## Investigations in Number, Data, and Space

##### v1
###### Usability
Our Review Process

Title ISBN Edition Publisher Year
9780328687121
9780328687176
9780328697564
9780328687145
9780328697533
9780328687169
9780328697557
9780328687138
9780328697526
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### Overall Summary

The instructional materials reviewed for Grade 2 do not meet expectations for alignment. The materials do not devote the large majority of time to grade-level work, but the materials can be utilized to appropriately assess grade-level content. There is little explicit connection made to the progressions of learning in the standards. Since the materials do not meet the expectations for focus and coherence in gateway 1, they were not reviewed for gateway 2.

###### Alignment
Does Not Meet Expectations
Not Rated

### Focus & Coherence

The instructional materials reviewed for Grade 2 do not meet expectations for focus on major work and coherence. The materials do not devote the large majority of time to grade-level work but the materials can be utilized to appropriately assess grade-level content. There is little explicit connection made to the progressions of learning in the standards.

##### Gateway 1
Does Not Meet Expectations

#### Criterion 1.1: Focus

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

The instructional materials reviewed for Grade 2 meet the expectations for assessing material at the grade level. Although there are multiple units and sessions noted that align to and/or assess standards that are beyond Grade 2, the inclusion of these sessions and units is either Mathematically appropriate or, where not appropriate, their omission would not significantly alter the structure of the materials.

##### Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Grade 2 meet expectations for assessment because above grade-level assessment items, and their accompanying lessons, sessions, or units, could be modified or omitted without significantly impacting the underlying structure of the instructional materials. For this indicator, all of the identified assessments and end-of-unit assessments for the nine units were reviewed. Units and sessions accompanying above grade-level assessment items are noted in the following list.

• In unit 4, session 2.8 assesses how to: order, represent, and describe a set of numerical data; describe what the data show about the group surveyed; interpret a data representation, including a line plot; compare two sets of data; sort a set of data by two attributes at one time; and use a Venn diagram to represent a sorted set of data. These expectations are appropriate for Grade 2 students, but there are seven sessions that should be omitted from the materials because of the way in which they assess the expectations. In the CCSSM, 2.MD.D.9 states “Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units,” and 2.MD.D.10 states, “Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.” Session 2.1 should be omitted because it uses number of pockets as the measure instead of lengths of objects, and there are problems with more than four categories. Session 2.2 continues to use measures other than lengths of objects. sessions 2.3 through 2.7 also use measures other than lengths of objects and expects displays of data other than the ones addressed in 2.MD.D.10. The omission of these sessions from the session does not significantly impact the underlying structure of the materials.
• The end-of-unit assessment for unit 5 assesses how to: find the value of one quantity in a constant ratio situation given the value of the other; connect the numbers in a table to the situation they represent; describe how the two numbers in the row of a table are connected to the situation the table represents; and determine the element of a repeating pattern associated with a particular counting number in an AB or AAB pattern. These expectations most closely align to 6.RP.A.3, “Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations” and 4.OA.C.5, “Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.”. Since these expectations are two or more grade levels above the current one, the entire unit with 11 sessions should be omitted, but the omission of the entire unit does not significantly impact the structure of the materials.
• In unit 7, there are several sessions that use fraction notation and fractions as numbers. Investigation 2 is consistent with the partitioning required in 2.G.A.3, “Partition circles and rectangles into two three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc. and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.”, Sessions 1.1 to 1.4 and 2.1 to 2.6 use fraction notation. Unit fractions are included in these lessons but not introduced as unit fractions, 1/b, which aligns to 3.NF.A.1, “Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b,” and Session 1.4 introduces mixed numbers which aligns to 4.NF.B, “Build fractions from unit fractions.”. This unit occurs towards the end of the year, so it may be Mathematically reasonable to introduce fraction notation if the students are ready. Teachers should either omit the references to fraction notation in the lessons and assessments or make sure they are also introducing the concept of a unit fraction along with fraction notation.

