High School - Gateway 2

The instructional materials reviewed for this series meet the expectation that the materials support the intentional development of students' conceptual understanding of key mathematical concepts. The instructional materials engage students in an inquiry-based investigative structure, allowing for students to develop their reasoning and critical thinking skills as it applies to their conceptual development in mathematics. The Core-Plus Mathematics integrated series promotes mathematical reasoning through various components, including Think About This Situation found at the beginning of every lesson and the Summarize the Mathematics prompts found throughout each lesson. Students further develop conceptual understanding by working collaboratively with their peers and sharing their ideas aloud during class discussions, as indicated in the instructional materials.

Core-Plus Mathematics provides sets of On Your Own (OYO) homework tasks, which include both contextual and non-contextual work with the mathematical concepts developed in that lesson. One type of task in OYO sets are the Connections. These tasks help students to build links between mathematical and statistical topics they have studied in the lesson and to connect those topics with other mathematics that they know. Additionally, it is recommended that student solutions to Connections tasks be discussed in class, which provides students with opportunities to compare and discuss student work and synthesize key ideas into deeper conceptual understanding. The final lesson in each unit, Looking Back, offers students the opportunity to review and to synthesize the key mathematical concepts developed in the unit.

Concepts build over many lessons within and between each course in the series. Specific examples are:

• A-REI.10: In Unit 3 of Course 1, Lesson 2 explores lines of best fit for data showing the change in percentage of male and female doctors in the U.S. since 1960. The implication is that the number of female doctors will soon equal and, perhaps, surpass the number of male doctors. The materials present a structured discussion and mathematical analysis of the possible implications of using these graphs for making assumptions. The conversation also includes the use of inequalities to create a more robust view of the situation.
• F-IF.A: In Course 1, Unit 5 begins to develop students' conceptual understanding of rate of change through a graphical analysis of the growth of an exponential situation and then follows up at the end of the investigation with students comparing tables, graphs, and rules for two similar situations and sharing out with the class.
• A-SSE.1 and A-REI.10: In Course 2, Unit 1 builds students' conceptual knowledge through first introducing multivariable linear equations in Lesson 2 and then having students express given relationships in equivalent forms. The task in Lesson 3 is to find one pair (x,y) of values that satisfies two linear equations.
• G-SRT.6: Unit 7 of Course 2 on trigonometry builds on concepts from Course 1 about the rigid nature of triangles and how certain relationships in a triangle can fix the shape and/or size of the triangle. Investigation 1 of Lesson 1 uses a series of application problems that review what students have previously learned. Students are then introduced to an angle in “standard position” on the plane and asked to find ratios of sides from several triangles formed by a given angle and points on the line created by the angle. Standards from N-Q, A-REI, and F-IF are addressed as students discover the defined trigonometric ratios.
• A-REI.A: In Unit 1 of Course 3, Lesson 3 begins Investigation 2 with a description of a reasoning process that encourages students and teachers (page T58) to utilize a process of reasoning in solving problems.
• G-CO and G-SRT: In Course 3, Unit 3 develops students' conceptual understanding of similarity through analysis of Escher works and tiling patterns and how different components relate to one another.

Materials of Core-Plus Mathematics meet the expectation for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific standards or clusters. Overall, the clusters and standards that specifically relate to procedural skills and fluencies are addressed.

Students develop procedural skills, including creating graphs and tables and writing rules and equations, through investigative work in a variety of contextual examples. Additionally, students are given the opportunity to check themselves with the Check Your Understanding sections found toward the end of the investigation. These problems are usually based in contexts but not directly related to the work throughout the investigation which enables students to engage in the mathematics through multiple contexts.

One type of task in On Your Own sets are the Just-In-Time Review and Distributed Practice. These tasks provide opportunities for Just In Time review of concepts and skills needed in the following lesson and distributed practice of mathematical skills to maintain procedural fluency. A clock icon near the solution in the Teacher Guide designates Just In Time review tasks.

The materials offer sufficient opportunities for students to understand the procedures, and examples of how the materials address select cluster(s) or standard(s) that specifically relate to procedural skill and fluency include:

• A-SSE.1b: In Unit 5 of Course 1, Lesson 1 introduces the procedure for compounding interest and gives students the opportunity to interpret both the expression for the formula and the parts of the expression.
• A-SSE.2: There are many opportunities for students to use the structure of an expression to identify ways to rewrite it. Within Unit 7 of Course 1, page 497 offers ample opportunity for students to work with and identify patterns in the multiplication of linear expressions and the creation of quadratics.
• A-SSE.3a: Within Unit 5 of Course 2, pages 336-344 offer many opportunities to develop procedural fluency with factoring a quadratic expression to reveal the zeros of the function it defines. There are more opportunities on page 348. Page 355 gives students an opportunity to provide justifications for each step of deriving the quadratic formula from the standard form of a quadratic equation.
• A-SSE.3c: Within Unit 5 of Course 2, Lesson 3 has many opportunities to use the properties of exponents to transform expressions for exponential functions, and the lesson also addresses rewriting polynomial, logarithmic and exponential expressions. Lesson 4 provides more opportunities for students to engage with this standard.
• G-GPE.4: In Unit 3 of Course 2, Lesson 1 uses coordinates to prove simple geometric theorems algebraically and begins with a detailed example that highlights how to use coordinates when creating a proof.
• G-SRT.5: In Unit 3 of Course 3, Lesson 1 gives ample opportunities for students to work with and develop using similarity criteria for triangles to solve problems and prove relationships in geometric figures.

