Report for High School
Overall Summary
The instructional materials reviewed partially meet the expectations for alignment to the CCSSM for high school, Gateways 1 and 2. In Gateway 1, the instructional materials partially meet the expectations for focus and coherence, and the instructional materials show strengths in allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites and fostering coherence through meaningful connections in a single course and throughout the series. In Gateway 2, the instructional materials meet the expectation for reflecting the balances in the Standards and helping students meet the Standards' rigorous expectations by giving appropriate attention to procedural skills, conceptual understanding, and applications. Also in Gateway 2, the instructional materials partially meet the expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice.
High School
Alignment
Usability
Overview of Gateway 1
Focus & Coherence
Criterion 1.1: Focus & Coherence
The instructional materials reviewed for this series partially meet the expectations for focus and coherence. The materials focus the students' time on the widely applicable prerequisites, or WAPs, and make meaningful connections in a single course and throughout the series. There are some standards which are not fully developed throughout the series because some aspects are never addressed or there are specific methods/content identified that are not addressed. The materials do not always attend to the full intent of the modeling process when applied to the modeling standards, require students to engage in mathematics at a level of sophistication appropriate to high school, or explicitly identify and build on knowledge from Grades 6-8 to the High School Standards.
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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.
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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.
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The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation that, when used as designed, they allow students to spend the majority of their time on the widely applicable prerequisites (WAPs). The materials allow students to spend the majority of their time on the content from CCSSM widely applicable prerequisites for a range of college majors, post-secondary programs, and careers.
- Units 1, 2, and 4 of Integrated Mathematics I are focused on the WAPs from Algebra and Functions as they relate to linear and exponential relationships. The Function WAPs continue to be supported through Unit 3 with the Geometry Congruence standards included in the WAPs. The Geometry Congruence WAPs are also the focus of most of Unit 5 and 6 of Integrated Mathematics I, along with the G-SRT WAPs in Unit 6.
- Integrated Mathematics II continues the focus on the Algebra and Functions WAPs in Units 2 and 3 for quadratic functions and equations. Math II also includes a focus on the N-RN WAPs in Unit 1 with additional work on the Algebra WAPs.
- Units 1-3 and 5 within Integrated Mathematics III also focus on the Algebra and Functions WAPs in regard to polynomials, radical, rational, and logarithmic equations and functions. The Geometry WAPs can also be found as the focus in Unit 4.
- A-SSE evidence is found in Integrated Mathematics I Activities 1, 2, 18, and 20; Integrated Mathematics II Activities 4, 8, and 14; and Integrated Mathematics III Activities 2, 3, and 11.
- Every cluster of F-IF, Interpreting Functions, contains WAPs. Integrated I Units 2 and 4, Integrated II Units 2 and 3, and Integrated III Units 2 and 3 and Lessons within Unit 5 are designed to engage the students in these WAPs.
- F-BF.1 is addressed in Integrated I Unit 1 Lesson 5 and Unit 4 Lesson 20 as well as in Integrated II Unit 1 Lesson 6.
- F-LE.1 is addressed in Integrated I Unit 4 Lessons 17, 20, and 21.
- S-ID.2 and S-ID.7 are found in Integrated Mathematics I in Lessons 35, 36, and 37 and in Integrated Mathematics III in Lesson 31.
- S-IC.1 can be found in Integrated Mathematics III in Lessons 29 and 32.
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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.
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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.
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The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectations that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and required by the Standards.
All conceptual categories are addressed over each of the courses. Connections between conceptual categories are explicitly stated throughout the three courses in both the teacher and student edition and are labeled as “Point of Integration” (i.e. Integrated Mathematics III Lesson 17-1). In the Algebra and Function domains, the coherence was evident within the domain and across conceptual categories throughout the series as seen in:
A-REI: Reasoning with Equations and Inequalities
- Activity 19 in Integrated Mathematics I connects A-CED and A-REI by having students write two variable equations and graphing those equations in order to understand the solutions to graphs of equations.
- Activity 15 in Integrated Mathematics II connects the A-SSE standards to the A-REI standards by having students rewrite quadratic equations based on their given structure in order to solve and graph quadratic equations.
- Across the series, students understanding of equations and inequalities begins in Integrated Mathematics I Activities 2 and 4. Integrated Mathematics II applies reasoning of equations to solving quadratics (Activity 12). Integrated Mathematics III extends the reasoning to rational equations and inequalities as students are to solve both algebraically and graphically (Activity 13).
F-IF.B: Interpret functions that arise in applications in terms of the context
- The series also connects new content and skills to those learned in previous years. For example: F.IF.B spirals throughout the series. Specifically, F.IF.4, is introduced in Integrated Mathematics I Unit 4 Activity 24 where the students interpret key features of any graph or table. Integrated Mathematics III Unit 3 Activity 15 interprets key features of logarithmic functions.
Connections are explicitly stated for teachers through the Activity Standards Focus and the lists of Common Core State Standards for each Activity.