*Evidence updated 10/27/15

#### Criterion 1.2: Coherence

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.

The instructional materials reviewed for Grade 2 do not meet expectations for time spent of the major clusters of the grade. The materials are not aligned to the major work of Grade 2 and do not devote the large majority of class time to the major clusters.

##### Indicator {{'1b' | indicatorName}}
Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Grade 2 do not meet expectations for majority of class time spent on the major clusters of the grade. According to the unit instructional plans, there are 166 days of lessons and assessments. Between units aligned to supporting and additional work or content from future grade levels, 51 lessons are not aligned to major work. This leaves 115 sessions on the major work, or 69%. Examples of misaligned standards that would decrease the focus percentage include:

• Unit 1: 13 of the 27 sessions more appropriately align with standards from Grade 1, specifically 1.1, adding to ten; lesson 2.1 which involves counting, but in Grade 2 this should extend to counting within 1,000 and the examples are far less; session 2.6 involves adding 1 and 2 to a given number.
• Unit 1: Lessons 3.1-3.5 align with Grade 1 rather than Grade 2, as the work is all combinations of ten.
• Unit 3: Session 4.2 addresses telling time to the hour and half hour, which is addressed in Grade 1.
• Unit 7: Sessions 1.2-1.4 address fractions and include fractions of a set, which is beyond the scope of Grade 2 and is more appropriate for Grade 4.
• Unit 5: Lessons in all three investigations are aligned to patterns, a Grade 4 standard.
• Unit 8: The daily practice on session 3.5 worksheet "What is the Fraction?" has an example of fifths. Grade 2 denominators (partitions) are limited to halves, thirds and fourths.
• Unit 9: Lessons 2.3-2.8 are not truly aligned to 2.MD.D.9 as the data for the line plots is not measurement lengths.
• There is a noted absence of two-step word problems. One-step instead of two-step problems are evident and a missed opportunity in unit 1, sessions 4.3, 4.5 and 4.8 and in unit 8, sessions 3.1, 3.2, 4.1 and 4.4.

#### Criterion 1.3: Coherence

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

The instructional materials reviewed for Grade 2 do not meet expectations for coherence in the grade. The materials are not coherent with the progressions because work from future grades interferes with a coherent and consistent progression with the standards.

##### Indicator {{'1c' | indicatorName}}
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Grade 2 partially meet expectations that supporting content enhances focus and coherence by engaging student in the major work of the grade. In some cases, the supporting work enhances and supports the major work of the grade level and in others, it does not. Examples that support a partial rating include:

• Unit 2, lesson 1.4 "Addition Combinations and Ways to Fill" asks the students to fill in a shape using pattern blocks, count the number of shapes used and record the number of each shape. This lesson engages students in the major work of 1.NBT.1 not the major work of Grade 2.
• Lessons in unit 4, 1.4A bar graphs connect to 2.OA.A.1 and 2.OA.B.1 through counting questions asked and provides contexts for solving problems (although numbers are small). Pictographs are absent.
• "How many teeth?" in unit 4 has the potential to go beyond 2.MD standards. In Grade 2, students should work with data in up to four categories, but a question posed to students in Kindergarten to Grade 5 in the sessions could go well beyond four categories and therefore, Grade 2 work.
• Unit 9, session 1.5, doesn't really align with 2.MD, as the activities do not fully align with standard measurement. Therefore it is not connected to 2.MD.A.1 (measuring length) as students do not use a ruler to generate measurement.
• Time to the five minutes is developed through a progression of daily routines throughout all units that span the curriculum. In unit 6, session 2.5, students begin practicing counting by fives to tell time, which supports their work in 2.NBT.A.2.
##### Indicator {{'1d' | indicatorName}}
The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Grade 2 do not meet expectations for viability of content for the scope of one year. The curriculum consists of 166 total sessions according to the provided pacing in the Investigations and Common Core State Standards Resource. Although this is a manageable number of days for a school year, there is a concern that major clusters, 2.MD.A (measure and estimate lengths) and part of 2.MD.B.5 (word problems involving measurement) are addressed in unit 9 and if a teacher runs out of time in the school year, these important clusters which are needed to prepare for Grade 3 may be shortchanged. Additionally, in that same unit, a great deal of time is devoted to non-standard measure, and only four lessons in investigation 3 actually hit the mark with the measurement of length using U.S. customary and metric measurement. Also, 2.NBT.A (place value and comparing numbers to 1000) is minimally touched upon. Three-digit place value is only addressed in four lessons in unit 6 (5A.2-5A.5), and this will definitely not prepare students for the computation and number work in Grade 3.

