# Mcgraw Hill Homework Answers Page 573

### Alignment

The Grade 4 *My Math* instructional materials meet the expectations for alignment to the CCSSM. The materials meet the expectations for Gateway 1 due to appropriately focusing on the major work of the grade and demonstrating coherence within the grade and across other grades. The instructional materials meet the expectations for Gateway 2 due to appropriately addressing rigor within the grade-level standards and attending to the specialized mathematical vocabulary. There are missed opportunities in the materials when it comes to attending to the full meaning of the standards for mathematical practices. Overall, the instructional materials address the content standards very well, attend to the specialized mathematical vocabulary, and do a nice job of identifying and partially integrating the practice standards.

GATEWAY ONE

### Focus & Coherence

MEETS EXPECTATIONS

Students and teachers using the materials as designed will devote a majority of time in Grade 4 on the major work of the grade. The materials are mostly coherent and consistent with the standards. Assessments only represent grade-level work. Seven percent of the lessons are on future grade-level content and are not clearly identified. At least 64% of the time is spent on the major work of the grade. Overall, the materials do provide a focus on the major work and the materials are coherent.

#### Focus

The materials for Grade 4 met the criteria for 1a and 1b. *My Math* does assess future grade level content. However, an online test generator is available, so points were not deducted At least 64% of the time is spent on the major work of the grade.

**Criterion 1a**

- 0
- 22

Materials do not **assess** topics **before **the grade level in which the topic should be introduced.

The Grade 4 *My Math* instructional materials assess topics from future grade levels, but an online test generator is available, so points were not deducted. The form assessments are featured in the digital companion. Six assessment forms exist for each chapter and an online test generator is available.

2/2

##### Indicator 1a

The instructional material ** assesses** the grade-level content

**, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.**

__and__,__if applicable__The Grade 4 *My Math* instructional materials assess future grade-level content on form assessments, however an online test generator is included in the materials for teachers to create their own assessments. With the inclusion of the digital test generator the Grade 4 *My Math* materials would not assess future grade level content if teachers created their own assessments.

- A test generator is included with the digital companion and teachers can build their own assessment, which would not assess future grade level material.
- The online assessments contain six form assessments per chapter.
- The assessment for chapter 12, on forms 2A and 2B, question 15 and forms 3A and 3B, question 16, assess combinations which is a Grade 7 expectation.
- All assessment forms and several questions in chapter 6 assess students using procedures for division, and at this level conceptual understanding should be emphasized.
- Assessment forms in chapter 7 have 5 questions using parenthesis, a Grade 5 expectation and 3 questions using equations at the Grade 6 level.
- Four benchmark tests are available online. Benchmark 1 (chapters 1-3), Benchmark 2 (chapters 4-6), Benchmark 3 (chapters 7-10), and Benchmark 4 (chapters 11-14).

**Criterion 1b**

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

Students and teachers using the materials as designed would devote the large majority of class time to the major work of the grade. Time spent on the major work was figured using days, lessons and chapters. At least 64% of the time is spent on the major work of the grade.

4/4

##### Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.

The Grade 4 instructional materials spend the majority of time on the major clusters of the grade. Grade 4 material for *My Math *is taught in 14 chapters scheduled to be taught in 160 days.

- Each chapter provides two days for review and assessment, which are included in the 160-day count.
- In the materials, 105 out of 160 days, or approximately 66%, are focused on the major work of the grade level.
- Nine of the 14 or about 64% of the time is spent on the major work of the grade.
- Three chapters (7, 11 and 12) or about 21% of the time is spent on supporting work, which is truly supporting the major work of the grade. This brings the time spent on the major work to closer to 85% of the time.
- Two chapters (13 and 14) of the 14 chapters, or about 14% of the time, is supporting work, which is treated separately.

#### Coherence

The instructional materials are mostly coherent and consistent with the standards. Seven lessons from future grade level content are present and are not clearly identified as such. The materials represent a year of viable content. Teachers using the materials would give their students extensive work in grade level problems, with 93% of the lessons representing grade-level work. Materials describe how the lessons connect with the grade-level standards and with prior and future standards. Overall coherence and consistency of the standards is achieved in Grade 4 *My Math*.

**Criterion 1c-1f**

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

2/2

##### Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Supporting content for Grade 4 *My Math* enhances focus and content by engaging students in the major work of the grade. Overall, the instructional materials do not miss opportunities to connect non-major clusters of standards to major clusters, and as a result, the supporting content does engage students in the major work of Grade 4.

- In Chapter 7, students must use operations (4.OA.A) to complete patterns (4.OA.C).
- In Chapter 11, lessons 2, 4, 6, 7, 9, 10 and 11 support the major work by using fractions and the four operations to solve problems.
- In chapter 12, lessons 4-6 are supporting the major work by using the four operations to solve problems.
- In chapter 13, lessons 2, 4 and 5 are using the four operations to solve problems.

2/2

##### Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The amount of content designated for Grade 4 *My Math* is viable for one school year. Overall, the amount of time needed to complete the lessons is appropriate for a school year of approximately 170-190 days.

- The materials are written for 160 days of a school year.
- Each chapter also has remediation and enrichment activities available plus chapter projects.
- The major work of the grade is the focus for 115 of the days.

