The instructional materials reviewed for the HMH Integrated series do not meet expectations for alignment to the CCSSM for high school. The materials do not meet the expectations for focus and coherence as they partially meet the expectations in the following areas: attending to the full intent of the mathematical content contained in the high school standards for all students, allowing students to fully learn each standard, and making meaningful connections in a single course and throughout the series. Since the materials did not meet the expectations for focus and coherence, evidence for rigor and the mathematical practices in Gateway 2 was not collected.
Focus & Coherence
DOES NOT MEET EXPECTATIONS
Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).
The instructional materials reviewed do not meet expectations for focus and coherence. The materials allow students to spend the majority of their time on the widely applicable prerequisites and attend to the full intent of much of the mathematical content contained in the high school standards for all students. However, the materials are lacking in the modeling process, and much work would need to be done in order to foster coherence.
The materials focus on the high school standards.*
The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
The instructional materials reviewed partially meet the expectations for attending 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 the non-plus standards, but there were some instances where the full intent of the standard was not met.
Examples where the full intent of the standards were met:
- N-RN.1: This series allows students to fully engage in this standard in Math 2, Module 3. On page 100, the lesson requires students to compare/contrast radicand restrictions on even and odd roots, as well as to reason about the order in which they should calculate the root or power of the base number. Teachers are provided with questioning strategies to help students analyze and reason with rational exponents in Math 2 TE on pages 99-100. In “Explain the Error and Communicate Mathematical Ideas” on page 105, students engage in higher-order thinking strategies through error analysis and abstract analysis.
- G-CO.9-11: The series allows students to fully engage in these standards in Math 2, Modules 14-16. Students complete multiple styles of proofs for theorems involving lines, triangles and parallelogram properties, including fill in the blank paragraph proofs, flow proofs, and 2-column proofs. Students are also given the opportunity to write a plan for various proofs of different theorems and to create their own proofs in various styles.
Examples where the full intent of the standards were not met:
- N-RN.3: This standard was addressed in Math 2, Module 3 on page 108. Students are required to identify whether the sum and product rules for two rational, two irrational, or one rational and one irrational number are sometimes, always, or never true. However, they are not required to explain why this is so as indicated by the standard.
- N-Q.2: This standard was addressed in Math 1 and Math 2 but was only partially met because there was no evidence found indicating students define their own quantities as called for in the standard. In Math 3, Module 19, on page 1111, one example asks students to define their own quantities, but students are provided with pre-labeled and numbered tables and charts thus taking away the opportunity to determine quantities.
- A-SSE.1 In Math 2, Lesson 4.1 places a considerable focus on mechanics and evaluating a polynomial expression to answer a real-world problem. This section does not address “interpreting its parts” as required by the standard nor is there evidence of this part of the standard being addressed in any other lessons/tasks within the series.
- A-SSE.3a,b: The standard is only partially addressed in Math 2, Module 9 on page 374. Evidence was not found that the materials require students to choose a form in order to reveal information. That is, students were directed to produce specific forms without opportunity for student choice.
- A-SSE.3c: The standard is identified in Math 3, Module 10; however, students do not need to use properties of exponents to transform any of the expressions. Students are identifying information or evaluating in the examples and problem set.
- F-IF.8a: In Math 2, Module 9, students are given ample opportunity to practice working with quadratic functions through completing the square with and without context. However, all context questions only asked students to find maximum values, not minimum values. Additionally, students do not use the processes of factoring or completing the square to show the symmetry of the graph.
- F-TF.2: Students did work with the concepts of the unit circle in Math 3, Module 18, on pages 927-940. The standard indicates that students should explain how the unit circle is an extension of the trigonometric functions, which was not found in this series.
- G-CO.10: Most of this standard is addressed in Math 2, Module 15, with the exception of Triangle Midpoint Theorem. Students do verify and work problems where the third side is parallel to the segment connecting the midpoints of the other two sides, but they do not find or work with the idea that the length of the segment connecting the midpoints of the other two sides is half the length of the third side.
- S-ID.4: This standard was addressed in Math 3, Module 21 on page 1057. Students were asked why standard deviation and mean are not appropriate statistics to use with skewed distributions, but students were not asked to determine if such procedures were or were not appropriate for the data sets, as indicated by the standard. Students were also limited in the exposure of technology, calculators only, used for this standard.
- S-CP.4: In Math 2, Module 22, the lesson places the focus on “finding the probability from a two-way table of data;” however, nowhere within this section are students prompted to “construct and interpret the two-way frequency table of data …” as indicated by the standard. It is not until Math 2, Module 23, that students are “involved” in creating such a table by being given a supporting matching activity on page 1254, but the students never fully construct a two-way table.