##### Indicator {{'1e' | indicatorName}}
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.

The instructional materials reviewed for Grade 2 do not meet expectations for consistency with the progressions. The materials do not develop according to the progressions, nor do they give students extensive work with grade level problems. Detailed explanations of the content addressed appears in in the front matter of each unit in a section called Mathematics in this Unit. Within this section, the Looking Back and Looking Forward pieces explain content from entering school to future grades within the program. There are not explicit connections to CCSSM, however. Some examples of misalignment with the indicator as a whole include:

• Unit 6, session 5.A2 and 5.A3 have math focus points shaped by the cluster heading for 2.NBT
• Unit 6, investigation 1 is about story problems. This is an extension of 2.OA.A.1 with furthering knowledge of addition and subtraction properties and extension of place value.
• Future standards are included that weaken an appropriate grade by grade progression, as in unit 5 "Growing Patterns: Ratio and Equal Groups."
• Lessons and quality problems are noticeably missing in the series relating to 2.MD.D.9 and would need to be included in order to qualify as extensive work with grade level problems.
• Work in 2.NBT.B.7 is limited in frequency (five lessons in unit 8; lessons 5A.1-5A.5). More work is needed in this area.
• Throughout, the review team noted the absence of two-step word problems in all instructional materials.
##### Indicator {{'1f' | indicatorName}}
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.

The instructional materials reviewed for Grade 2 partially meet expectations for fostering coherence through connections at a single grade. The materials include some instances where learning objectives are shaped by cluster headings and include some problems that connect clusters and domains. Examples for both arguments are below:

• All of the lessons in investigation 1 contain objectives that indicate developing strategies for accurate measurement and iterating units to measure length. However, instead of identifying work with standard units, all of the objectives in the investigation refer to non-standard measure. In investigations 2 and 3, the objectives become more visibly shaped by the cluster heading. Objectives can be found which include statements such as "using a ruler as a standard measurement tool," and "identifying strategies for accurate measurement." However, noticeably missing from the objectives in investigations 2 and 3 is the word "estimate."
• Unit 5 addresses topics aligned to Grade 6 standards, such as describing the relationship between two quantities in a constant ratio situation and using tables to represent ratio relationship between two quantities as well as finding the value of one quantity in a constant ratio situation.
• Unit 7 includes work from Grades 3-5 in the NF domain, such as finding equal parts of a group and naming them with fractions, finding one half of a set, solving problems about finding halves of quantities in different contexts and solving problems that result in mixed numbers.
• Problems involving dollars, dimes and pennies (2.MD.C.8) are connected with the place value learning of 100s, 10s and 1s (2.NBT.A.1) and in fluently adding and subtracting within 100 using strategies based on place value (2.NBT.B.5).
• In Unit 1, sessions 2.3 and 2.4 illustrate that a dime is a bundle of ten pennies.
• Unit 3 sessions 3.6 and 3.7 lessons involve counting money and trading into groups of dimes and also grouping objects into tens to count them by tens.
• Unit 3 investigation 4 lessons on place value (4.1-4.5) focus on grouping tens and ones and in the student activity pages 67-70 for those lessons, there are money problems involving dimes and pennies included, thereby making a natural connection. Also, student activity page 74 connects place value and money.
• Unit 6 Investigation 3 involves applying place value understanding to computation of 2-digit numbers. In investigation 3, "Get to 100" is introduced in session 3.1 and is then connected with "Collect $1.00" in the next session. Unroll a square (subtraction from 100) is introduced in session 3.4 and then immediately following that, connecting money to place value is the "Spend$1.00" activity. In the discussion in Session 3.6, "How many 10's in 100?" a connection between 10 tens in 100 and 10 dimes in a dollar is specifically demonstrated.

