2.1, a Developing Understanding Task, focuses on conceptual understanding as 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 various representations (table, graph, and equation). In Math 2, Lesson 2.2, a Solidifying Understanding Task, students discern when it is appropriate to represent a situation with a discrete or continuous model, thus deepening conceptual understanding. This task also has students practice modeling with mathematics by connecting the type of change (linear or exponential) with the nature of that change (discrete or continuous) which develops students’ procedural skill and fluency. Throughout both tasks, problems are presented to students within real-world contexts (medicine metabolized within a dog’s bloodstream, library re-shelving efficiency, e-book download rate, savings accounts, pool filling, pool draining, etc), so students learn the mathematical concepts and procedures through the application of the mathematics.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of overarching mathematical practices (MPs 1 and 6), in connection to the high school content standards. Overall, MP1 and MP6 are used to enrich the mathematical content and are not treated as individual mathematical practices. Throughout the materials, students are expected to make sense of problems and persevere in solving them while attending to precision. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples where students make sense of problems and persevere in solving them include:

• Math 1, Lesson 3.1: Students are able to make sense of creating graphs given a situation. Students are already familiar with graphing rate of change and continuous and non-continuous situations. This task addresses domain and step functions. Students persevere in creating graphs by analyzing what is happening during each interval of time on their graph. 

• Math 2, Lesson 4.1: Students write piecewise linear functions and interpret piecewise functions presented in algebraic form. Throughout the task, students use the piecewise-defined function to make sense of the problem to identify key features of the function including average rates of change, domain, and other relative information connecting the context to the function. 

• Math 3, Lesson 7.2: Students interpret a given function to answer questions about high tide, low tide, and the time between tide events. Students make sense of the problem by using multiple representations such as graphs, tables, the unit circle, and the meaning of the parameters of a periodic equation to answer these questions. 

Examples where students attend to precision include:

• Math 1, Lesson 5.5: Students represent constraints in the context of a pet sitting business with inequalities and with systems of inequalities. Students find the point of intersection and must interpret its meaning in the context of cats and dogs. Students must attend to the language in the constraints. Students must recognize that time is measured in both minutes and hours in the constraints which require them to attend to the units they choose to use. 

• Math 2, Lesson 7.1: Students attend to the precision of language by using correct mathematical vocabulary when describing and illustrating their process for finding the center of rotation of a figure consisting of several image/pre-image pairs of points.

• Math 3, Lesson 5.1: Students visualize two-dimensional cross sections of three-dimensional objects and draw the cross sections of those objects. Students are encouraged to use precise language as they work through the task in order to precisely identify components of the three-dimensional objects used.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. Overall, MP2 and MP3 are used to enrich the mathematical content found in the materials, and these practices are not treated as isolated experiences for the students. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples where students reason abstractly and quantitatively include:

• In Math 1, Lesson 5.1, students solve systems of linear equations using a variety of strategies. Then students use the solution to make decisions based on a context. Students decontextualize the contexts when working with a system of equations to represent pricing plans for each company. Students must attend to units and the meaning of operations used in the equations. Students then contextualize the solution to verify it makes sense in the context. Students reason abstractly and quantitatively which company to hire by using a table, a graph, and/or Algebra.

• In Math 2, Lesson 1.4, students write a function for a given context. Students must reason quantitatively to generate the function based on a numeric pattern. Students are required to consider two parameters, perimeter and area, and they must write the function for the area using one variable. Students will decontextualize the model in order to identify the key features and then relate the key features to the story context. 

• In Math 3, Lesson 3.2, students identify the characteristics and graph the basic cubic function. Students will 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 practice set, students reason quantitatively by substituting in values to compare different power functions. They reason abstractly by making generalizations based on their knowledge of exponents.

Examples where students construct viable arguments and critique the reasoning of others:

• In Math 1, Lesson 3.2, students explain why they either agree or disagree with each observation Sierra made. Students listen to the reasoning of others and decide whether the reasoning makes sense. Students also justify or explain flaws in Sierra’s observations. 

• In Math 2, Lesson 6.3, students read through Mia and Mason’s conjectures about similar polygons and decide which they believe are true. Students are also presented “explanations” from either Mia or Mason and must write an argument deciding whether they agree. 

• In Math 3, Lesson 3.9, students find patterns in the end behavior of functions and describe the end behavior of functions using the appropriate notation. Throughout the task students will use prior knowledge, make conjectures, and develop a series of statements to describe the relationship between expressions as the input values approach various quantities. Students justify conclusions and respond to others by listening, asking clarifying questions, and commenting on the reasoning of others. 

