High School - Gateway 1

The instructional materials reviewed for the series meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Overall, the instructional materials address almost all of the non-plus standards, and almost all parts of them, across the series.

Below are examples of standards that are addressed across the series:

• In Course 2 Units 1, 4 and 8, students use functions fitted to data to solve problems in the context of the data, investigate the effect of outliers and influential points on regression lines, and summarize categorical data (S-ID). In Course 3, Units 1 and 4, students extend their ability to reason statistically and investigate samples and variation (S-IC).
• The standards from A-SSE are represented throughout the series. In Course 1, students explore linear, quadratic, inverse variation, and exponential patterns of change. In Course 2, students analyze and use linear, exponential, and quadratic functions in realistic situations. In Course 3, students’ understanding is extended with graphing linear, quadratic, and inverse variation functions; solving inequalities graphically; solving quadratic equations algebraically; graphing linear equations in two variables; and solving systems of linear equations in two variables.

There is one standard, G-GPE.2, that is not addressed within the three courses in the series, and there is one standard, F-TF.8, that is partially addressed in the instructional materials. For F-TF.8, problem 11 on page 68 in Course 3 gives students the opportunity to prove the Pythagorean identity, but there are not other opportunities in the three courses of the series for students to use that identity to find trigonometric ratios of an angle given one trigonometric ratio of the angle and the quadrant in which the angle lies.

The instructional materials reviewed for the Core-Plus Mathematics integrated series meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. The Core-Plus Mathematics integrated series addresses mathematical modeling throughout the entire series. Through various components of the curriculum, including Investigations, Summarize the Mathematics, and Think About the Situation, students are able to explore all facets of the modeling process.

Particular instances where students have opportunities to experience the entire modeling process include:

• Course 1, Unit 5, Lesson 2, Exponential Decay: Among the standards listed for this lesson are the modeling standards N-Q.1; A-CED.2; F-IF.4, 5, 7; F-BF.1; all of F-LE except F-LE.4; and S-ID.6. The lesson begins with a discussion of the 2010 BP oil spill in the Gulf of Mexico. The students are asked to think about a simulation experiment intended to model the cleanup efforts. Students work through the experiment and are shown two possible ways to analyze the results (linear and exponential). Students then complete the experiments, design and do experiments and then find equations to model data from given experiments which leads to an understanding of when and why an exponential model may be the best. Students do a ball bounce rebound experiment that will lead to an exponential result, are asked to analyze their results and support their thinking with regards to the best model. Results are verified by using different types of balls and students use a “NOW-NEXT” approach to developing a pattern. Students then move to a problem looking at prescription drug level decay in the body. After each experiment students are asked to check their understanding of what they have just observed. Students are presented with a set of real world situations and are asked to write equations to model exponential decay, then verify and justify their thinking.
• Course 2, Unit 8 Lesson 1, Probability Distributions: Among the standards listed for this lesson are the modeling standards S-ID.5; S-CD.1 - 5, 6, 8; and S-MD.1-3, 5, 6. The lesson begins with a discussion of physical characteristics as determined by genes. Students are asked how to determine the probability one might have any particular characteristic. The discussion then leads to the multiplication rule, and students are given opportunities to formulate ideas about when and how this rule can and should be applied. This leads to the concept of conditional probability. The students are given situations to explore which lead to the definition of conditional probability. They are then given situations where they have to understand and identify dependent and independent events and determine the probability of such events and varying conditions. Students work in groups and must be ready to explain and defend their work at each stage. In one application problem (problem 1, page 536) they are given information about using the tire valves on a car to determine if a parking ticket is warranted. The last section of this question asks if they think the judge ruled the owner was guilty or not guilty based on their analysis of the data.
• Course 3, Unit 2, Lesson 2, Inequalities in two variables: Among the standards listed for this lesson are the modeling standards A-SSE.1; A-CED.1, 2, 3; F-IF.4, 5, 7; and F-LE.1, 5. The lesson starts with a discussion of an assembly plant that must assemble and test two types of video game systems. The plant must maximize profit while staying within available time constraints. Students are asked to think about the situation and what might be needed to determine the best use of time. The discussion leads to finding equations which could be used together to find an answer. The students are then given a situation with which they are familiar from previous courses, selling tickets to a concert. After they have developed the inequalities they are asked to graph them and determine how they could use the graphs to determine “a feasible solution area.” They are asked to explain and defend their reasoning. At this point the idea of linear programming is introduced as a way to bring profit into the picture. At the end of the lesson they are asked to go back and work through the video game plant problem presented at the start of the lesson. There are two possible solutions for the system, and they must chose and defend their choice.