- S-IC.4: This standard is addressed in Math 3, Module 23 on page 1123. The definition and formulas of margin of error are introduced to students. In the work that immediately follows, the margin of error is given and then used to determine appropriate sample size. Then, in the Lesson Performance Task on page 1132, students are asked to find the margin of error but only after being given the formula. Students do not develop a margin of error from the use of simulation models as indicated by the standard.
The materials attend to the full intent of the modeling process when applied to the modeling standards.
Two problems within the materials provide opportunities for students to engage with the modeling process.
- F-LE.1b/S-ID.6a: On page 508 of Math 2, the Lesson Performance Task provides students with a simple table and asks them to find a function to model the data as well as to justify the model and make predictions about their model. Students are given the opportunity to identify variables, select essential features, and formulate and analyze a model. Students will need to justify their model and the predictions made, so students will need to extrapolate to obtain an answer.
- G-GMD.3: On page 1180 of Math 2, Computer Screens, the task offers students an opportunity to engage with the modeling process as the task has multiple possible answers. Students must identify measurements, formulate models to represent the situation, and report on the conclusion. Student have the opportunity to explore, justify, and critique multiple answers in this task, so this task engages students in the full modeling process.
The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
The materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.
The materials, when used as designed, meet the expectations for allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers. Overall, the majority of the instructional materials addressed widely applicable prerequisite standards (WAP’s), and the large quantity of duplicated/repeated lessons throughout the series covering non-WAP standards was not found to be distracting.
In Math 1, a majority of the material was spent on WAPs, and the course mainly addressed WAPs from Algebra, F-IF, and G-CO.1. In Math 2, a majority of the new material was spent on WAPs, and the course mainly addressed WAPs from F-IF and Algebra along with standards from N-RN, G-CO.9,10, and G-SRT.B,C. In Math 3, less than a majority of of the new material was spent on WAPs, and the course mainly addressed WAPs from F-IF.
This series is inconsistent in addressing the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers. The series spends limited time on lessons where the Number and Quantity, Statistics and Probability, and Geometry WAPs are the focus of lessons. The time spent on WAPs decreases as the series progresses, as does the time spent in the these categories.
The materials, when used as designed, allow students to fully learn each standard.
The materials, when used as designed, partially meet the expectations for allowing students to fully learn each non-plus standard. Overall, the series addressed many of the standards in a way that would allow students to learn the standards fully. However, there are cases where the standards are not fully addressed or where the instructional time devoted to the standard was insufficient.
Throughout the series, students do not prove or derive theorems, properties, or formulas when required by the CCSSM. Students are routinely shown the process to derive, given fill-in-the-blank proofs, or just given the information, but students do not independently construct proofs or derive formulas without significant direct instruction or support. The following are examples where students are not able to fully learn the standard because they are not engaged in the process of proving or deriving as prescribed in the standards.
- G-C.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Students were not given the opportunity to independently complete the derivation indicated by the standard.
- G-GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Students were not given the opportunity to independently complete the derivation indicated by the standard.
- G-GPE.2: Derive the equation of a parabola given a focus and directrix. Students followed a prescriptive, fill-in-the-blanks derivation provided by the book.
- A-REI.5: Students complete a fill-in-the-blank proof but are not required to do an independent proof of this standard.
- F-TF. 8: Math 2, Module 18 page 982, students are asked to confirm the Pythagorean identity using different values for theta, but they do not formally prove the Pythagorean Identity, as indicated by the standard. On page 990 in problem 19, students are asked to show the theorem is true, but not provide a formal proof.
The materials, when used as designed enable students to have exposure to each of the standards, but they do not always allow students to fully learn each standard. Examples of standards that are not fully developed within the curriculum include the following:
- N-RN.2: Math 2, Module 3 page 108, students are required to identify whether the sum and product rules for two rational, two irrational or one rational and one irrational number are sometimes, always or never true; however, they are not required to explain why this is so.
- F-IF.7c: Math 2, Lesson 7.2 lays the foundation for using zeros to plot x-intercepts and construct a rough graph (of a quadratic function). On page 270 of that lesson, the material notes a perfect square trinomial that could be used to introduce the concept of duplicity/multiplicity of a zero, but nothing is indicated in the facilitation notes to encourage such a discussion. Math 3, Lesson 5.4 on page 295, reinforces the connection between a zero and an x-intercept. The idea of duplicity is discussed in problem 3 on page 296. The remainder of the lesson does not relate the concepts of end behavior and the x-intercepts to sketch the rough graph; rather, the materials encourage students to develop a sign chart. While essential elements toward learning this standard are introduced, the series does not give students an opportunity to fully learn the standard.
