High School - Gateway 1

The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation that the materials attend to the full intent of the mathematical content contained in the high school standards for all students. In general, the series included the majority of all of the non-plus standards, but there were instances where aspects of standards were not met. While standards are listed at the beginning of each text on page XXIII, not all the standards that were actually taught in that respective text were a part of that comprehensive list.

The following standards are identified as having been met across the Integrated Mathematics I, Integrated Mathematics II and Integrated Mathematics III materials.

• There was evidence to indicate that all aspects of the non-plus standards from the following domains and clusters were addressed: N-RN.A, N-RN.B, N-Q.A, N-CN.A, N-CN.B, and N-CN.C.
• Within the Algebra conceptual category, the domains of Arithmetic with Polynomials and Rational Expressions (A-APR) and Creating Equations (A-CED) are well represented by the instructional materials of the series. The following exemplifies this:
  • A-CED.1 was thoroughly represented throughout the series: creating and solving linear equations and inequalities in one variable in Integrated Mathematics I Lessons 1, 2, and 4; creating and solving exponential equations in one variable in Integrated Mathematics I Lesson 19 and Integrated Mathematics III Lesson 16; creating and solving quadratic equations in one variable in Integrated Mathematics I Lesson 1 and Integrated Mathematics II Lesson 10; and creating and solving simple rational equations in one variable in Integrated Mathematics III Lesson 13.
• There was evidence to indicate that all aspects of the non-plus standards from the following domains and clusters were addressed: F-IF.A, F-IF.B, F-BF.A F-BF.B, F-LE.B, and F-TF.A.
• Throughout the series, students are given multiple opportunities to work with linear, quadratic, exponential, polynomial and trigonometric functions utilizing tables, equations, graphs, sigma notation, and comparisons of these functions in multiple problems.
• All aspects of all non-plus Geometry standards within the domains of Congruence (G-CO), Similarity, Right Triangle, and Trigonometry (G-SRT), Circle (G-C), Geometric Measurement and Dimension (G-GMD), and Modeling with Geometry (G-MG) were addressed throughout the three course series.
  • An example of a Geometry standard that was exemplary in terms of attending to the use of various mathematical tools was G-CO.2. The students in Integrated Mathematics I Unit 5 were given the opportunity to use tracing paper in Lesson 25-1 and geometric software in Lesson 25-3 to represent, describe, and compare transformations in the plane.
• There was evidence to indicate that all aspects of the non-plus standards from the following domains and clusters were addressed: S-ID.C, S-IC.B, S-CP.B, S-MD.A, and S-MD.B.

The following standards are identified as having not been met or partially met in this series.

• A-SSE.3: There are several examples within the series where students are to “produce an equivalent form of an expression” (Integrated Mathematics I Lesson 20; Integrated Mathematics II Lesson 1, Lesson 5, and Lesson 12); however, the students are not “choosing” an equivalent expression in order to explain properties. In each problem students are told how to rewrite an expression to reveal a specific quantity, yet the students do not determine how the expressions should be rewritten in order to gain more understanding about a specific quantity within the expression.
• A-REI.3: The Integrated Mathematics I materials contains many examples of students “solving linear equations and inequalities in one variable” (Integrated Mathematics I Lesson 3 and Lesson 4); however, the equations and inequalities to be solved do not contain “coefficients represented by letters.” There are examples of problems where students solve a formula for a given variable in which the formula contains only letters. However, the letters do not stand for coefficients in the formulas; they stand for other variables (Integrated Mathematics I Lesson 3 problems 3, 4, and 6-10a).
• A-REI.10: Problems within the lessons of the Integrated Mathematics I and Integrated Mathematics II materials imply that students must understand that “the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane,” but students are not explicitly taught this nor do they have to explain that they understand this concept. Students are simply asked to use the concept when solving problems (Integrated Mathematics I Lesson 6 and Integrated Mathematics II Lesson 11).
• For standard F-LE.1(a), “Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals,” students have opportunities to explore linear and exponential functions over intervals, as well as within larger areas to determine growth of these functions, as seen in Integrated Mathematics I Unit 4 Lessons 20 and 21 where students work with these functions, but do not prove the idea of equal differences over equal intervals.
• For standard F-TF.5- “Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline”- students use and discuss trigonometric functions to model situations with regards to amplitude, frequency, and midline; however, students do not choose the function. In each situation the functions are given and students identify the previously mentioned parameters of the functions. This is seen in Integrated Mathematics III Unit 5 Lesson 27 Pg. 395-403. In these scenarios, students work with the same function in different problems with changing parameters; however, students do not choose the function to model phenomena.
• Standard G-GPE.7 was partially met. Students were given the opportunity to use coordinates to compute perimeters of polygons and areas of triangles in the Integrated Mathematics I Unit 3 Lesson 14-1 and Lesson 14-2. No evidence was found that students are given the opportunity to compute areas of rectangles using coordinates.
• For standard S-ID.5, “... Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). …,” no evidence was found for joint frequencies.