Course 1 provides an introduction to modeling linear relationships in Unit 3, modeling discrete mathematics in Unit 4, and modeling probability in Unit 8, and the following are instances where different parts of the modeling process are highlighted for students:

• Unit 3, Lesson 1, Investigation 3 allows for students to use technology to address A-SSE.1, F-IF.6 and F-BF.2 through manipulating various parts of an expression (Time Flies, pages 163 - 164) to find a rule to model situations that appear to be linear in nature. Extension opportunities are suggested for students to collect their own data by selecting a nearby airline hub and search its schedule for nonstop flights.
• Unit 5, Lesson 1, Investigation 1 has students explore a variety of situations involving exponential growth to address A-CED.1, A-CED.2, A-REI.10, F-LE.1 and F-LE.2 (Pay It Forward, pages 290 – 293) and develops student understanding and skill in recognizing and modeling these patterns.

Course 2 further develops modeling through geometric transformations in Unit 3, optimization in Unit 6, and probability in Unit 8, and the following are instances where different parts of the modeling process are highlighted for students:

• Unit 1, Lesson 3, within Think About This Situation multiple representations, graphing and symbolic with equations, are used to introduce systems of equations within a context to address A-SSE.1, A-CED.1-3, A-REI.11 and F-BF.1 (pages 50-53). Students are asked to identify parts of the problem and solutions, eventually leading to the focus of the standard, a solution to a system being an (x, y) value.
• Unit 7, Lesson 2, Investigation 3 has students explore, in a real-life setting and with software, triangles that are possible when two sides and an angle opposite one of those sides are given to address G-MG.1 and G-MG.3 (Propping Open a Cold Frame Box, pages 498 – 501). Students develop criteria for identifying the conditions under which this given information determines two, one or no triangles.

Course 3 attends to students' modeling capabilities through linear programming in Unit 2, polynomial functions in Unit 5, periodic functions in Unit 6, and recursion and iteration in Unit 7, and the following is an instance where different parts of the modeling process are highlighted for students:

• Unit 2, Lesson 2 permits students to investigate multiple scenarios that can be analyzed with linear programming for standards A-CED.1-3 and introduces students to various methods for doing so, including graphing and creating grids. Balancing Astronaut Diets (page 134) analyzes nutritional values as used as an example for A-CED.3. Summarize the Mathematics after Investigation 2 (page 136) requires students to compare the different problems to analyze common features.

The materials for this series, when used as designed, meet the expectation for allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, post-secondary programs, and careers. The following examples show how the standards/clusters specified in the Publisher's Criteria as Widely Applicable Prerequisites (WAPs) are addressed across the entire series.

• The Algebra standards are included throughout the series. Evidence is found in Course 1, Units 1, 3, 4, 5 and 7; Course 2, Units 1, 2 and 5; and Course 3, Units 1, 2, 5, 7 and 8.
• The Function standards are included throughout the series. Evidence is found in Course 1, Units 1, 3, 5 and 7; Course 2, Units 1 and 5; and Course 3, Units 2, 5, 6, 7 and 8.
• A variety of functions are interpreted and analyzed. Course 1 focuses on linear, exponential, and quadratic functions. Course 2 reviews and extends to power, non-linear, and trigonometric functions, and Course 3 focuses on polynomial, rational, circular and inverse functions.

Prerequisite material was mostly limited to Course 1: Unit 1, Lesson 1; Unit 2, Lesson 1; and Unit 3, Lesson 1. This material was identified by the publisher within the Unit Planning Guide as optional content, depending on students' prior learning experiences.

The instructional materials reviewed for the Core-Plus Mathematics integrated series meet the expectation for requiring students to engage in mathematics at a level of sophistication appropriate to high school.

Students engage in investigations throughout each unit that ground the standards in real-world contexts appropriate for high school use. The following are examples from various Units and Lessons within Courses 1, 2 and 3 that highlight how the series uses different types of numbers, different forms of equations, and different tools throughout that are appropriate to high school.

• In Unit 3 of Course 1, Lessons 2 and 3 have students working with linear equations and inequalities. During Lesson 2, students see equations in various forms, such as part a of the Check Your Understanding on page 194, and they also work with inequalities in various forms with non-integer coefficients and non-integer solutions, for an example, see problem 5 on page 196.
• In Unit 7 of Course 1, Lesson 2 has students determine equivalent quadratic expressions that are initially written in different forms, such as part a of Check Your Understanding on page 494. In Lesson 3, students solve quadratic equations that are not written in the same form, such as standard, factored, or vertex, or do not have integer coefficients, and even when the coefficients are integers, there are some equations that have irrational numbers as solutions. For an example of an equation with irrational solutions, see problem 8 on page 519.
• In Unit 4 of Course 2, Lesson 2 offers students multiple opportunities to analyze sets of data through least squares regression and correlation. The data sets have different sizes which are appropriate to high school, and the numbers within the data sets also vary. The least squares regression lines that are created do not just have integers as coefficients.
• In Unit 5 of Course 2, Lesson 3 gives students the opportunity to work with common logarithms and exponential equations. In this setting, the materials do not restrict exponents to integer values, and some equations have non-integer solutions, for an example, see problems 7-9 on page 385.
• In Course 3, Unit 2 engages students with multiple types of inequalities, for an example, see problem 3 on page 116. The inequalities are written in various forms throughout the unit, as on page 110, and the coefficients of the variables within the inequalities are not always integers, as in problem 1 on page 110. Even when the coefficients are integers, the solutions to the inequalities are not always integers, for an example, see problem 6e on page 114.
• In Unit 5 of Course 3, Lesson 3 engages students with rational functions. In this lesson, students have to create their own rational functions given other types of functions where the coefficients of the variables are not integers, and the solutions to the rational functions are also not integers. Also, in this lesson, students are presented with varying tools, such as manually drawing graphs or using technology to create graphs, that are all appropriate to high school to help them solve the problems.