1/2

##### Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

Grade 4 *My Math* materials are partially consistent with the progressions in the standards. Future grade level content is not clearly identified. There are extensive grade-level problems and concepts are explicitly related to prior knowledge.

Materials mostly develop according to the grade-by-grade progressions in the standards.

- There are eight lessons, which deal with future grade-level content and those are not identified as off grade-level work.
- The content in chapter 6, lessons 5, 6, 7, 9 and 10, use the division procedures, but at this level conceptual understanding should be emphasized.
- The content in chapter 7, lessons 7, 8 and 9, are expressions at the Grade 6 level.
- Each chapter has a section at the beginning called "What's the Math in this Chapter?" On these pages, the progression from grade to grade is shown.
- In each chapter there is also a spot for coherence, which lists what happened before, now and next in the standards. An example of this can be found in chapter 6, lesson 1 on page 329A.

Materials give students extensive work with grade-level standards.

- The chapters in this book contain a check my progress section to make sure students are ready to move on.
- Only one lesson (lesson 7, chapter 2) exists on subtracting across zeros, which may not be enough for this concept.
- Differentiated instruction activities are available in the teacher edition for students who are approaching level, on level and above level.
- Grade level practice is evident in the "Practice the Strategy," "Apply the Strategy," and "Review the Strategy" within each lesson.
- There are 119 lessons over 160 days.
- Of these, 111 or 93% of lessons provide work with grade-level problems.
- The content in chapter 6, lessons 5, 6, 7, 9 and 10, use procedures for division, but at this level conceptual understanding should be emphasized.
- The content in chapter 7, lessons 8 and 9, are expressions at the Grade 6 level.

Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

- The "Am I Ready?" section at the start of each chapter is focused on knowledge that is truly prior knowledge either from prior grade work or from previous work in grade 4. All prior knowledge is grade appropriate.
- The materials relate grade level concepts explicitly to prior knowledge from earlier grades.
- In the teacher guide, each chapter contains a section called "Where's the Math in this Chapter?" with information on what students should already know prior to entering Grade 4. Also, each lesson in the chapter has a clearly identified section on coherence which states previous standards needed.
- Each chapter begins with a readiness quiz. This quiz can be taken in the student edition under "Am I Ready?" or online.
- Each lesson begins with a review problem of the day to review prior knowledge. For example, in chapter 4, page 197B contains the "review problem of the day."

2/2

##### Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

Grade 4 materials foster coherence through connections at a single grade level. Overall, the materials do include learning objectives that are visibly shaped by the CCSSM cluster headings, and the materials connect two or more clusters in a domain or two or more domains in a grade when appropriate.

Materials include learning objectives visibly shaped by CCSSM cluster headings.

- Each standard is taught in the sequence that the CCSSM is written.
- In the chapter overview of the teacher edition, each lesson is identified as major, supporting, or additional work. Also, the learning objective is listed below.
- For example, chapter 4 focuses on major work of 4.NBT.A and 4.NBT.B. Lesson 2 has students estimating products by rounding and lesson 5 has students multiplying by a two-digit number.
- Each lesson identifies the domain, cluster, objective, and any additional objectives that are addressed in the lesson.

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

- The content in chapters 2, 3, 5 and 6 connect 4.OA and 4.NBT.
- The content in chapter 7 incorporates both 4.OA and 4.G.
- The content in chapter 8 uses 4.NF and 4.NBT standards.
- The content in chapter 14 combines 4.MD and 4.G.
- The content in lesson 2 in chapter 9 connects 4.NF.B.3 with 4.NBT, 4.NF.B.3.A and 4.NF.D.3.D.
- The content in lesson 3 in chapter 14 connects 4.MD.C.5.A with 4.G.A.1.

GATEWAY TWO

### Rigor And Mathematical Practices

MEETS EXPECTATIONS

The Grade 4 *My Math* instructional materials meet the expectations for rigor and mathematical practices. The instructional materials meet the expectations for the criterion on rigor and balance with a perfect score; however, the materials only partially meet the expectations of the criterion on practice-content connections due to not fully attending to the meaning of each MP standard. Overall, the instructional materials are strong in regards to rigor, identifying mathematical practices, and the language of mathematics.

#### Rigor and Balance

The instructional materials reviewed for Grade 4 *My Math* meet expectations for rigor and balance. The instructional materials give appropriate attention to conceptual understanding, procedural skill and fluency, and application, and the materials address these three aspects with balance, not always treating them separately and not always together. Overall, the instructional materials reflect the balances in the CCSSM which helps students meet rigorous expectations by developing conceptual understanding, procedural skill and fluency, and application.

**Criterion 2a-2d**

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

2/2

##### Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Grade 4 *My Math* materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

- The content in chapters 1-6 and 8-10 specifically and fully address standards which are explicitly outlined as conceptual standards (4.NBT.A, 4.NBT.B and 4.NF.A).
- Of the 119 lessons, 51 are focused specifically on the conceptual understanding standards.
- Most lessons in the series have a section "Investigate the Math" which targets conceptual understanding. This is contained in the online lesson presentation. For example, page 491B, teacher edition.
- All lessons in the series have a section "Talk Math" which targets conceptual understanding. This is contained in the online lesson presentation. For example, page 491-492, teacher edition.
- The majority of the homework contains problems which provide students the opportunity to view and to demonstrate their conceptual understanding. For example, pages 83, 661 and 673-674, teacher/student edition.
- Procedures for division is emphasized in Grade 4. More attention is needed for the conceptual understanding of division.
- Procedures for multiplication is emphasized in Grade 4. More attention is needed for the conceptual understanding of multiplication.