- A-CED.4: Math 1 Lesson 2.3 focuses on the linear case- offering scaffolded (fill-in-the-blank) examples and 25 exercises. The series does not include any focus on this standard for the quadratic case.
- A-REI.10: In Math 1, Module 6 on page 202, students are not given opportunity to investigate the standard or to develop their own understanding that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve, but are instead told with an example.
- G-CO.1: Math 1 Lesson 16.2 focuses on “angle.” Although the materials appear to define angle at the top of page 790, the convex part of the angle is not noted. The definition is provided not giving students an opportunity to develop a precise definition. Attempts at defining “parallel lines” in Math 1 Lesson 17.1 and “perpendicular lines” in Math 1 Lesson 17.2 are not precise; rather, the material lists properties in place of a precise definition. Here, too, students are not encouraged to develop a precise definition. Thus, the series does not allow students to fully learn this standard.
- G-CO.4: In Math 1, Module 17, the definitions of rotations, reflections, and translations are given and used but not developed as required by the standards. For example, students are told to find the midpoint between image and pre-image points to find the line of reflection instead of using the line of reflection to confirm it is the midpoint and therefore proving it is a reflection. This is also seen with angles of rotations; students are told to use angles to make the rotations but are not asked to verify a rotation by measuring or finding angles.
- G-GPE.6: Math 3 Lesson 2.6 introduces examples on a number line and a coordinate plane. In each case, the approach involves partitioning a horizontal and/or vertical distance, applying the appropriate ratio, and determining the final point. A variety of ratios are used in examples and exercises. Students are also introduced to the concept of partitioning via constructions. Moreover, students are encouraged to confirm the point of partition using the distance formula. The series does not introduce the student to any algebraic/numerical approach and, in turn, does not ask the student to determine the ratio if given a partitioned segment.
- S-CP.3: Math 2, Lesson 23.1 addresses the standard although it is not indicated as such in the table of contents. While students engage in calculating conditional probabilities using formulas, there was not any evidence of attending to students “understanding” of the concept or why the formula works.
The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The materials reviewed meet the expectations for requiring students to engage in mathematics at a level of sophistication appropriate to high school. The series regularly uses age appropriate contexts to engage students in the mathematics while exposing students to various types of real numbers. The context of the series did not advance in sophistication as the series progressed, but the context was not beyond or below high school student understanding. The series provides age-appropriate, mathematical contexts by including scenarios that are relevant to students such as maximizing profit, perimeter and area of shapes, probabilities of games, and projectile motions with sports.
The following are examples of the age appropriate contexts included in the instructional materials:
- In Math 1, Lesson 1.1, Solving Equations pages 11-12: The real-world situations pertain to topics that are interesting to high school students: population, overtime pay, weight of a catch, a student discount bus card, saving to buy a bike, radius of Earth and Mars, and one “typical Algebra age problem.”
- Math 2 Lesson 9.3, Using the Quadratic Formula to Solve Equations: The Lesson Performance Task requires students to decide if the center fielder is able to catch the ball using the height and initial speed of balls hit to that player knowing the speed and position of the center fielder.
- In Math 3 Lesson 13.4 Compound Interest, the Lesson Performance Task focuses on grandparents establishing a college fund and references the average annual in-state college tuition. Students are asked to write a model for the value of the investment over time and a model for the growth of college tuition and use the models to evaluate the sufficiency of the investment.
The materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.
The materials partially meet the expectation for being mathematically coherent and making meaningful connections in a single course and throughout the series, where appropriate and where required by the standards. Although this is an integrated mathematics series, the books do not work to build connections between and amongst domains and clusters. Rather, topics are typically taught in isolation within the materials. Repeated lessons throughout the series do not build on or extend students knowledge to reach higher order thinking on the identified standards within these lessons. Repeated lessons detract from students’ learning as they are distracting and unnecessary.
However, a few standards were connected well through multiple books and modules, in a way that built upon itself and extended students to higher levels of thinking. These examples include:
- G-GPE.1,2: From Math 2 to Math 3, lessons build on one another to develop an understanding of not only circles but other conic sections as well.
- S-ID.6: From Math 1 to Math 3, students are expected to fit data with linear and exponential functions then extends fitting of data to include quadratic functions, with a final extension of this concept to fit data with sine functions.
- F-BF.2: In Math 1, students work with arithmetic sequences to prepare for lessons in the following module dealing with linear functions.
