High School - Gateway 2

The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation that the materials support the intentional development of students' conceptual understanding of key mathematical concepts. The instructional materials include whole- and small-group opportunities for exploration or demonstration of conceptual understandings. The materials often provide students with opportunities to justify, explain, and critique the reasoning of others. The instructional materials promote mathematical reasoning through various components, including Discussion Group Tips, Teach notes found at the beginning of every lesson, and the Suggested Pacing; each unit balances directed, guided, and investigative activities within the unit. 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 by the Think-Pair-Share, Sharing, and Responding teacher directions.

The following are specific standards for which the materials fully met the expectation for developing conceptual understanding:

• A-APR.B: Integrated Mathematics II Activity 9 introduces factoring polynomials with and without modeling. Activity 11 emphasizes the definition of a parabola and how the equation relates to a quadratic function. The foundation for solving has been built for students to develop the relationship between zeros and factors of quadratics as seen in Activity 12. Students are given many opportunities to determine which method of finding zeros is appropriate. On pages 164-166 Integrated Mathematics II makes explicit connections between solutions of a graph, the factors of a quadratics, and the meaning of zeroes.
• F-IF.A: Integrated I Unit 2 Lesson 5-1 provides students with the opportunity to analyze relations (represented in a table, graph, or diagram) to determine if they are functions, and Lesson 5-3 builds on this conceptual understanding by allowing the student the opportunity to use and interpret function notation.
• G-SRT.7: Integrated Mathematics II Activity 24 Lesson 2 problem 8 has students discover the relationship between the sine and cosine of complementary angles using a single right triangle. Students provide conjectures about the relationship and then determine whether the conjecture is true or false.
• G-C.2: Integrated Math II Activity 30 Lesson 2 provides students with an opportunity to discover and describe the relationship between chords in a circle and between chords and the diameter of a circle. After this, students use the relationships for proof writing.
• G-SRT.2: Integrated Mathematics II Activity 19 Lesson 3 defines similarity in terms of similarity transformations. Students write the similarity transformations required to prove similarity. Then in practice problem 13 students must use similarity transformations to explain whether or not two figures are similar and describe two separate sequences of similarity transformations that could be used. Problem 13 provides students with an opportunity to explain their conceptual understanding of similarity, and extra practice with this concept is provided through the Activity Practice and the Additional Unit Practice in the digital Teacher Resources.
• F-LE.1: In Integrated I Unit 4 Lesson 19-3 students are given two tables to discuss, and they make connections between the pattern of bacteria growth and the two given functions. As the students complete the table and use a graphing calculator to graph each function, they are led through a series of questions to enhance their conceptual understanding while guiding them to predict and confirm which pattern of bacteria growth is linear and which is exponential.

The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation for providing intentional opportunities for students to develop procedural skills and fluency. Learning targets are clearly articulated at the start of each lesson, and students are guided through a series of problems and structured responses to give them the opportunity to build procedural skills and fluencies. Lessons follow a clear progression of the stated learning targets, designed to give students several opportunities to practice the designated skills through either investigative or guided instruction followed by step-by-step example problems with similar Try These problems. Before the Lesson Practice problems there is a short Check for Understanding, which both provide more opportunities for students to develop procedural skills.

At the end of each Activity there is an additional Activity Practice on the concepts within the lessons. The Teacher Digital Resources also contain additional problems for each lesson that can be used to give students more practice and to build fluency.

Some highlights of strong development of procedural skills and fluency include:

• A-APR.1: This standard is addressed in Integrated Mathematics II Activity 4 and 5. Students are given extensive opportunities to develop procedural fluency of operations with polynomials.
• A-APR.6: Activity 10 Integrated Mathematics III provides multiple opportunities for students to develop procedural fluency with rational expressions.
• F-BF.3: Materials emphasize transformations of functions, and this is evident in the amount of practice the materials provide across the series. For several types of functions, students practice graphing a transformed function, write in words how f(x) is transformed to g(x), and write transformed functions in terms of other graphed functions, e.g. transformations of quadratic function problems can be found in Integrated Mathematics II Unit 3 Lesson 15-1 through Lesson 15-3.
• G-GPE.5: Activity 16 in Integrated Mathematics I includes practice and fluency of determining the slope of parallel and perpendicular lines.
• G-CO.1: Throughout Integrated Mathematics I students are provided many opportunities to build fluency regarding precise geometric definitions.
• G-SRT.5: Across all three courses, students are provided several Activities in which they build fluency with solving problems using similarity and congruence criteria.