2/2

##### Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The Grade 4 *My Math* materials give attention throughout the year to individual standards, which sets an expectation of procedural skill and fluency. Lessons contain multiple examples of fluency practice pages.

- Lessons contain multiple examples of fluency practice pages.
- In the student edition fluency practice pages in chapters 2 and 6. For example, chapter 2, pages 119-120, and chapter 6, pages 399-400.
- Homework contains multiple opportunities for students to practice fluency.
- A "Fact Dash" game is available online with the student login to practice fluency. Students can select the operation and number facts.
- Each chapter in the online teacher edition has additional fluency pages available for printing.
- "Sail through the Math" is an app game for fluency is available for purchase ($1.99).
- Seven lessons out of 119 address 4.NBT.B.4 and are in chapter 2.
- Procedural skills are present in the majority of the lessons. For example, page 169, teacher/student edition, contains procedural skill.

2/2

##### Indicator 2c

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

Grade 4 My Math materials are designed so teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work.

- The teacher edition states "Math in My World," "HOT (Higher Order Thinking) Problems," and "Real-World Problem Solving Readers" address application.
- While "Real-World Problem Solving Readers" are available to provide additional problems, they were not reviewed by EdReports.org as they are not included in the basic package with the student and teacher editions and were therefore considered supplementary.
- The majority of lessons begin with "Math in My World" which uses real-world problems to introduce concepts. For example, chapter 7, lesson 2, page 419.
- The majority of "HOT Problems" address application (for example, pages 488, 508 and 514, teacher/student edition).
- Real-world problems are found in the majority of lessons and homework assignments.
- "Count-down to Common Core" provides performance tasks requiring application of the standards.

2/2

##### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The Grade 4 My Math instructional materials meet the expectations for balance. Overall, the three aspects of rigor are neither always treated together nor always treated separately within the materials, and there is a balance of the three aspects of rigor within the grade.

- At the beginning of each lesson a "rigor" section exists to identify levels of complexity by problem or exercise number. For example, chapter 10, lesson 6 has four problems for conceptual learning (understand concepts), eight problems for fluency/procedural skill (apply concepts), and five problems for application (extend concepts).

#### Mathematical Practice-Content Connections

The Grade 4 *My Math* instructional materials partially meet the expectations for practice-content connections. The materials meet expectations for identifying the practice standards, prompting students to construct viable arguments, and explicitly attending to the specialized language of mathematics. Attending to mathematical vocabulary is a strength of the materials. However, the materials only partially meet the expectations for attending to the full meaning of each practice standard and engaging students in mathematical reasoning. Overall, in order to meet the expectations for meaningfully connecting the CCSSM and the Standards for Mathematical Practice, the instructional materials should carefully attend to the full meaning of every practice standard, especially MP3 in regards to students critiquing the reasoning of other students.

**Criterion 2e-2g**

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

2/2

##### Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

In the Grade 4 *My Math*, the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout Grade 4. Overall, the instructional materials do not over-identify or under-identify the MPs, and the MPs are used within and throughout the grade.

- The teacher edition, pages T22-T24 state the MPs and the corresponding pages.
- The practices are identified throughout all 119 lessons. Each lesson has three to four practices as a focus.
- The student edition does indicate which MP the student is working in the lesson and in the homework.

1/2

##### Indicator 2f

Materials carefully attend to the full meaning of each practice standard

The Grade 4 *My Math* instructional materials partially meet the expectations for carefully attending to the full meaning of each practice standard. Overall, the instructional materials carefully attend to the full meaning of some of the practice standards but not for all of them. Some practice standards do not fully address the intent/context of the MPs. Some examples include:

- MP1, make sense of problems and persevere in solving problems: pages 501-502, 515-516, and 531B, teacher/student edition.
- MP5, use appropriate tools: pages 499-500, 519-520, and 567B, teacher/student edition.
- MP6, attend to precision: pages 491A, 515-516, 521-522, and 545, teacher/student edition.
- Some of the practice standards fully address the intent/context of the MP. Overall, standards for MPs 2, 4, 7 and 8 were well developed.

##### Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

2/2

##### Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Grade 4 *My Math* meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, the materials consistently allow students to construct viable arguments and prompt students to analyze other students' arguments.

- Materials provide opportunities for students to construct viable arguments independent of the teacher.
- The majority of "HOT Problems" have students constructing viable arguments. For example, pages 110, 258 and 582, teacher/student edition.
- Some problems in the homework have students constructing viable arguments. For example, pages 504, 662, 679 and 844.
- More time is given to constructing arguments than analyzing the arguments of others.

1/2

##### Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Grade 4 *My Math* partially meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, the materials do not consistently assist teachers in having students construct viable arguments or analyze other students' arguments.

- Teacher materials do not consistently provide true opportunities for students to construct arguments or analyze the arguments of others.
- Pages 23 - 24, 757--758, and 787-788 provide opportunities for students to construct arguments.
- Pages 79A, 153, and 711-712 provide an opportunity for students to construct an argument and analyze the arguments of others.
- Pages 19 - 20, 198, 593-594, 599-600, and 671-672 do not provide opportunities for students to construct arguments or analyze the arguments of others as stated in the teacher edition.