The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. Throughout the series, there are a number of lessons that contain a variety of components of the modeling process described in the CCSSM. Students are provided scaffolded questions to help guide them through the process of modeling a function with an equation or graph and reasoning from that model. However, throughout the series, students do not have an opportunity to authentically engage in the full modeling process.

Examples of where the modeling process is incomplete are:

• In Integrated Mathematics I Activity 37-1 students are constructing representations of univariate data, describing characteristics, comparing distributions and identifying similarities and differences. Scaffolding of portions of the modeling process is present, students are given graphs,tables, and histograms, and questions are scaffolded to direct students to predesigned outcomes, rather than allowing students to determine what information to gather and use. Standards addressed S-ID.1, S-ID.2, S-ID.3, S-ID.4
• In Integrated Mathematics II Unit 5 modeling standards S-CP.2, S-CP.3, S-CP.4, S-CP.6, and S-CP.7 are cited for Lesson 28-1. Practice problem 11, labeled “Model with mathematics,” addresses the various aspects of the modeling process, yet the students are not given the opportunity to collect data and formulate their own model. A table is given in which the students are asked to compute probabilities and then write a newspaper article to interpret, validate and report the data.
• In Integrated Mathematics III Unit 6, modeling standards S-ID.1, S-ID.2, S-ID.3, S-ID.4 and S-IC.1 are cited for Lessons 31-1 through 31-4, and there are many problems labeled as modeling problems. However, the students are not given the opportunity to design the experiment and then record and analyze their results.
• In Integrated Mathematics III Unit 3 on Page 199 students work with standard F-LE.4. Students are provided opportunities to explore and work with exponential functions; however, they are not given the opportunity to interpret these functions within a context.

Though problems labeled as “Model with mathematics” occur throughout the series, these problems are application problems. More information on these problems is included in 2C.

The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation for providing students with opportunities to work with all high school standards without distracting students with prerequisite or additional topics. The materials, when used as designed, allow students to fully learn most, but not all, standards. (Those standards that were not attended to by the materials, as noted in indicator 1ai, are not mentioned here.)

The following are some examples of how the materials, when used as designed, do not allow students to fully learn each standard.