Materials of Core-Plus Mathematics meet the expectation 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. The Core-Plus Mathematics integrated series was written using a context-based approach to mathematics, and accordingly, it includes numerous opportunities for students to work through a variety of contextual applications for each of the mathematical concepts addressed in the series. Students practice with multiple ideas throughout the different investigations within a lesson, as well as independently in the "On Your Own" student practice sets.

One type of task in On Your Own (OYO) sets are the Applications. These tasks provide opportunities for students to apply their understanding of the ideas they have learned in the lesson. The series includes numerous applications across the series, and examples of select domain(s), cluster(s), or standard(s) that specifically relate to applications include, but are not limited to:

• A-SSE, F-IF, F-BF, and F-LE: In Course 1, Unit 1 analyzes patterns of change using tables, graphs and algebraic rules in a variety of contextualized situations, including bungee jumping, price setting, and income.
• G-SRT: In Course 2, Unit 7 develops students' contextualized understanding of trigonometric functions through calculating distances using the angle of elevation.
• G-CO, G-C, and G-MG: In Course 3, Unit 6 develops students' contextualized understandings of circular motion and periodic functions through first understanding circles and their properties (notably tangent lines) and then applying these concepts to pulleys and sprockets to study angular and linear velocity.
• S-ID.2: In Unit 2 of Course 1, Lesson 1 develops statistics with regard to distribution of data and appropriate shape. Graphs, tables, and discussion through questioning often involve contextual problems which help students make sense of the data and conclusions.
• A-APR.B: In Unit 5 of Course 2, Lesson 1 connects parabolas, their zeros, and factors to contextual situations.
• F-IF.B: In Unit 2 of Course 3, Lesson 1 utilizes line graphs, parabolas, and polynomials to introduce functions and their parts. This introduction quickly leads to interpreting solutions for the functions within the contexts given for the problems.

The instructional materials reviewed for the Core-Plus Mathematics integrated series meet the expectation for not always treating the three aspects of rigor together nor always treating them separately. Overall, the three aspects of rigor are balanced with respect to the standards being addressed. The series provides students the opportunity to develop procedural skills and conceptual understandings through contextualized applications. The materials include a variety of different contexts with which mathematical topics can be applied and promote students' conceptual development in understanding the meaning of the mathematics.

In addition to the classroom investigations, Core-Plus Mathematics integrated provides sets of On Your Own homework tasks, which are designed to engage students in applications and conceptual understandings of their evolving mathematical knowledge. The following are examples of balancing the three aspects of rigor in the instructional materials:

• In Unit 1 of Course 1, Lesson 3 begins with a section on patterns of change. The use of symbols, shapes, tables, graphs, discussions and contexts within this lesson offers students the opportunity to engage with and develop each aspect of rigor as they work.
• In Course 1, Unit 3 balances the aspects of rigor by establishing a foundation of conceptual understanding for linear growth and creating and using algebraic rules to express and solve for information about contextualized problems, including prices and earnings.
• In Course 2, Unit 5 balances the aspects of rigor by developing procedural fluency with solving nonlinear equations and using students' prior conceptual understanding of systems of equations to analyze real-world examples of nonlinear systems of equations, including supply and demand.
• In Unit 3 of Course 2, Lesson 1 uses technology and algebraic expressions to represent geometric shapes and ideas in the coordinate plane. Initially, there seems to be only connections to conceptual understanding and engaging applications, but procedural skill and fluency are built in through continuing work on page 172.
• In Unit 1 of Course 3, Lesson 2 uses parallel lines and transversals and geometric figures to work through understanding of proof. Throughout this section, procedural skills and conceptual understandings are treated separately as appropriate.

The materials meet the expectation of supporting the intentional development of reasoning and explaining (MP2 and MP3), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, 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.

Listed below are examples of where MP2 is used to enrich the mathematical content:

• In Course 1, Unit 2, Lesson 2, Investigation 4 introduces students to the idea of standard deviation as a measure of variability, after working on quantitative variability in Box Plots and Histograms. The investigation looks first at distance on the coordinate plane then moves, in a structured way, from the distance formula to an abstract understanding of deviation. All along the way, students are asked to make and explain the connections between abstract numbers and the quantities they represent.
• In Course 3, Unit 5, Lesson 1, the On Your Own on page 339 contains a lengthy situation where students are asked to look at a rule for a function and determine output values for certain inputs. They are then asked to give a reasonable domain and range for the situation and to explain why a given income function makes sense in the situation. They are then given an expense function for the situation and asked to use all known information to answer questions about the profit prospects for the company.