2/2

##### Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The Grade 4 *My Math* instructional materials explicitly attend to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics that are consistently present throughout the materials.

- The special language of mathematics is a strength of the series.
- Individual vocabulary cards are found at the beginning of each chapter in the student edition.
- Vocabulary checks are included in some homework assignments. For example, chapter 10, lesson 1, page 636.
- Vocabulary assessments can be created online.
- Virtual word walls are available online.
- "Match the Pairs" is an interactive vocabulary component.
- "Check my Progress" assesses vocabulary.
- Each chapter begins with a foldable supporting vocabulary development.
- At the beginning some chapters contain "My Math Words." For example, chapter 5, page 274.
- The teacher, student, and online editions contain extensive glossaries in English and Spanish.
- Lessons contain mathematical terminology.

GATEWAY THREE

### Usability

MEETS EXPECTATIONS

The Grade 4 materials reviewed meet the expectations for usability. Foremost, the materials met the criterion for use and design. The underlying design of the materials makes a distinction between problems (labeled as Math in My World and Guided Practice) and exercises (labeled as Independent Practice and Practice). Each problem or exercise has a purpose. The design is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The materials reviewed partially meet the criterion for teacher planning and learning. Materials contain a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. The materials rarely contain full adult level explanations to help teachers improve their own knowledge. The materials only provide the role of grade-level mathematics within the grade level before and the grade-level after, but not the entire K-5 series spectrum. The materials meet the criterion for assessment. One strength is that the materials provide strategies for gathering information about students’ prior knowledge within and across grade levels. Materials provide strategies for teachers to identify and address common student errors and misconceptions. On the other hand, a note of concern is that students performing below grade-level may never be held accountable for grade-level standards, if the teachers follow the sequence provided for struggling students. The materials meet expectations for the criterion for differentiated instruction**. **Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners. Each lesson has strategies for struggling, advanced, and ELL learners. A strong point is that the materials attempt to provide a balanced portrayal of various demographic and personal characteristics. Overall, the Grade 4 material partially meets the criterion for usability.

**Criterion 3a-3e**

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The materials meet the criterion for use and design. The underlying design of the materials makes a distinction between problems (labeled as Math in My World and Guided Practice) and exercises (labeled as Independent Practice and Practice). The difference between the problems/exercises is: in solving problems students learn new mathematics, whereas in working exercises students apply what they have already learned in order to build mastery. Each problem or exercise has a purpose. The design of assignments is not haphazard; exercises are given in intentional sequences. Furthermore, the design is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

Additionally, the manipulatives and/or models accurately and consistently represent the mathematical objectives. Overall, the materials reviewed for Grade 4 meet the expectations for this criterion.

2/2

##### Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The underlying design of the materials does distinguish between problems and exercises, thereby meeting the expectations for this indicator.

- The design of the materials distinguishes between problems and exercises. In My Math, exercises are labeled as Independent Practice and Practice while problems are labeled Math in My World and Guided Practice. For example, page 513 SE is an Independent Practice, and pages 511–512 SE are Math in My World and Guided Practice.
- All lessons contain practice exercises that allow students to apply what they have learned. These exercises are broken down into approaching level, on level, and beyond level.
- Test Practice is used to diagnose student errors. In Chapter 4, students have to draw area models to multiply.
- In the Write About It section, exercises provide opportunity for students to reflect on a topic and build understanding needed to answer the chapter Essential Question.
- Math in My World provides the new mathematics. The independent practice gives a chance for the child to practice the new learning with guidance from a teacher. The homework provides yet another opportunity to use the new skill on your own. Examples are on pages 135–138 and 37–40 in the Teacher’s Edition.

2/2

##### Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.

The design of assignments is not haphazard; exercises do seem to be given in intentional sequences, which meets the expectations for this indicator.

- The problems in My Math are purposefully designed and organized. Connections are made and directly stated in the “What’s the Math in this Chapter?” and “Making Connections” sections at the beginning of each chapter in the Teacher’s Edition.
- Teachers are also directed in the lessons to point out the connections. Examples occur on pages 4–6, 189E–189F, 321E–321F and 478 in the Teacher’s Edition.
- The sequence in lessons usually goes from a more concrete example using manipulatives or models to more abstract types of problems.
- Due to the intentional layout of the materials, the “What’s the Math in This Chapter” section provides the reason for lessons being presented as they are. It shows the link between prior knowledge and new information to be learned.
- The design of the assignments is done in an intentional sequence. For example, the sequence starts with place value, addition and subtraction, and then moves into multiplication and division before working on primes and factors which then segue to fractions.

2/2

##### Indicator 3c

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

There is a variety in what students are asked to produce, which meets the expectations for this indicator.

- Students are asked to produce many types of answers throughout the work they do.
- Math in My World usually has the student work through the math providing a model first and then moves on to students producing answers using other strategies besides always using a model.
- Some opportunities are given in the problem-solving sections for students to create arguments about someone else’s work. Examples include pages 237–238.
- Students are required to justify their answers on some of the Independent Practice and Homework. Examples include pages 503–504.
- Students are asked to produce a variety of work throughout the materials. For example, Independent Practice found on page 513 SE.
- Math in My World, page 517 SE, helps students see math in a real world context and students are producing solutions in a grade-appropriate manner.
- Throughout the materials, students are using mathematical models, for example page 539 SE, problems 2–7. During some HOT problems students are asked to explain their thinking, for example, page 804 SE, problem 26.