• S-IC.2: "Decide if a specified model is consistent with results from a given data-generating process …” In Integrated Mathematics III Lesson 30-1 Pg. 429-440 and Lesson 33-1 Pg. 481-492 evidence is found of students conducting simulations to determine results of outcomes. Although teacher notes assist with the implementation of this standard, students do not decide what to look for or what outcomes to study.
• S-CP.4: Evidence was found of students completing but not constructing two way frequency tables in many different places; however, there is one instance where students construct a two way frequency table and then complete it in Integrated Mathematics II Lesson 25-3 page 371 Problem 10.
• S-ID.4: “Use the mean and standard deviation of a data set… Use calculators, spreadsheets, and tables to estimate areas under the normal curve.” Students are not asked to use spreadsheets for solving even though spreadsheets are mentioned in Integrated Mathematics III Page 458.
• A-APR.4: The standard states to prove polynomial identities and use them to describe numerical relationships. In Integrated Mathematics III page 49 the students are walked through the verification of a polynomial identity, but they are not given practice to prove any identities.
• A-REI.5: The process of solving a system of linear equations using the elimination method or linear combination method is found within Lesson 10 of Integrated Mathematics I; however, this standard says students must “prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.” Students are asked to use the concept, but they never prove it nor do the teacher instructions state that the teacher is proving the concept.
• A-REI.10: The standard states to understand that the graph of an equation in two variables is the set of all its solutions plotted on a coordinate plane. In Integrated Mathematics I pg. 107, students are asked to verify whether or not specific ordered pairs are on the same line. On pg. 109 problem 3d students are asked whether all of the values make sense for the graph. There is no prior practice found to help students understand that a graph contains all the possible solutions.
• F-IF.9: There are five problems that provide opportunities for students to practice comparing properties of two functions. In Integrated III Unit 1 Lesson 6-3 problems 1-3 and problems 7-8 students were given two functions in an algebraic, graphical, numerical table, or verbal representations to compare their properties.
• F-TF.2: The teacher note in Unit 5 Lesson 24-1 under Differentiating Instruction suggests that the teacher facilitates a discussion of the relationship between the unit circle and trigonometric functions, but students are not given the opportunity to explain how the unit circle enables extension of trigonometric functions..
• G-SRT.7: This standard states that students must “explain and use the relationship between the sine and cosine of complementary angles” found in Activity 24-2 of Integrated Mathematics II. There are three problems within Activity 24-2 and one question on the Embedded Assessment 2 after Activity 24 that provide opportunities for students to engage with this standard..
• Within the Modeling with Geometry standards, G-MG.3 asks students to “apply geometric methods to solve design problems.” Problem 12 of Activity 34 Practice and problems 2-4 of Lesson 36-1 in Mathematics II, as well as problem 5 of Lesson 20-1 in Mathematics III, provide students with some practice toward this standard..

Overall students are given the opportunity to work with all the non-plus standards and do not distract students with prerequisite or additional topics. However, there are a few missed opportunities for students to fully learn the aspects of each standard.

The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation for requiring students to engage in mathematics at a level of sophistication appropriate to high school. The materials use age appropriate contexts and apply key takeaways from grades 6 - 8. However, students do not have many opportunities to work with a variety of real numbers appropriate for high school.

Some instances of where students are engaged with the content appropriate to high school include:

• In Integrated Mathematics I Activity 6 students identify linear functions by comparing rates of change and practice applying ratios and proportional reasoning. In Integrated Mathematics I Activity 17 students are continuing their application of proportional reasoning as they compare rates of change applied to arithmetic and geometric series. Integrated Mathematics III Activity 17 continues the application of ratio and proportional reasoning as students use variables to represent coordinates of points.
• The Integrated Mathematics II Unit 3 Embedded Assessment 1 titled “Diving Competition” asks students to compare Juan and Benjamin’s dives based on given quadratic functions. The student are asked to graph each function then interpret features of those graphs to critique each men’s dives. This example is relevant contextually for high school students and includes mostly rational and some irrational solutions. Embedded assessments provide multiple opportunities to apply basic function concepts across the series.

There are instances where students are not engaged with the content appropriate to high school. These examples include the following:

• Integrated Mathematics I uses mostly whole numbers and as the series continues through Integrated Mathematics II and III the numbers expand to include all subsets of rational numbers. There were few problem sets that included irrational numbers.
  • In Integrated Mathematics 1 Activity 12 angle measures are given in whole number values. Angle measures continue to be given as whole number values in Integrated Mathematics II Activity 31 Embedded Assessment 1 with the exception of a few angle measurements given as a whole number and 5 tenths.
  • In Integrated Mathematics I Activity 14 irrational numbers appear in final answers (using the distance formula in); however, students do not complete calculations using irrational numbers within the Geometry domain. Irrational numbers appear in Integrated Mathematics II Activity 23 when working with side lengths of special right triangles, yet students are only doing basic calculations involving multiplying a radical by a whole number.

There are problems that involve rational numbers in the form of fractions or decimals.