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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)
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The plus standards are identified in the series as (+) standards. They are explicitly identified at the beginning of each book where the list of CCSS are given for that individual book and then again within each unit they occur. The plus standards, when included, coherently support the mathematics which all students should study in order to be college and career ready.
Integrated I does not list any plus standards.
Integrated II lists N-CN.8, 9, F-BF.1c, S-CP.8, and S-MD.7.
Integrated III lists N-CN.9, A-APR.5, A-APR.7, F-IF.7, F-BF.B, F-BF.C, F-BF.D, F-BF.5, F-TF.3, F-TF.4, G-SRT.9, G-SRT.10, G-SRT.11, and S-CP.9.
The plus standards support the development of mathematics throughout the series:
- In Activity 7 of Integrated Mathematics III the standards addressed include the plus standards F-BF.4b and F-BF.4c, and these plus standards are connected to standards to F-BF.4 and F-BF.4a as they relate to inverse functions. The teacher edition states, “in Activity 7, students work with inverse functions. They verify that two functions are inverses of each other by showing that the functions undo each other when the output of one is used as the input for the other. They find the inverse of a given function by interchanging the domain and range of the function, at times restricting the domain of a function so that it is invertible. They explore the symmetry of the graphs of inverse functions” (which involves the plus standards).
- In Activity 6 of Integrated Mathematics II the standards addressed include F-BF.1b and F-BF.1c. It is also noted that while F-BF.1c is introduced in this activity, it is also addressed in higher level mathematics courses. Students begin work with the composition of functions in this series but do not explore it completely to keep the focus on the non-plus standards.
- The Integrated Mathematics Series indicates that plus standards S-MD.7, S-CP.8, and S-CP.9 are in the materials and evidence is noted to support the learning that students should have to be college and career ready.
- In Integrated Mathematics III Unit 6 Lesson 28-1 students are learning about combinations and permutations to compute probabilities. This is a plus standard. It has been included at the start of the unit, just before students are working with Random Samplings. The idea of combinations and permutations can be used to enhance student understanding of random samplings. This could be left out, and students could continue with random samplings and still build an understanding of these. The idea of using combinations and permutations continue to be used throughout the unit, yet these problems could be omitted.
- In Unit 2 Lessons 7-1 and 7-2 students are developing an understanding of functions and their inverses. Two plus standards have been included, F-BF.4b and F-BF.4c, where students verify inverses by composition and by reading values from a graph or table. These ideas are included very naturally yet could be omitted from the lesson.
Overview of Gateway 2
Rigor & Mathematical Practices
Gateway 2
Criterion 2.1: Rigor
The instructional materials reviewed for the Springboard Integrated Mathematics series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. Overall, all three elements of rigor are thoroughly attended to and interwoven in a way that focuses on the needs of a specific standard as well as balancing procedural skills, application, and conceptual understanding.
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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.
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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.
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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.
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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.
Criterion 2.2: Math Practices
The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation that the materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The materials support the intentional development of reasoning and explaining (MPs 2 and 3) and seeing structure and generalizing (MPs 7 and 8). The materials also support the development of attending to precision (MP6), but they do not consistently support the intentional development of making sense of problems and persevering in solving them (MP1), modeling with mathematics (MP4), and choosing and using appropriate tools strategically (MP5).
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The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation of supporting the intentional development of overarching, mathematical practices of MP1 and MP6, in connection to the high school content standards, as required by the Mathematical Practice Standards. The materials develop MP6, yet not MP1, to the full intent of the standards.
In each lesson, the materials have problems labeled with a Mathematical Practice Standard. Although the materials list out the content standards in their entirety and often expound upon them, the materials do not do so with the Mathematical Practice Standards. Instructional support/strategies appear in the teacher edition and “are called out [in the student edition] so students are reminded to apply them as they respond to problems and applications.”
The materials do not develop MP 1 to the full intent of the standard. Student development of strategies to solve problems is not evident in the majority of lessons.
- Problems labeled “Make Sense of Problems and Persevere in Solving Them,” as well as other unlabeled problems, provide students with opportunities to “explain how knowing …” (Integrated Math III Activity 2 Practice problem 27) or “describe what you observe” (Integrated Math I Activity 17 Practice problem 25), but students engage with routine problems rather than engaging in making sense of non-routine problems.
- Problems throughout the series tell students what information in the problem to use. This interferes with students’ opportunity to explain to themselves the meaning of a problem and look for entry points to its solution.
- In Integrated Math I Activity 17 Practice problem 25 students are working with a Lucas sequence to determine what other terms of the sequence may be and are given explicit information on how to determine this.
- In Integrated Math III Activity 2 Practice problem 27 students are asked to explain how knowledge of knowing two terms of a geometric sequence is sufficient to find the other terms in the sequence.
- In Integrated Math 1 Activity 17 embedded assessments and Integrated Math III Activity 2 students are given guidance as to how to approach the problem, eliminating the need to make sense of the problems.
There are some examples of problems, whether labeled as MP1 or not, where students are allowed to determine their own method for solving the problem to reach a single correct response.