2/2

##### Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The manipulatives are almost always faithful representations of the mathematical objectives they represent and, when appropriate, are connected to written models meeting the expectations for this indicator.

- Manipulatives are appropriately used and explained in both the student book (for example, page 561 build it) and on homework assignments (for example, page 565 homework helper).
- The digital website has tools containing virtual manipulatives.
- Page 657 SE is an example of using base ten blocks to represent fractions.
- Another example is page 527 SE in Homework Helper where the number line is used to compare and order fractions.
- The Teacher Edition contains Model the Math sections which provide teachers the materials needed to model the math. Examples are on page 241B, 445B, and 387B.
- For Math in My World, there is a visual model that is an appropriate representation.

##### Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The visual design is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

- The visual design of the print text has a consistent layout for each lesson and is not visually distracting but contains pictures and models that support student learning and engage students.
- The pictures provided are meant to support and engage student learning. For example, page 845 SE, Math in My World, the picture of the sandbox is designed so the students can visualize the sandbox while determining the area.
- The online text is identical to the print version.
- The student editions are free from clutter and aesthetically pleasing.
- There are minimal distractions on student and teacher pages.

**Criterion 3f-3l**

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The materials reviewed partially meet the criterion for teacher planning and learning. The materials partially support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Materials contain a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. The materials rarely contain full adult-level explanations to help teachers improve their own knowledge. The materials only provide the role of grade-level mathematics within the grade level before and the grade-level after, but not the entire K-5 spectrum. Overall, the materials reviewed for Grade 4 partially meet the expectations for this criterion.

1/2

##### Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The materials partially support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

- Created teacher lesson presentations that follow the text lesson are available online.
- Lessons contain a Problem of the Day, which reviews learning from other units.
- The teacher edition has alert boxes to notify teachers of common mistakes students make.
- Throughout the lessons, there are scripted parts and questions for the teacher to ask.
- Some quality questions are provided to help guide the students’ mathematical development. Some examples of quality questions can be found on pages 17–18, 79B, 445–446, and 907–908 in the Teacher’s Edition.
- Some of the questions provided for the teacher are just reciting facts and do not guide the students to further understand the mathematical concepts. Examples of these types of questions can be found on pages 167B, 243–244, 353–354, and 637–638 in the Teacher’s Edition.
- Quality questions are also included in the independent practice and Homework pages.
- Some of the suggestions to guide students’ mathematical development only partially get the teacher to the target. For example, on page 141 TE of Investigate the Math, teachers are to ask students to prove that division can be thought of as repeated subtraction. Then during the model and extend portion, teachers are not given clear directions for an investigative activity or how to help their students construct viable arguments.

2/2

##### Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

Materials meet the expectations for containing a teacher edition that has ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The teacher edition contains ample annotations and suggestions. Common Errors are noted (for example, page 581 TE, 827–828 TE, 595–596 TE, and 115–116 TE) and ways to incorporate mathematical practices are also noted (for example, page 579 TE).

Each lesson contains differentiation suggestions and ELL support (for example, TE page 583A). • Exit tickets identify common misconceptions/errors (for example, page 584 TE).

Each lesson has an online presentation to mirror instruction in the text to support and enhance student learning.

In the Teacher’s Edition there are many scripted parts of the lesson with expected or anticipated student responses to help guide the teacher.

There are answer keys for most of the student problem sets, exit tickets, homework and tests. Some problems state "see students’ work" or "see students’ models" as the answer key (some examples of this are on pages 115–116 TE, 351–352 TE).

In the Online piece, there is a section called PD. Under PD, there is a section called content videos. These videos do help understand the mathematics a little better and also give the teacher a little better understanding of how to present the information to the students.

Each lesson contains a section called Diagnose Student Errors to help the teachers understand where students may have misunderstandings. For example, see pages 145–146 TE and 15 –152 TE.

1/2

##### Indicator 3h

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The materials partially meet the expectations for containing a teacher edition (in print or clearly distinguished and accessible as such in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematical concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

- The print materials contain a section before each chapter called “What’s the Math in this Chapter” for example, Chapter 1, page 1c. This section has a brief overview of the mathematics and what students are expected to understand and to do. This section does not go into the depth needed for teachers to expand their content knowledge.
- The professional development available online contains videos explaining mathematical practice in action and content videos.
- A few examples of more advanced mathematical concepts are presented in the beyond level work. Even with these pieces there is not an explanation of the mathematics to help further the instructors’ knowledge.

1/2

##### Indicator 3i

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The materials do contain a teacher edition (in print or clearly distinguished and accessible as such in digital materials) that partially explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for Kindergarten through Grade 12.

- The Teacher’s Edition contains a section "What's the Math?” in this Chapter that explains how grade-level mathematics is connected to the previous grade and the next grades (for example, pages 189E–189F TE and 553E–553F TE).
- Since this is a K-5 series the expectation is not to connect to the entire K-12 sequence but to at least the K-5. While the materials do connect the mathematics to the prior knowledge and what students will do next with these skills, there is not a clear path outlined to explain the path from K-5.