The materials reviewed for Open Up High School Mathematics Integrated 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. Overall, MP4 and MP5 are used to enrich the mathematical content and are not treated as individual practices. Throughout the materials, students model with mathematics and use tools strategically. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples where students engage in modeling with mathematics include:

• In Math 1, Lesson 5.4, students use systems of equations, tables and graphs to model the start-up costs of a new business and the space available to board cats and dogs. Students write equations for the constraints in different forms and then identify what information each form reveals in the context of the situation. 

• In Math 2, Lesson 10.3, students use Venn diagrams to model the situation, analyze the data, and write various probability statements (unions, intersections, and complements) and then apply the Addition Rule and interpret the answer in terms of the model.

• In Math 3, Lesson 6.1 students develop ways of thinking about the location of points around a circle, which become fundamental in their understanding of trigonometric functions, radian measure, and the unit circle. Students develop expressions to model the height of a rider at a particular angle. Students apply mathematics they know (right triangle trigonometry) to model motion around a Ferris wheel based on angles of rotation. Students use points to model the location of the rider.

Examples where students choose appropriate tools strategically include:

• In Math 1, Lesson 9.1 students strategically use graphing technology to interpret the correlation coefficient of a linear fit. Using technology, students also alter the data to see the effect on the scatter plot and correlation coefficient. 

• In Math 2, Lesson 3.1, students solve quadratic equations graphically and using different algebraic techniques and make connections between solving quadratic equations and graphing quadratic functions. Throughout the student tasks, students have options for which method they would like to use. Sometimes the problems leave it up to the student to choose which method they would like to use, and specific methods are called out to be used in other problems. Students solve quadratic equations by factoring, completing the square, using inspection, graphing, and using symmetry. Students will have opportunities to select appropriate tools to use while solving and graphing. They can hand sketch graphs or construct graphs using graphing calculators or other digital platforms.

• In Math 3, Lesson 2.5, students solve exponential equations which would require the use of logarithms using tables and graphs. In the last part of the task, students solve systems of linear and exponential equations using a method of their choice. Students may choose to use calculators or other technology with base 10 logarithmic and exponential functions to complete the problems. Students are encouraged to 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 Integrated series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. Overall, MP7 and MP8 are used to enrich the mathematical content, and these practices are not treated as isolated experiences for the students. There is an increasing expectation that these practices will lead students to experience the full intent of the standards. 

The materials frequently take a task from a previous course and add a new contextual layer to the mathematics, such as Math 1, Lesson 5.2, “Pet Sitters” and Math 3, Lesson 1.1, “Brutus Bites Back.” Students are constantly extending the structures used when solving problems that build on one another and, as a result, are able to solve increasingly complex problems. In the instructional materials repeated reasoning based on similar structures allows for increasingly complex mathematical concepts to be developed from simpler ones.

Examples where students look for and make use of structure include:

• In Math 1, Lesson 2.1, builds upon students’ previous experiences with arithmetic and geometric sequences to extend to the broader class of linear and exponential functions with continuous domains. Students use tables, graphs, and equations to create mathematical models for contextual situations. Students continue to define linear and exponential functions by their patterns of growth. Students repeatedly use similarities and differences between situations to define the appropriate expression structure. 

• In Math 2, Lesson 4.3, students learn how to graph, write, and create linear absolute value functions by looking at structure and making sense of piecewise defined functions. They connect prior understandings of transformations, domain, linear functions, and piecewise functions and share strategies for how to go from one representation to another in order to graph and write equations for absolute value piecewise functions. 

• In Math 3, Lesson 3.7, students use their background knowledge of quadratic functions and end behavior to extend their understanding of polynomials in general. The polynomials in this task are easily factorable and allow students opportunities to solidify their understanding of end behavior, the Fundamental Theorem of Algebra, and the multiplicity of a given root, and what that would look like graphically. 

Examples where students look for and express regularity in repeated reasoning:

• In Math 1, Lesson 5.9, students practice solving systems of linear equations by obtaining systems of equivalent equations. By the end of the task, through repeated practice, students develop a procedure for solving a system of equations by elimination. 

• In Math 2, Lesson 3.2, students work with specific examples of quadratic functions in order to identify a process for locating the x-intercepts relative to the axis of symmetry. Students use the repeated reasoning after working the specific examples to develop the quadratic formula.

• In Math 3, Lesson 6.6, students calculate the x and y coordinates for stakes placed on concentric circles as well as the arc length on each circle placed around an archeological site. In problems 3 and 4 students look for patterns in the tables created and the processes used to calculate points and identify what the repeated patterns imply.