- Integrated Mathematics I Lesson 14-2 problem 14 allows students to use multiple solution pathways as they determine coordinates of a point according to specified criteria.
- Integrated Mathematics II Lesson 14-4 problem 18 allows students to use graphing or the algebraic equation to determine maximum profit.
MP6 is addressed and fully developed throughout the materials and sometimes specifically identified in a problem. Students are often asked to use definitions, use units appropriately, and communicate understanding clearly in writing and/or orally. The teacher wrap and lesson answers in the teacher edition support the use of appropriate units and mathematical terminology throughout the series.
- Integrated Mathematics II Unit 2 Lesson 10-1 problem 24 is labeled with MP6. This problem gives the students six functions and ask the students to determine whether or not each function is a quadratic. The students are using the definition of a quadratic function to determine which of the functions are 2nd degree of the polynomial functions within these six functions.
- Throughout the series, the materials foster use of mathematical conventions and nomenclature.
- In Integrated Mathematics II Lesson 11-3 students make a prediction of a quadratic application of a parachute on a model rocket. Here students are asked to use a table to make predictions about the height of the rocket at certain times and create a scatter plot from the table on a coordinate grid.
- In Integrated Mathematics III Activity 7 students attend to precision as they graph inverse functions using the correct function notation, create and analyze graphs of inverse functions, and solve for them algebraically.
- Students are expected to use units and descriptions of solutions throughout the series, and often these are not identified as supporting MP6. Teachers notes will often encourage teachers to look for and encourage precision in student responses and in the solutions to lessons and problem sets. Teachers would need to facilitate conversations for students regarding precision and use of units to help students develop a connection between the importance of units, precision, and their importance to problem solving and building understanding. This is not always made clear in the teacher notes.
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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.
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The instructional materials reviewed for the Springboard Integrated Mathematics series partially meet the expectation that materials support the intentional development of addressing mathematical modeling and using tools (MP4 and MP5), in connection to the high school content standards, as required by the MPs. Overall, MP4 and MP5 are either not consistently used to enrich the content or are not building increasing expectations across the series to develop these standards to the full intent.
Introductory or inquiry problems are presented consistently as a series of steps and are labeled telling students exactly what to do in each step. For example in Integrated Mathematics I Lesson 21-1 the following is found:
- Let “t” equal the number of years.
- Write an expression where “t” represent the amount of money in the account after “t” years.
- Evaluate the expression for t=6 to confirm that the expression is correct.
- Evaluate the expression for t=10.
Examples where MP4 or MP5 are fully developed or are used to enrich the mathematics include the following:
- MP 4: In Integrated Mathematics III Activity 31 Problem 8 students determine the graphical display that will model the distribution of the data.
- MP 5: In Integrated Mathematics III Unit 4 embedded assessment after activity 18 is one contextual problem with seven questions called “Location Matters.” Within these seven questions the student is given an opportunity to analyze linear and quadratic functions by modeling on a coordinate grid in order to draw a conclusion as to where to recommend the location of the grocery store in terms of an ordered pair.
- MP 4: In Integrated Mathematics I Embedded Assessment #2 students use tables, equations, and graphs to represent linear functions.
- MP 4: In Integrated Mathematics II Activity 16-2 students choose how to model possible volumes of a candle when given its height.
Examples where MP4 or MP5 are not fully developed or are not used to enrich the mathematics include include the following:
- MP 5: In Integrated Mathematics II Lesson 10-1 problem 25 students compare their paper-pencil graph of a quadratic function modeling a real-world situation with the graph of the function using a calculator, which does not provide students the opportunity to choose their tools appropriately.
- MP 5: In Integrated Mathematics III Lesson 30-1 practice problems continue to tell the students what tool to use, such as “using a fair coin.”
- MP4: Integrated Mathematics II Lesson 25-2 Problem 2 is labeled “Model with mathematics” and reads: “Mr. Torres catches a bus each morning for work. The bus runs every 20 minutes. If he arrives at his bus stop at a random time, what is the probability that he will have to wait 5 minutes or more? Assume the bus stops for an insignificant amount of time. This number lines represents elapsed time. Point B is when the next bus will arrive.” Students are given the model, a labeled number line, to use.
- MP4: Students are often provided with models for problems or given a direction to use a specific model to solve a problem. Students are rarely given opportunities to devise models, determine the effectiveness of their model, or the opportunity to revise their solution or their model.
- MP5: Integrated Mathematics III Unit 3 Lesson 16-1 problem 4 is labeled “Use appropriate tools strategically.” The problem ask the students to find the solution to when public college tuition reaches the current private tuition based on two functions. The students are instructed to use both the graphing and table feature of a calculator; they are not given any choice as to which tools they will use.
- MP5: Integrated Mathematics II Activity 10 is titled “Modeling with a Quadratic Function.” Students examine quadratic functions presented with real world situations; however, the student is given directions to either graph or create a table. Students are not given the opportunity to identify important quantities on their own, discovering relationships using tools of their choosing.
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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.