##### Indicator 3j

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The materials do provide a list of lessons in the teacher edition (in print or clearly distinguished/accessible as such in digital materials), cross-referencing the standards covered and providing pacing guidance on the estimated instructional time for each lesson, chapter and unit.

- Each unit begins with a sequence of learning. Lessons are listed with CCSSM, MPs and objectives as well as a suggested pacing (for example, chapter 9 553A–553D).
- There is a pacing guide on page vi in the Teacher’s Edition that explains how many days each chapter is expected to take.
- The teacher edition has a chart on page T17–T24 which aligns each lesson to the CCSSM and to the MPs.
- At the beginning of each chapter there is a more in depth overview of the lessons and the standards covered in the chapter.
- Additionally, each chapter begins with pages laying out the chapter, which includes the length of instructional time for the lessons and chapter. For example, page 1A-B TE.

##### Indicator 3k

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The materials contain some strategies for informing parents or caregivers about the mathematics program and some suggestions for how they can help support student progress and achievement.

- Students can share each unit’s "My Common Core State Standards," which states the CCSSM and MPs for the unit (SE 554), with their parents. This could be torn out and sent to parents; however it is not in very friendly language.
- Teachers or students can print a parent letter, found online, for each unit explaining the standards being taught, suggested activities and vocabulary.
- A ConnectED Parent Letter is available online. This online resource introduces ConnectED to parents.
- The online portion has a place for students and parents/caregivers to check progress and watch videos for homework help.

##### Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials contain limited explanations of the instructional approaches of the program and identification of the research-based strategies within the teaching materials.

- Each unit identifies a 21st century skill for the unit (for example pages, 1E TE and 553G TE).
- Each unit is based on an Essential Question.
- Lessons contain ideas for differentiation and ELL support.
- Detailed explanations of instructional approaches are not found.
- Each chapter also has a foldable, which is explained, including how to use it and what the math is in the foldable, chapter 1, pages 9-10 TE.

**Criterion 3m-3q**

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The materials reviewed meet expectations for the criterion of assessment for Grade 4. A strength is that the materials provide strategies for gathering information about students’ prior knowledge within and across grade levels. Materials provide strategies for teachers to identify and address common student errors and misconceptions. The materials offer formative and summative assessments. Although some rubrics are provided, suggestions for follow-up are not provided. A note of concern is student’s performing below grade-level may never be held accountable for grade-level standards, if the teachers follow the sequence provided for struggling students. Overall, the materials reviewed for the Grade 4 meet the expectations for the assessment criterion.

DAVID KESTENBAUM, HOST:

Since the very early days of PLANET MONEY, we have regularly gotten emails from people asking us please, please, do a show about this thing, this thing I had to buy that seems weirdly expensive - textbooks.

You got the textbook there?

KENDALL RAYDUN: I do, yeah. Let me grab it. Yeah, it wasn't cheap, that's for sure.

KESTENBAUM: This is Kendall Raydun. She goes to school at American River College in California.

What's it called? What's the title? Can you read it?

RAYDUN: It is "College Physics."

KESTENBAUM: And how much was the textbook? How much...

RAYDUN: It is about $310.

KESTENBAUM: It was the most expensive book she'd ever seen. Really, she thought, $310 for a book with some online stuff?

RAYDUN: So I was actually kind of irritated, you know? So I ended up - you know, at some point, though, I just needed the book. You know, we were starting class and homework and all that stuff. So I just went and bought it, put it on a credit card.

KESTENBAUM: Kendall is not questioning the value of an education or that in the long run she'll be earning more as a result of this class and the price of that book will be a drop in the bucket. It just seemed like a lot of money for what it was. And something truly strange has been going on in the textbook market. There's this chart that gets cited a lot, maybe you've even seen it. It's from a government report on textbook prices and shows the price of new textbooks over the past decade, and it is a very steep line. The prices of new textbooks have been going up like crazy, faster than clothing, food, cars, even health care.

(SOUNDBITE OF MUSIC)

KESTENBAUM: Hello, and welcome to PLANET MONEY. I'm David Kestenbaum.

JACOB GOLDSTEIN, BYLINE: And I'm Jacob Goldstein. Today on the show, by popular demand, why are textbooks so expensive? We talked to students and textbook authors and analysts and publishers. And after all that, we think we found a pretty satisfying answer.

(SOUNDBITE OF MUSIC)

GOLDSTEIN: Let's start with a guy named James Koch. He taught economics at Old Dominion University for a long time. And he never thought about the economics of textbooks until at some point students started coming to him and asking, excuse me, professor, do we really need this book you assigned for class? It's kind of expensive. Koch said, yes, you need it. It's a good book. But this got him thinking about the odd nature of the textbook market.

KESTENBAUM: One thing in particular struck him, which is that he was the one who was choosing the books for class. He thought about it carefully. The book was well-written. He knew it was clear and knew it had good problems at the end of the chapters. But he realized there was one key thing he did not know about it.

JAMES KOCH: I did not know how much it cost.

GOLDSTEIN: Normally, Koch says, the person deciding to buy something is also the person paying for that thing. In this case, that relationship was severed. The person choosing was not paying. Economists call this the principal agent problem. The principal is the person with the money. The agent is the guy figuring out how to spend it.