The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation of 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. Throughout the series, often in sections labeled “Model with mathematics,” the materials provide opportunities for students to use concepts and skills in problems designed to model real world situations. Single-step and multi-step contextual problems are used in different class settings (individual, small group, and/or whole group) to engage students in applications.

Examples of real-world applications include:

• A-REI.11: Students are given opportunities for applications as seen in Integrated I Activity 10, “A Tale of Two Trucks.” Students use several methods, including graphing, to solve and classify a system of equations. In addition, in Integrated I Activity 19 students examine exponential functions by analyzing their graphs as well as comparing rates of change of linear and exponential functions determined by their graphs.
• F-IF.B: Integrated I Unit 4 Lesson 24-1 through Lesson 24-4 includes several contextual problems that develop interpreting functions in terms of the context. The lessons include a variety of single and multi-­step contextual problems involving such topics as temperature change (polynomial function), hiking trail (linear function), rocket launch (quadratic function), parking garage cost (greatest integer function), and marching band formation (radical function). Other examples can be found across the courses in the series. For example, in Integrated II Unit 2 Lesson 10-1 students use a basketball court to interpret a quadratic function in terms of the court. Another example in Integrated III occurs in Unit 2 Lesson 8-1 where students are given the opportunity to interpret a radical function in the context of the hull speed of a boat. Contextual problems asking the students to interpret a function can be found throughout all courses of the series.
• G-SRT.8: Integrated Mathematics II Activity 21 - Activity 24 include examples of application problems for the Pythagorean Theorem and Trigonometric Ratios. Application problems in these Activities involve architectural design, height of structures, framing of pictures, dimensions of flat screen televisions, distance on a baseball field, and quilt patch dimensions.
• Statistical concepts are presented within contextual settings requiring students to interpret data and make sense of their conclusions. For example, in Integrated Mathematics I Lesson 37-1 measures of central tendency are compared when analyzing the dot plots, histograms, and box and whisker plots to determine the impact of human activity on wildlife in the “home ranges” of certain animals. Polls and voting are used to provide context for how to make inferences from population samples.

The application problems are often scaffolded, especially as they apply to the modeling process.

The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation of providing balance among conceptual understanding, procedural skill and fluency, and application. All three aspects of rigor are present in the materials of the series, and balance among the three aspects is present within a course and throughout the series. In some application problems and in problems building conceptual understanding throughout the series, students are required to use multiple representations and written explanations to support their work and justify their thinking in order to demonstrate their understanding of procedures, skills, and concepts. The Activities generally provide opportunities for students to develop conceptual understanding through exploratory problems, often within a context, in the first lesson of the unit. Next, there are opportunities for students to develop fluency and understanding through application in the subsequent Activity lessons. The balance of procedural skill development and application is not rigid throughout the materials and changes based on the targeted concept.

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

• Integrated Mathematics I Unit 1 Activity 1 builds on students previous work of writing numeric and algebraic expressions to representing situations and by using tables and graphs to examine the relationship between two quantities. Activity 2 builds on this knowledge to provide opportunities for procedural skill and fluency as they solve linear equations in one variable, including multi-step equations.
• Integrated Mathematics I Unit 2 uses previous learning from Activity 7 when students are connecting the features of linear functions to their meaning in context.
• Integrated Mathematics II Unit 2 focuses on quadratic functions and equations. Students are given opportunities to develop fluency of writing quadratic equations. Activity 9 reinforces factoring of quadratic expressions using multiple methods, including algebra tiles. Activity 10 expands students’ understanding of functions though modeling in real life applications and includes questions meant to lead students to conceptual understanding.
• F-IF.2: Integrated I Unit 2 Lesson 8-1 contains a scenario where a young lady, Annalise, needs to determine the price of one pound of coho salmon from several supermarket receipts. The students are given the opportunity to use function notation to write the function that gives the total price f(x) for x pounds of coho salmon, evaluate their function for 4, 12, and 26 pounds of salmon, and interpret an additional receipt found to show that the total price on the receipt is correct for the 60 pounds purchased. The students use their function to discovered that the customer was incorrectly charged and are asked to explain their findings.
• G-SRT.7: Students are asked to explain and use the relationship between the sine and cosine of complementary angles. Activity 24 of Integrated Mathematics I builds students conceptual understanding by having them explain the relationship. This follows from Activity 22 where students are building a procedural understanding of basic functions and their transformations through Activity 23, where students continue to build procedural understanding of an increasing number of different graphs, thus balancing procedural fluency and application of these graphs.