Listed below are examples of where MP3 is used to enrich the mathematical content:

• In Course 1, Unit 6 on page 374, students are asked to reason through determining triangle congruence using corresponding parts of triangles. In doing so, students are provided with reasoning and justifications and are asked to explain if the reasoning given is correct and why it is correct.
• In Course 3, Unit 3, Lesson 1, Investigation 1 gives information and a diagram showing possible similar shapes to students, and they are asked how they might test to see if the shapes were similar and how to go about proving their methods. Students are asked to compare answers with others and resolve differences.

Although not explicitly labeled in the majority of cases, MP2 and MP3 are sometimes used together to enhance the content. Listed below are examples of where both MP2 and MP3 are used to enhance the content.

• In Course 1, Unit 2, Lesson 1, On your own sections are often used to address reasoning and explaining. In this case, students are asked to describe and connect distributions to shapes and context.
• In Course 2, Unit 5, problems 30-32 on page 353 ask students to predict "common errors" when expanding quadratic expressions and offer explanations for how to help students understand the errors, and students also justify how they themselves decide on the best method to solve a quadratic equation. Then, problem 32 asks students to explain how concrete examples can help to make sense of factoring quadratic expressions, which requires students to make connections between quantities and abstract mathematical concepts.

The materials meet the expectation of supporting the intentional development of modeling and using tools (MP4 and MP5), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP4 and MP5 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, students are expected to model with mathematics and use tools strategically.

Listed below are examples of where MP4 is used to enrich the mathematical content:

• In Course 1, Unit 8 is entitled "Patterns in Chance." There are multiple opportunities for students to engage with MP4 during the unit, and in particular, the Check Your Understanding on pages 557-558 engages students in making assumptions and approximations explicit, deciding whether data are consistent with a proposed model, and revising the proposed model if that is deemed necessary.
• In Course 2, Unit 7, Lesson 3 requires students to analyze a sailing situation by first modeling it with a diagram that represents the problem. Then, students use trigonometry to find unknown information regarding the path of the ship.

Listed below are examples of where MP5 is used to enrich the mathematical content:

• On page 420 of Unit 6 in Course 1, students are encouraged to use interactive geometry software or other tools to investigate properties of regular polygons, including central angle measurements and interior/exterior angle measurements.
• In Lesson 2 of Unit 6 in Course 3, problems 5-7 of Investigation 4 engage students in a context where many tools could be utilized to help students solve the problem. In problems 5 and 6, students could draw graphs of the context manually or with technology to model the context, or they could select various physical objects to help them make sense of the problem. In problem 7, students could use various technological tools to create generalizations about the scenario.

The materials meet the expectation of supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, 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.

Listed below are a few examples of where MP7 is used to enrich the mathematical content:

• In Course 1, Unit 5, Lesson 1, the Summarize the Mathematics on page 303 asks students to explain how they would chose a regression model for a data set based on the patterns they found in the data.
• In Course 2, Unit 1, Lesson 3, part d of Summarize the Mathematics has students explain how they might tell the number of solutions for a system of linear equations just by looking at the equations.
• In Course 3, Unit 2, Lesson 1, Investigation 2 on pages 112-113 prompts students to discuss what the graph of a quadratic function would look like by using the expression only. In particular, students find the number and values of zeros of the function and speculate on regions that would satisfy quadratic inequalities related to the given quadratic functions.

Listed below are a few examples of where MP8 is used to enrich the mathematical content:

• In Course 1, Unit 5, Lesson 2, students use repeated applications of the Pythagorean Theorem to find the formula for the length of the diagonal of any square.
• In Course 2, Unit 8, Lesson 3, Investigation 3 has students use repeated trials to develop the formula for Expected Value.
• In Course 3, Unit 5, Lesson 1, Investigation 2 gives students the opportunity to work through an income/cost/profit situation for which they have previously developed a set of equations. By repeatedly operating with the equations in different scenarios, the students develop “rules” for how to operate with polynomial expressions and the implications of the operations on the resulting polynomial.

Although not explicity labeled in the majority of cases, MP7 and MP8 are sometimes used together to enhance the content.  Listed below are examples of where both MP7 and MP8 are used to enhance the content.  

• In Course 2 Unit 3, Lesson 2, students have opportunities to work with transformations of coordinates on a plane that creates the structure for what later becomes a set of mathematical rules for geometric transformations.
• In Course 3, Unit 3, Lesson 1, students engage with designs created using similar figures. Initially, students are prompted to examine a specific design to determine the structure that exists among the similar figures in the design. Then, after repeatedly examining the structure of designs based on similar figures, students are asked to determine sufficient conditions for knowing when two figures are similar.