KESTENBAUM: I call it the someone-else's money problem. It's someone spending someone else's money.

GOLDSTEIN: Is it really true professors don't think about price when they're looking at textbooks for the class?

KOCH: I've been in higher education now for - what? - about four decades, and I never have had a single textbook sales person come into my office and talk about price. They're always talking about, gee, we have this new coverage of these new topics. We have this new DVD. We have these ways that you can test students and give them quizzes and keep track of their progress. It's always what's in the textbook package, never about the price.

KESTENBAUM: It's odd for a salesperson not to talk about price.

KOCH: Well, it's not odd when you think that they're talking to a person who doesn't have to pay it.

KESTENBAUM: (Laughter) That's the problem, huh?

GOLDSTEIN: This can lead to higher prices really in two different ways. I mean, first, the simple one is publishers just have less of an incentive to keep prices down to compete on price. But the second one is actually more interesting to me. This market can lead to a fancier textbook than you'd get otherwise, a textbook loaded up with stuff that students might not need because remember it's the professor who's the customer. So publishers go to professors and say, hey, buy our textbook. It comes with online quizzes and slides and a workbook. I mean, I remember when I was in college, like, at the back of the biology textbook there was this DVD sitting there that I never once used. But I paid for it in the price of the book.

KESTENBAUM: I asked Koch about this particular economics textbook I happened to have in my desk. It is the best-selling economics textbook in America by Harvard economist Greg Mankiw. You can buy it on Amazon for $286.36. So I ask Koch does what you're saying mean that Greg Mankiw, your fellow economist, is making more money than he should off this book? It was an awkward question, but I had to ask. Koch did not want to weigh in, so we did something more awkward. We called up Greg Mankiw himself.

GREG MANKIW: This is Greg Mankiw. I'm a professor of economics at Harvard University.

KESTENBAUM: Let me just start by saying thank you for agreeing to come in and be grilled about the price of your textbook.

MANKIW: It's my pleasure, I think.

(LAUGHTER)

KESTENBAUM: I wanted to know what did the author of an economics textbook think about the very market that that textbook was in. The principal agent problem is right there in his book. It's on page 462. So we ran the idea by him that professors who pick the books aren't paying attention to price. He agreed this was not ideal, but he didn't see it as a big problem.

MANKIW: I don't think it's as unusual a market as you suggest. I think there's lots of other markets that are similar. When you go get a medical operation, you often rely on the advice of a doctor, even though you're paying - you or the insurance company's paying the price. When you go get your car fixed, you're relying on the mechanic to pick out the parts for you. When you're building a new house or a new extension on your house, you're relying on your contractor to find the right parts and look out for your best interests. So I think there's lots of situations where we rely on someone else to help us make an informed decision. And certainly the textbook market is that way.

GOLDSTEIN: I feel like the examples you chose - health care, paying a contractor to work on your house, going to a mechanic - are all the things that people hate paying for for this reason. I mean, people who get their homes renovated talk about what a nightmare it is, and health care is this famously broken market.

MANKIW: Well, I think it's - whenever you have a principal agent problem, there's a risk that the agent, which in this case is the professor, doesn't do due diligence and doesn't do their job correctly and look for the best interests of the principal, which in this case is the student. But a good professor would do that.

GOLDSTEIN: Mankiw says professors do think a lot about what's best for the students.

MANKIW: The biggest expenditure for students is not the expenditure of money, but it's the expenditure of their time. They're spending a lot of time - and, of course, they should spend a lot of time on their course if they're going to get a lot out of it. And I want their time to be used as productively as it can be. And giving them the best book to read that I can is far more important than saving them a few dollars. So if somebody comes in and says I have a book that's - it's not as good, but it's going to save your students $30, I'm going to say given that they're going to spend 40, 50, 60, 70 hours reading this book over the course of a semester, am I really going to skimp on a textbook to get something that's inferior? I don't think so.

KESTENBAUM: So, OK, we have this theory about the someone-else's-money problem driving up the cost of college textbooks. It makes sense, but it seems hard to test, right? You need some parallel universe where the people choosing the textbooks were paying for them so you could see what difference it makes.

GOLDSTEIN: Fortunately for us, that parallel universe exists. It's called high school.

KESTENBAUM: Very nice. High school textbooks are chosen and paid for by the local and state governments. It turns out that forces publishers to keep the cost down. Jonathan Helliwell is a financial analyst at Panmure Gordon, one of the oldest brokerages in England. He says publishers earn much smaller profit margins on high school textbooks than they do on college textbooks.

JONATHAN HELLIWELL: School book publishers make about 5 to 10 percent profit margins and college textbook publishers make 20 percent - can make 20-25 percent profit margins.

KESTENBAUM: And you think one reason is that the people picking the books aren't thinking about cost.

HELLIWELL: I'd say that's the biggest reason is who's the customer and what's the effect of the buying process.

KESTENBAUM: Has anyone in the publishing business ever acknowledged to you that they have this advantage, that the professors aren't thinking about cost?

HELLIWELL: (Laughter) They haven't argued about it when I put it to them. Let's put it that way.

GOLDSTEIN: The someone-else's-money theory is powerful, but it leaves out this one big thing that just about everybody we talked to agreed was a major factor driving up prices, and it's something that oddly does not seem like it would make prices go up.

