## Bridges In Mathematics

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

Showing:

## Report for Kindergarten

### Overall Summary

The instructional materials reviewed for Kindergarten are aligned to the CCSSM. Most of the assessments are focused on grade-level standards, and the materials spend the majority of the time on the major work of the grade. The materials are also coherent. The materials follow the progression of the standards and connect the mathematics within the grade level although at times off-grade level content is not identified. There is also coherence within units of each grade. The Kindergarten materials include all three aspects of rigor, and there is a balance of the aspects of rigor. The MPs are used to enrich the learning, but the materials do not always attend to the full meaning of each MP and additional teacher assistance in engaging students in constructing viable arguments and analyzing the arguments of others is needed. Overall, the materials are aligned to the CCSSM.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The materials reviewed for Kindergarten meet the expectations for Gateway 1. These materials do not assess above-grade level content, and they spend the majority of the time on the major clusters of each grade level. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are partially consistent with the mathematical progression in the standards, and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the Kindergarten materials are focused and follow a coherent plan.

##### Gateway 1
Meets 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 Kindergarten meet the expectations for assessing grade-level content. Overall, the instructional materials can be modified without substantially affecting the integrity of the materials so that they do not assess content from future grades within the summative assessments provided. Summative assessments considered during the review for this indicator include unit post-assessments and Number Corner assessments that require mastery of a skill.

##### 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 assessment materials reviewed for Kindergarten meet expectations focus within assessment. Content from future grades was found to be introduced; however, above grade-level assessment items, and their accompanying lessons, could be modified or omitted without significantly impacting the underlying structure of the instructional materials.

For this indicator, the Kindergarten Assessment Map found on pages 12 – 14 in the Assessment Overview section was used to identify “summative” assessments. The Assessment Map indicates when mastery of each standard is expected and where the mastery standard is assessed. Based on the Assessment Map, the following were considered to be the summative assessments and were reviewed for Indicator 1a:

• Number Corner Checkups 1 – 4
• the Comprehensive Growth Assessment
• Select Unit Checkpoints
• Unit 2 Module 1, Session 5, Count and Compare Checkpoint
• Unit 3 M1, Session 4, Beat You to Ten Checkpoint
• Unit 4 M3, Session 3, Counting and Writing Numbers Checkpoint
• Unit 5 M1, Session 4, Sort and Count Checkpoint
• Unit 5 M3, Session 4, 2-D Shapes and Their Attributes Checkpoint
• Unit 6 M1, Session 4, Cylinder Tens and Ones Checkpoint
• Unit 6 M2, Session 4, 3-D Shapes and Their Attributes Checkpoint

Assessments are student observation/interview or written in nature. The Comprehensive Growth Assessment (CGA) and all of the Number Corner Quarterly Checkups are fully aligned to the Kindergarten CCSSM. In the Number Corner Quarterly Checkups, several skills/concepts in the K.CC cluster are benchmarked and assessed throughout the year. For example, K.CC.1 (Count to 100 by 1s) is assessed to 20 on NCCU1, to 60 on NCCU3, and to 100 on NCCU4.

The Unit Assessment Checkpoints that contain above grade-level or content not specifically required by the standards are noted in the following list:

• In the Unit 4 Module 3 Session 3 Counting and Writing Numbers Checkpoint, Prompt 4, students are asked to count backward from a number (4 – 9) until they reach zero. Counting backwards is not an explicit K CCSSM expectation; however, it makes mathematical sense to address it as a precursor to subtraction where counting backward is a necessary skill. This skill is identified in the assessment scoring guide as “Supports K.CC.”
• In the Unit 5 Module 3 Session 4 Two-Dimensional Shapes & Their Attributes Checkpoint, students are expected to identify a rhombus and a trapezoid. Those shapes are not specifically identified in K.G.2; however, it makes sense to include them since they are shapes in the pattern block set students use throughout the unit.
• In the Unit 6 Module 1 Session 4 Cylinder Tens & Ones Checkpoint, students are asked to create a cylinder using a strip of paper. Question 2 asks students to estimate how many unifix cubes they think it will hold and then to fill the cylinder with unifix cubes without counting them. After dumping out the cubes and arranging them into tens and ones, they are asked to count them (Question 5) and then compare the actual number of cubes to their estimate to determine if their estimate was more or less than the actual number of cubes (Question 6). Since teachers are instructed to provide each student with 40 cubes, it is reasonable to believe that the number of cubes students will be expected to count will exceed the limit of 20 designated in K.CC.5 and subsequently, students may be comparing numbers greater than 20. Additionally, the estimation of quantities is not a Kindergarten expectation. Adjusting the size of the cylinder to ensure the counting of smaller quantities and eliminating the estimation portion of the assessment would be an easy fix and would not affect the integrity of the unit.
• In the Unit 6 Module 2 Session 4 3-Dimensional Shapes and Their Attributes Checkpoint, the majority of the student observations are aligned to the K Geometry Standards: K.G.1, K.G.2, K.G.3, and K.G.4. However, in the observational task in which students use polydrons to build 3-D shapes, the teacher is to document if a student has successfully built rectangular prisms, triangular prisms and pyramids. This expectation is more appropriately aligned to 1.G.2 and not to K.G.5 (model shapes in the world by building shapes from components, e.g., sticks and clay balls, and drawing shapes) because the 3-D shapes identified in the K Geometry standards are limited to cube, cone, sphere and cylinder. However, as long as the focus is on building shapes and not naming them, this would be acceptable as most of these shapes are introduced within the K-2 grade band.

#### 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 Kindergarten meet the expectations for focus on the major clusters of each grade. Students and teachers using the materials as designated will devote the majority of class time to major clusters of the grade.

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

The instructional materials reviewed for Kindergarten meet the expectations for focus by spending the majority of class time on the major clusters of the grade. All sessions (lessons), except summative and pre-assessment sessions, were counted as 60 minutes of time. Number Corner activities were counted and assigned 20 minutes of time. When sessions or Number Corner activities focused on supporting clusters and clearly supported major clusters of the grade, they were counted. Reviewers looked individually at each session and Number Corner in order to determine alignment with major clusters and supporting clusters. Optional Daily Practice pages and Home Connection pages were not considered for this indicator because they did not appear to be a required component of the sessions.

When looking at the modules (chapters) and instructional time, when considering both sessions and Number Corners together, approximately 90 percent of the time is spent on major work of the grade.

• Units – 8 out of 8 units spend the majority of the unit on major clusters of the grade, which equals 100 percent. Each unit devotes most of the instructional time to major clusters of the grade.
• Modules (chapters) – 28 out of 32 modules spend the majority of the time on major clusters of the grade, which equals approximately 88 percent. Units 2, 5, 7 and 8 had three Modules that focused on major work of the grade, and all other units had all four Modules focused on major work of the grade.
• Bridges Sessions (lessons) – 143 out of 160 sessions focus on major clusters of the grade, which equals approximately 89 percent. Major work is not the focus of the following sessions:
• Unit 1, Module 4, Sessions 1, 2, 3 and 4
• Unit 2, Module 4, Sessions 1, 2, 3 and 4
• Unit 4, Module 3, Session 2
• Unit 5, Module 1, Session 1
• Unit 5, Module 2, Session 5
• Unit 5, Module 4, Sessions 2, 3 and 4
• Unit 6, Module 2, Session 4
• Unit 8, Module 4, Sessions 4 and 5
• Bridges sessions require 60 minutes. A total of 143 sessions are focused on major work of the grade. Bridges sessions devote 8,580 minutes of 9,600 minutes to major work of the grade. A total of 155 days of Number Corner activities address major work of the grade. Number Corner activities are 20 minutes each adding another 3,100 minutes to this total. In all 11,680 of 13,000 minutes, approximately 90 percent, is devoted to major work of the grade.

#### Criterion 1.3: Coherence

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

The instructional materials reviewed for Kindergarten meet the expectations for coherence. The materials use supporting content as a way to continue working with the major work of the grade. For example, students count shapes in categories and then compare the quantities. The materials include a full program of study that is viable content for a school year, including 160 days of lessons and assessment. All students are given extensive work on grade-level problems, even students who are struggling, and this work progresses mathematically. However, future grade-level content is not consistently identified. These instructional materials are visibly shaped by the cluster headings in the standards; for example, one session is called "Classify Objects Into Categories." Connections are made between domains and clusters within the grade level. For instance, materials make connections between counting and cardinality and measurement and data. Overall, the Kindergarten materials support coherence and are consistent with the progressions in 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 Kindergarten meet expectations that supporting content enhances focus and coherence by engaging students in the major work of the grade.