- F-LE.2: From Math 1 to Math 3, lessons are designed to build students’ abilities to work with geometric sequences and extend these ideas to geometric series, while preparing students to connect concepts to exponential functions.
- N-RN.1: From Math 2 to Math 3, lessons build on one another to help students understand and connect concepts of rational exponents and radicals.
Algebra and Functions standards, and occasionally Number and Quantity standards, are taught in conjunction with each other, but the book does not make a specific effort to draw attention to connections between the categories. Geometry and Statistics and Probability standards are normally taught in isolation. For example, in Math 2, two lessons were found that focus on concepts from Geometry and Algebra in the same lesson. The module performance tasks do combine multiple standards, but not to draw connections between the concepts; instead, they produce a culminating application of procedures covered in the module.
Examples of connections not made in the materials include:
- Math 1, Lesson 2.3, (A-CED.A.4) does not extend Properties of Equality from Lesson 1.1 and Lesson 2.2 where the emphasis was on solving equations. Instead the lesson focuses on "rearranging;" however, students are not "using the same reasoning as in solving equations," which is only introduced minimally in Lesson 2.3. Module 17 does not connect the concepts of functions and transformations as is done in the Geometry Progression. Making this connection in Math 1 would support students as they work with transformations of increasingly complex function transformations in Math 2 and Math 3.
- Math 3: Lesson 5.4 Explain 2 page 299 connects cubic functions to the volume formula again, this time through standard application of the volume of a prism. Again, this is presented as new material and not explicitly connected to the material in Math 2 Module 21. This is also a missed opportunity to connect quadratic functions to surface area from Module 3, Visualizing Solids.
Almost a quarter of the lessons in the series are exact repeats of each other. Students are not given the opportunity to build on foundations because the lessons that should provide foundation are just repeated from course to course, and careful connections are unable to be built. The following are lessons that are repeated across multiple courses along with the standards that they address:
- S-ID.6b: Math 1, 10.2 and Math 2, 10.2; Math 1, 15.2 and Math 2, 10.3
- A-REI.3: Math 1, 13.3 and Math 2, 2.2; Math 1, 13.4 and Math 2, 2.3
- F-IF.7e: Math 1, 14.4 and Math 2, 10.2
- G-CO.9: Math 1, 19.1 and Math 2, 14.1; Math 1, 19.2 and Math 2, 14.2; Math 1, 19.3 and Math 2, 14.3 and Math 3, 1.1; Math 1, 19.4 and Math 2, 14.4 and Math 3, 1.2
- G-CO.10: Math 1, 22.1 and Math 2, 15.1; Math 1, 22.2 and Math 2, 15.2; Math 2, 2.3 and Math 2, 15.3; Math 1, 23.1 and Math 2, 15.4; Math 1, 23.2 and Math 2, 15.5
- G-CO.11: Math 1, 24.1 and Math 2, 15.6 and Math 3, 1.4; Math 1, 24.2 and Math 2, 15.7; Math 1, 25.4 and Math 3, 2.4
- G-GPE.5: Math 1, 25.1 and Math 3, 2.1; Math 1, 25.2 and Math 3, 2.2
- G-GPE.4: Math 1, 25.3 and Math 3, 2.3
- G-GPE.7: Math 1, 25.2 and Math 3, 2.5
- G-GPE.6: Math 2, 17.2 and Math 3, 2.6
- G-GMD.3: Math 2, 21.5 and Math 3, 4.1
- G-C.2: Math 2, 19.1 and Math 3, 24.1; Math 2, 19.3 and Math 3, 24.3; Math 2, 19.4 and Math 3, 24.4; Math 2, 19.5 and Math 3, 24.5
- G-C.3: Math 2, 19.2 and Math 3, 24.2
- G-GMD.1: Math 2, 20.1 and Math 3, 25.1
- G-C.5: Math 2, 20.3 and Math 3, 25.3
- G-C.1: Math 2, 20.2 and Math 3, 25.2
- G-SRT.8: Math 2, 18.4 and Math 3, 17.1
- F-TF.8: Math 2, 18.5 and Math 3, 18.3
The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
The instructional materials reviewed do not meet the expectations that the materials explicitly identify and build on knowledge from Grades 6-8 to the High School Standards. There is limited evidence of how standards from Grades 6-8 are extended or built upon to develop the high school standards. When the materials focus on standards from Grades 6-8, the standards are not explicitly identified as middle school standards, and they are taught as new lessons rather than reviewed or used to build toward understanding of high school standards. Some lessons include introductory activities that have been designed to allow students to use previous learning. These activities allow students to use a middle grades approach to solving a problem before learning the high school content.