KESTENBAUM: It is the used textbook market. Twenty years ago, the used book market was local. Basically, you'd go to your college bookstore. If they had a used copy, you could buy it. If not, you were stuck. The internet, of course, changed all that. Now when a student in Florida finishes her class, she can sell her book on eBay to someone in Montana or just get a book on Amazon. Here, I'm looking at the page for Mankiw's textbook - new version, $286; used, $227. If you don't mind one that's a few years old, you can get the previous edition used for $26.

GOLDSTEIN: All of this means that if you're a publisher or a textbook author, you have this really short window to make money off your book because after the first semester, all the students who bought your book are going to turn around and sell it to the next batch of students.

KESTENBAUM: Robert Frank is an economist at Cornell who's written a textbook with former Fed Chairman Ben Bernanke.

ROBERT FRANK: It used to be your new edition would come out. Then, the next year, it would sell half as many copies as the first year and then half again on the third year. Now it's you sell copies - if you sell any at all in the first year - and it's done.

KESTENBAUM: Nothing the next year?

FRANK: Almost nothing the next year.

KESTENBAUM: So if you're a textbook company faced with this problem, what do you do? You're selling fewer books. How are you going to cover your costs? Well, you raise prices.

GOLDSTEIN: And this, it seems, is what textbook companies did, but raising prices just made things worse. People came up with new ways to avoid those higher prices - textbook rentals, illegal downloads. Some students just skip buying books altogether.

KESTENBAUM: Which meant the textbook companies were selling even fewer books, so they raised the price for new books again. You can see where this is going. One textbook salesman I talked to called it a spiral of destruction.

GOLDSTEIN: So you have this battle. On the one hand, the price of new textbooks going up and up and up. On the other hand, students finding lots of ways around buying new, full-priced books. Who's winning?

KESTENBAUM: Turns out, there's data on this, comes from the National Association of College Stores. They do these student surveys asking, how much do you actually spend on textbooks? We talked to Rich Hershman there. He went through the numbers with us.

In 2007, how much did students spend?

RICH HERSHMAN: Seven hundred and two dollars.

KESTENBAUM: 2009 - two years later?

HERSHMAN: Reported spending - $667.

KESTENBAUM: Oh, so it went down?

HERSHMAN: Yes, it did.

KESTENBAUM: 2011?

HERSHMAN: Six hundred and fifty-five dollars.

KESTENBAUM: Down again, went up a bit the next year, then down even further.

HERSHMAN: So what we've seen, essentially, is flat to declining spending from students - what the students are saying they're spending on required course materials over the last five-six years.

GOLDSTEIN: Students have, basically, fought the publishers to a draw on this one.

KESTENBAUM: And yet, everyone seems to feel like they're losing - students who have to pay $300 for new physics textbooks, professors who are hearing their students complain, textbook sales people who once felt proud that they were helping further education but are now embarrassed to be out selling these really expensive books, which may be one reason why when I finally sat down with David Levin, the president and CEO of one of the biggest textbook companies around, McGraw-Hill Education, he did not want to talk about textbooks. I kept asking about books. He kept talking about educational software. He saw it as a way out of this big spiral of destruction and rising prices - electronic, interactive versions of textbooks.

DAVID LEVIN: We've got now 500 engineers building those full time, about $150 million a year going into product creation. Your grandfather's textbook company didn't do that. It sat astride a business which was simple - produce books, and that's what it did. We are ploughing huge resources into creating a new set of instructional materials which help students and help instructors and do so at a much lower price than it's ever been seen before.

KESTENBAUM: You really don't want to talk about books anymore.

LEVIN: I - we don't. This isn't - this is not - it's not very interesting to us.

GOLDSTEIN: Digital textbooks are cheaper than traditional textbooks, easy to update, they don't weigh anything. But for students, there is this one big drawback. You can't sell them back to the bookstore or to anyone at the end of the semester. There is no used market for digital textbooks.

KESTENBAUM: They just got to find a way to keep students from downloading them illegally.

(SOUNDBITE OF MUSIC)

GOLDSTEIN: Our show today was produced by Phia Bennin. You can email us at planetmoney@npr.org. You can find us on Facebook or Twitter.

KESTENBAUM: Also special thanks to Eric Weil at Student Monitor for sharing your data on textbook spending and to all the students who talked to me about their expensive textbook purchases - Beverly Minton (ph), Jamie Pyce Phillips (ph), Josh Raymond, and Brittany Basile (ph).

Several of you, by the way, asked us to look into why there seemed to be this constant stream of new editions of textbooks. Were those really necessary, you asked, or just attempts to sell more new copies? So we asked this question. Robert Frank, the economist, told us that for the micro-economics textbooks that he writes, he did not see any good reason why it needed to be updated as often as it is. He said for macroeconomics, you would want to update it because, you know, there are things like the financial crisis that happened.

I also asked David Levin, the CEO of McGraw-Hill Education, about new editions. In particular, we were talking about calculus textbooks because, you know, the subject of calculus doesn't really change. He told me they published one calculus book. It had last been updated in 2011, five years after the previous edition. He said they added a bunch of new problems and revised some parts of the text. But he wrote, quote, "I can pretty much guarantee that we will never produce another print edition of this book."

All right, I think that's it. I'm David Kestenbaum.

GOLDSTEIN: And I'm Jacob Goldstein. Thanks for listening.

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