Supporting standard K.MD.1 is connected to K.CC and K.OA — major work of the grade — throughout the instructional materials. For example, in Unit 4, Module 3, Session 1, standard K.MD.1 supports major work of K.CC.6 by measuring and comparing lengths and then correlating the measurement into a number unit and comparing the quantities.

Supporting standard K.MD.2 is connected to the major work of K.CC.6 throughout the instructional materials. For example, in Unit 4, Module 3, Session 1 standard K.MD.2 supports major work of K.CC.6 by measuring and comparing lengths and then correlating the measurement into a number unit and comparing the quantities.

Supporting standard K.G.2 is connected to K.CC and K.OA, major work of the grade, throughout the instructional materials. For example, in Unit 5, Module 2, Sessions 1-3, after sorting, students are asked to count the number of shape cards in each category and then compare the quantities to find which group has the most or least, and combining amounts. This supports K.CC and cluster K.OA. Another example is found in Unit 6, Module 1, Session 2. In this session work in three-dimensional geometry is used as a vehicle to represent and solve addition situations. This supports K.OA. Also, in Unit 6, Module 2, Session 5, students use three-dimensional shapes as a vehicle to practice decomposing numbers and fluently adding within 5 (K.OA).

Supporting standard K.MD.3 is connected to K.CC and K.OA, major work of the grade, throughout the instructional materials. For example, in Unit 5, Module 2, Sessions 1-3, while sorting shape cards in various ways, students are asked to count the number in each category and then compare the quantities to find which group has the most or least, and in some sessions combining amounts. This supports standards in K.CC and K.OA. Also, in the March and May Calendar Collector, students examine data and connect it to clusters K.CC and K.OA through questions about counting, comparing and combining.

##### 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 Kindergarten meet the expectations for this indicator by providing a viable level of content for one school year. Overall, the materials have expectations for teachers and students that are reasonable.

• Materials provide for 160 days of instruction. Each Unit has 20 sessions = 20 days. There are 8 Units. (20x8=160)
• The prescribed daily instruction includes both unit session instruction and a Number Corners session. (170 days). There are no additional days built in for re-teaching.
• Assessments are incorporated into sessions and do not require an additional amount of time. Instead, they are embedded into module sessions one-on-one as a formative assessment.
• The Number Corner Assessments/Checkups (a total of 10 assessments, 1 interview and 1 written, in each of the following months: September, October, January, March and May) would require additional time to conduct a 7-10 minute interview with each student.
• A Comprehensive Growth Assessment is completed at the end of the year and will require additional number of days to administer.
• There are no additional time/days built in for additional Support, Intervention or Enrichment in the pacing guide. The Publisher recommends re-teaching of strategies, facts and skills take place in small groups while the rest of the class is at Work Places (math stations) or doing some other independent task. There is a concern that if a particular session’s activities take up most of the 60 minutes allotted, there will be no time for the remediation and enrichment to take place.
• Based on the Bridges Publisher Orientation Video and Guide provided to the reviewers, unit sessions are approximately 60 minutes of each instructional day.
• Each unit session contains: Problems & Investigations (whole group), Work Places (math stations), Assessments (not found in each session), and Home Connections (homework assignments not found in each session).
• Based on the introduction section in the Number Corners Teacher Guide, as well as the Bridges Publisher Orientation Video, Number Corners sessions are approximately 20 to 25 minutes of each instructional day.
• Approximately 80-85 minutes is spent on the Bridges and Number Corner activities daily.
##### 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 Kindergarten are partially consistent with the progressions in the standards. Although students are given extensive grade-level problems and connections to future work are made, future grade level content is not always clearly identified to the teacher or student.

At times, the session materials do not concentrate on the mathematics of the grade. Some of the sessions within each module focus on above grade-level concepts. Examples of this include addition and subtraction beyond 10, counting quantities beyond 20, the use of greater than and less than symbols, patterns, volume, and identifying and counting money amounts using coins. The inclusion of off-grade level concepts takes away from the number of sessions that could be spent more fully developing the work on the mathematics of the grade.