### Rigor & Mathematical Practices

Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One
Not Rated

#### Criterion 2.1: Rigor

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.
##### Indicator {{'2a' | indicatorName}}
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
##### Indicator {{'2b' | indicatorName}}
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
##### Indicator {{'2c' | indicatorName}}
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
##### Indicator {{'2d' | indicatorName}}
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.

#### Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
##### Indicator {{'2e' | indicatorName}}
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
##### Indicator {{'2f' | indicatorName}}
Materials carefully attend to the full meaning of each practice standard
##### Indicator {{'2g' | indicatorName}}
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
##### Indicator {{'2g.i' | indicatorName}}
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
##### Indicator {{'2g.ii' | indicatorName}}
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.
##### Indicator {{'2g.iii' | indicatorName}}
Materials explicitly attend to the specialized language of mathematics.

### Usability

This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two
Not Rated

#### Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
##### Indicator {{'3a' | indicatorName}}
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.
##### Indicator {{'3b' | indicatorName}}
Design of assignments is not haphazard: exercises are given in intentional sequences.
##### Indicator {{'3c' | indicatorName}}
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.
##### Indicator {{'3d' | indicatorName}}
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
##### Indicator {{'3e' | indicatorName}}
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

#### Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
##### Indicator {{'3g' | indicatorName}}
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.
##### Indicator {{'3h' | indicatorName}}
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.
##### Indicator {{'3i' | indicatorName}}
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.
##### Indicator {{'3j' | indicatorName}}
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).
##### Indicator {{'3k' | indicatorName}}
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

#### Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
##### Indicator {{'3n' | indicatorName}}
Materials provide strategies for teachers to identify and address common student errors and misconceptions.
##### Indicator {{'3o' | indicatorName}}
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.
##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
##### Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.
##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

#### Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

## Report Overview

### Summary of Alignment & Usability for Investigations in Number, Data, and Space | Math

#### Product Notes

Reviewed print materials include:

• Investigations and the Common Core State Standards resource
• Common Core Cards for each unit for each grade, Kindergarten through Grade 5
• Student activity book, Common Core Edition for each grade, Kindergarten through Grade 5
• Differentiation and Intervention Guide
• Investigation Digital Resources, CD-ROM which housed all print materials for each grade level Kindergarten through Grade 5

#### Math K-2

Investigations Grades K-2 does not meet the expectations for Alignment to the Common Core State Standards and Usability. While numerous units of material are provided, they do not spend the majority of instructional time on major work of the grades. The sequence in which topics are covered is not consistent with the logical structure as outlined by the CCSSM and address topics before the grade level introduced in the standards. Therefore, materials are lacking important connections between standards, clusters and/or domains where appropriate and required. Overall, the instructional materials included in this series lack mathematical focus and coherence.

##### Kindergarten
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated

#### Math 3-5

Investigations, Grades 3-5 does not meet the expectations for Alignment to the Common Core State Standards/Usability. While numerous units of material are provided, they do not spend the majority of instructional time on major work of the grades. The sequence in which topics are covered is not consistent with the logical structure as outlined by the CCSSM and address topics before the grade level introduced in the standards. Therefore, materials are lacking important connections between standards, clusters and/or domains where appropriate and required. Overall, the instructional materials included in this series lack mathematical focus and coherence.

###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated

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###### Usability
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