• In Integrated Mathematics I Activity 12 students calculate the difference between linear distance using 7.3 and 8.5.
• In Integrated Mathematics I Activity 13 students find the value of the variable when 4 ½ x = 9 or when 4c = 30.
• In Integrated Mathematics I Activity 16 students write the equation of a line parallel to 3x + 4y = 4 that contains the point (8, 1).
• In Integrated Mathematics II Activity 34 students are calculating volume with answers represented as decimals or fractions; however, the given dimensions are whole numbers.

The materials apply topics from grades 6-8 such as linear graphs, histograms, box and whisker plots, and scatter plots; however, problems are often scaffolded and lead students to answers rather than allowing open ended solutions. For example, in Integrated Mathematics II Unit 5 Lesson 25-3 students are given graphs, tables, or variables for problems posed. Students are directed to the solution that is to be reached, for example, as students work with probability and utilize given sample spaces, two-way frequency tables, and calculation of probabilities. These scaffolds provided in these problems direct students on representation and interpretation of solutions that leads students to a pre-determined outcome rather than giving students the opportunity to determine how to model the problem and ask the questions to determine how to interpret outcomes or solutions. The values are whole numbers, with the decimal value 0.5 being used twice, but final answers may have decimals.

The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. In the instructional materials, content from Grades 6 to 8 is present but not clearly identified and/or does not fully support the progressions of the high school standards. Connections between the non-plus standards and how those standards are built upon from Grades 6-8 is not clearly articulated for students and only partially articulated for teachers.

Grade 6-8 standards are listed for most units in each Teacher Edition’s "Getting Ready.” "Teacher to Teacher" sections give teachers information about how current topics relate to prior topics and provide information about how to encourage students to reach deeper understanding at a high school level.

• Integrated Mathematics I Unit 7 Activity 33 on page 567 the teacher notes in “Teacher to Teacher” include the following: “The initial section of this activity reintroduces students to informal analysis of numerical bivariate data in terms of form, direction, and strength of relationship. (Note: These topics were most likely introduced in eighth-grade mathematics.) Then, the correlation coefficient is introduced as a statistic that can communicate and evaluate strength and direction of linear relationships.”
• Integrated Mathematics III Unit 4 on page 240 in the teacher edition, the list of prerequisite standards covered in the “Getting Ready” section include the following standards: 8.G.8, G-CO.1, 7.G.5, G-CO.9, A-REI.4a, F-IF.4, F-IF.7a, and G-MG.1. Student exercises in the “Getting Ready” set include problems that address these standards. However, these standards are not directly connected to any work within the unit.
• Integrated III Unit 1 Arithmetic Sequences and Series, Getting Ready identifies 10 standards, two from Grades 6-8 (7.NS..3 and 8.EE.1) and 8 more high school standards. Of the eleven problems presented to students, four review the Grades 6-8 standards, and the remaining seven problems are reviewing High School.

The series indicates high school standards for lessons, but problems presented to students do not always align to high school standards.

• The Integrated Mathematics II Activity 33 learning targets are to “develop and apply the formulas for circumference and area of a circle.” (7.G.4) There are no related High School Standards that require developing and application of the formulas or the area and circumference of a circle as described.
• In Integrated Mathematics II Activity 1 students develop basic Properties of Exponents (8.EE.1). In Activity 2, students rewrite rational exponents as radical expressions which is a High School Cluster (N-RN.A). Activity 2 builds on the seventh grade content in Activity 1; however, there is no direct connection between the activities.

Examples of how lessons connect to middle school content, but do not explicitly indicate the connection, include:

• In Integrated Mathematics I Activity 14 students utilize the Pythagorean Theorem and coordinates on the coordinate plane (standards 8.G.6, 8.G.7, 8.G.8, 8.SP.2, and 8.SP.3) to find the lengths of line segments that make up sides of a geometric polygon. Students build on this understanding to derive the distance formula. These middle school standards are not listed in either the student or teacher materials.
• Integrated Mathematics II introduces students to multiplying polynomials through the use of repeated distributive property and the use of the area model, which builds on students prior understanding from middle school standards. (7.EE.1)