The materials meet the expectation for fostering coherence through meaningful connections in a single course and throughout the series. Overall, connections between and across multiple standards are made in meaningful ways.

In Core-Plus Mathematics Course 1, the teacher material clearly references units which refer back to middle school understandings within Unit 1, Lesson 1; Unit 2, Lesson 1; and Unit 3, Lesson 1 indicating within the Planning Guide that these lessons are optional, depending on students' middle school background.

Teachers are cautioned that the materials need to be taught in the order they appear to assure coherence because each subsequent topic will depend on previously covered material. The implementation guide (page 6) states, "The eight Core-Plus Mathematics units in each course should be taught in the order they have been developed to retain the learning progressions, coherence, and connections designed into the program."

The student and teacher materials often refer back to prior lessons to make connections and/or build understanding. Specific examples of connections between and among conceptual categories include:

• Course 1, Unit 1, Lesson 1, “Cause and Effect:” While stating it may be omitted if students come with a very strong middle school preparation, this lesson is an example of how these materials provide coherence. The lesson objectives ask students to “develop disposition to look for cause-and-effect” and “review and develop” skills covered in middle school. The lesson also foreshadows some change patterns that students will address later in this course, or in a subsequent course. The use of “patterns of change” for the opening Unit, which connects Algebra, Functions, and Statistics and Probability, starts the high school courses with a cohesive and coherent theme.
• F-LE.2: Course 1, Unit 1, Lesson 2, Investigation 1 asks students to interpret population change data including creating and analyzing tables to write Now-Next rules, as a precursor or foundation for recursive function rules, F-BF.2.
• A-SSE.3, A-APR.3, A-CED.1,2, A-REI.4,7,11, and F-BF.1: Course 2, Unit 5, Lesson 2, On Your Own, Connections pages 370-373, students are asked to find the number of solutions that might arise in solving a system of equations where some equations are non-linear. They are asked to recall the methods they previously used to solve systems of linear equations (tables, graphs and algebra) and apply those methods to solve new types of systems. They are asked to speculate on the possible number of solutions they may need to look for in each type of system.

Two examples of connections made within the courses are:

• S-ID.6: Within Course 1, Unit 2 "Patterns in Data" begins to build the conceptual connection between univariate and bivariate data. In Unit 3, students' build on their prior experiences with linear relationship to strengthen their ability to recognize data patterns, graphs, and problem situations that indicate such linearity conditions.
• Within Course 2, Unit 5, Lesson 2 Investigation 1, question 4 asks students to "recall from work with multivariable relations..." connecting this topic back to Unit 1 in that course.

Several examples of connections made between the courses are:

• Within Course 2, students learn about how to solve systems of linear equations with graphing, substitution and elimination in Unit 1, then learn how to apply matrices to solve linear equations in Unit 2, and revisit the concept of systems of equations with nonlinear equations in Unit 5, even explicitly suggesting the use of graphs to explore possible solutions (first brought up in Unit 1). This idea is further built upon within Course 3, in Unit 2 where students apply their knowledge of systems of equations to linear programming as on page 131.
• A-SSE.1 - 3; F-IF.2,4, and 8: Course 2, Unit 5 Lesson 1 Investigation 3 connects to previous content topics in discussing quadratic expressions as products of linear expressions and further connecting the distributive property to multiply those linear expressions and expand the quadratic. Quadratics are addressed again in Unit 5 of Course 3, and the book clearly states "in your previous work with linear, and quadratic polynomial functions...." before continuing on with a lesson on the zeroes of polynomial functions. (page 329)
• G-CO.2, 4, 6, 9, 12: In Course 3, pages T1C–T1E give the key geometric concepts and relationships from Courses 1 and 2 that will be needed to implement the unit.

An opportunity to make connections between standards is missed in Course 3, Unit 5, Lessons 2 and 4. The materials provide limited opportunities to connect solving quadratic equations that have complex solutions, N-CN.7, with graphing quadratic functions, F-IF.7a.