Within the Teacher Edition, the Tracking Your Learning Progression, found at the beginning of each module, informs teachers of what students should know before, during, and after the unit; however, there are no clearly articulated connections between 6-8 and high school concepts indicated for the teacher. No additional evidence was found of explicit indication of standards from Grades 6-8 in any of the provided material.
The first 14 Modules in Math 1 are primarily focused on middle grade standards even though listed as high school standards. Students work with linear functions and models and use quantitative reasoning, which are aligned to standards from 8.F and 8.EE. Connections are not made explicit here between content from Grades 6-8 and the HS standards addressed.
In Math 1, Modules 17-18, the concepts should be an extension of grade 8 work on congruence and transformations. There are no connections to previous learning made within the lessons or no standards from grade 8 are mentioned. Again, the materials reteach previous standards and extend but do not make connections nor move students to deepen their understanding.
Limited information and general statements were found in the margin of the teacher editions of the materials in Math Background or Learning Progressions. Math Background is intended to be professional development for teachers and sometimes relates lesson concepts to definitions, properties, or ideas from previous years. For example, in Math 1 on page 16, dimensional analysis is linked to the Multiplicative Identity Property and multiplication of fractions. Teachers are not told where either of these concepts was first introduced to students or how to expand this thinking to benefit the lesson for student understanding. The Learning Progression snippets offer teachers connections to middle school standards or previous standards addressed in the course. No direction as to where these concepts were covered, how to expand upon them, or connect with future concepts or skills is given.
The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.
All included plus (+) standards are explicitly identified at the beginning of each book in the Correlations section for Common Core State Standards and in the margins of the teacher’s editions at the lesson level. Many of these standards are addressed in isolation within a lesson in the module and can easily be omitted without interference with the rest of the sequence in the series. In cases where both plus and non-plus standards are addressed, the non-plus standards can be focused on while plus standards are used to enrich the learning. These standards are treated just like non-plus standards; limited connections are made and heavy scaffolding exists when the standards ask students to derive or prove.
Math 1: None found
- G-C.4 is found in Lesson 19.3 and is addressed in connection to non-plus standard G-C.2 as an example directly following the portion of G-C.2 that pertains to tangents of a circle. Problems relating to this standard are clearly identified in the Assignment Guide for teachers (page 1035) and can be omitted without detriment to the student should the teacher choose.
- G-GMD.2, found in Lesson 21.4, is addressed as the focus of the lesson and in connection with G-GMD.3, G-MG.1, and G-MG.2. It can be omitted or used as an extension to the non-plus standards.
- S-CP.8, found in Lesson 23.3, is addressed in connection with non-plus standards S-CP.3, S-CP.4, and S-CP.5. The multiplication rule is used to build on students’ knowledge.
- S-CP.9 is found in Lessons 22.2 and 22.3 and is addressed in a lesson focused on plus standard material. It can be omitted without disruption to the course, and it addresses permutations and combinations as a means of computing probabilities.
- S-MD.6 is found in Lesson 24.1 (also in Math 3) and is addressed in a lesson focused on plus standard material. It can be omitted without disruption to the course. The module introduces students to using probability in decision-making through relevant problems.
- S-MD.7 is found in Lesson 24.2 and is addressed within a lesson focused on S-CP.4. Analysis of decisions using probability is used to strengthen students’ understanding of two-way tables in probability.
- N-CN.8 is found in Lesson 6.4 and is addressed in connection with A-SSE.2 and multiple non-plus supporting standards about factoring polynomials.
- N-CN.9 is found in Lesson 7.2 and is addressed in connection with A-APR.2 and multiple non-plus supporting standards about finding complex solutions to polynomial equations.
- A-APR.5 is found in Lesson 6.3 and is addressed as the primary standard with supporting non-plus standards. It can be omitted without risk of loss of learning to the student. The standard is addressed again in Lesson 21.1 as a supporting standard in a non-plus focused lesson on probability distributions.
- A-APR.7 is found in Lessons 9.1 and 9.2 and is addressed as the primary standard with supporting non-plus standards. These lessons are essential to success in Lesson 9.3, which addresses non-plus standards.
- G-SRT.10 is found in Lessons 17.1-17.3. It is first addressed as a supporting standard to G-SRT.8 in 17.1 as an application of the area formula. It can be omitted without impacting a student’s success. Lessons 17.2 and 17.3 focus on the plus standards and can be omitted also.
Rigor And Mathematical Practices
DID NOT REVIEW (Learn why)
DID NOT REVIEW (Learn why)