The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation that materials support the intentional development of reasoning and explaining (MP2 and MP3), in connection to the high school content standards, as required by the MPs. Overall, the majority of the time MP2 and MP3 are used to enrich the mathematical content inherently found in the material, and across the series there are increasing expectations for MP2 and MP3 to the full intent of the standards. Throughout the materials, students are expected to reason abstractly and quantitatively, as well as, construct viable arguments and critique the reasoning of others.

Examples of MP2 Reason Abstractly and Quantitatively include the following:

• In Integrated I Lesson 15-1 problem 7 students write and justify two statements based on a given figure. Students must understand the relationship the mathematical representations within the context of the problem..
• In Integrated II Activity 17 problem 19 students explain the conditions under which a system of equations, involving a coefficient of a, would have no real solution, exactly one real solution, and two real solutions. In order for students to answer they must be able to represent a system symbolically and graphically and determine what the a could represent for the various solutions.
• In Integrated III Lesson 20-1 problem 5 students design a shipping container of specific shapes and express the surface area algebraically.
• In Integrated I Activity 2-4 students must choose values to determine infinite solutions of an equation.
• In Integrated II Activity 8-2 students must reason abstractly to justify that a product of factors is equal to the sum of the squares.
• In Integrated III Activity 3-4 students are given opportunities to determine specific values in generating Pythagorean Triples.

Examples of MP3 Construct Viable Arguments and Critique the Reasoning of Others include the following:

• In Integrated I Activity 19 problem 23 students determine reasonableness and justify their response.
• In Integrated II Lesson 19-4 problem 12 students justify whether or not they agree with a given solution. Students are critiquing the reasoning for a given solution and constructing arguments to support the critique.
• In Integrated I Activity 2-3 students justify their choice of two phone plans by comparing equations that model the cost of the two phone plans. Students create equations based on the information given regarding the cost of telephone plans, and they determine when the two equations are equal to each other to determine when the two plans have the same cost. Students are then asked to make a choice of plans and justify their choice.
• In Integrated III Activity 13-1 students make conjectures about when a rational equation is likely to have extraneous solutions.

The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation that materials support the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the MPs. MP7 and MP8 are used to enrich the mathematical content inherently found in the materials and are not treated as isolated experiences for the students.

Some examples of MP7 and MP8 in the series are as follows:

• MP 8: In Integrated Mathematics I Activities 1 and 2 students use repeated reasoning to make predictions as they represent a pattern in geometric figures using a table, a sequence, and an expression.
• MP 7 and 8: Integrated Mathematics I Activity 29 guides students on an exploration of angle measures to develop the Triangle Sum Theorem. The activity has students "use repeated reasoning to generalize” a rule for determining the sum of the interior angles of a triangle and then “make use of structure” to calculate individual angle measures of triangles using algebraic expressions and equations.
• MP 8: Integrated Mathematics II Unit 1 Lesson 1-1 problem 7 requires students to look at the pattern observed in a table of expanded forms to discover the shortcut in the pattern called the Product of Powers Property.
• MP 8: In Integrated II Activity 12 students make use of structure as they solve quadratic equations by using square roots, completing the square, and applying the quadratic formula. Students identify the commonalities of each of these methods and choose the appropriate method to a variety of problems.
• MP 7 and 8: In Integrated Mathematics III Lesson 14-2 students use a table to look at patterns and show the relationship between exponential functions of base 10 and the common logarithmic function.
• MP 7 and 8: In Integrated Mathematics III Lesson 28-2 students create a list of possibilities in a series of problems to help students make a connection to the idea of combinations and the formula for combinations.