In some cases, the above-grade level content is identified as such by the publishers, and in other cases it is not. On the first page of every session, the Skills & Concepts are listed along with the standard to which it has been aligned by the publisher. In some cases, this alerts the user to the inclusion of off-grade level concepts. Examples include:

• Unit 5, Module 1, Exploring Shapes Overview, page 1, the publisher describes the work in Module 1 as “extending the range of their counting and comparing skills” which somewhat signifies that students will be moving beyond the expected range of Kindergarten standards.
• Unit 5, Module 1, Session 3 warm-ups include counting backward from 20 which is not a Kindergarten standard, but the publisher alerts teachers to this by aligning it to “supports K.CC.”
• Unit 5, Module 4, Session 5- students make a quilt following an AB pattern which publisher identifies as “create and extend simple repetitive patterns with up to 3 elements. The publisher alerts teachers by aligning it to “supports K.OA.”
• In the Unit 6 Introduction, page ii, the publishers state that “a mastery of the forward and backward counting sequences, one-to-one correspondence, and cardinality helps students correctly determine sums and differences as they begin to solve addition and subtraction tasks,” thereby explaining an inclusion of counting backwards throughout the unit even though it is not a Kindergarten expectation.

In other cases, the above-grade level concepts are not identified as such within the sessions in the "Skills and Concepts" listing or at the beginning of the Units in the "Skills Across the Grade Levels" sections. Examples of unidentified above-grade level content include:

• Unit 3, Module 1, Session 2 and Session : Counting by 2’s is a skip counting strategy/skill that is not introduced until Grade 2 (2.NBT.2).
• Unit 4, Module 3 Sessions 2 -5: These focus on above-grade level content using standard units of measurement. The lessons are worded with the language of the standard for measurement in first grade (1.MD.1).
• Unit 4, Module 4, Session 2 and Session 5: Both sessions involve counting by 5’s, which is a skip counting strategy/skill that is not introduced until Grade 2 (2.NBT.2).
• Unit 5, Module 1, Session 5: This focuses on addition to 20, number combinations, and comparing numbers. This goes beyond K.OA expectations of adding to 10 as it continues on to 20, asking questions such as “how many more to 20?”
• Unit 6, Module 1, Sessions 3 and 4: The comparison of the cylinders as to which one holds more is a volume activity which is more appropriate for Grade 4.
• Unit 6, Module 2: Activities involve three-dimensional shapes that go beyond the shape expectations as outlined in the K.G cluster. The activities in this module include drawing and identifying rectangular prisms, triangular prisms and pyramids.
• Unit 6, Module 3, Session 4: Money is a Grade 2 standard, but is not specifically identified as such by the publisher. In this session, students are determining the total value of a collection of coins.
• Unit, 8, Module 4, Sessions 4 and 5: Students use repetitive patterns while completing a double Irish chain frog quilt. This moves beyond the grade-level work.

Materials provide students opportunities to work with grade-level problems. The majority of differentiation/support provided is on grade-level. Extension activities are embedded within sessions and allow students to engage more deeply with grade-level work. Additional extension activities are also provided online.

##### 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 Kindergarten meet the expectations for fostering coherence through connections at a single grade, where appropriate and when the standards require. The standards are referred to throughout the materials. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings and include problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.

Instructional materials shaped by cluster headings include the following examples:

• Unit 5, Module 2, Session 3, "Sorting Shapes by Sides, and Corners," is shaped by K.G.B.
• Unit 6, Module 3, "Exploring the Teen Numbers," is shaped by the K.NBT cluster heading.
• Unit 7, Module 1, Session 1, “Compare Weights,” is shaped by the K.MD.A cluster heading.
• Unit 8, Module 1, “Catching, Counting, and Comparing,” is shaped by the K.OA.A cluster heading.
• The Unit 5, Module 1, Session 1 learning objectives include "Classify objects into categories," which is visibly shaped by K.MD.B.
• The Unit 5, Module 3, Session 1 learning objectives include "Classify objects into given categories" and "Count the number of objects in each category," which are visibly shaped by K.MD.B cluster heading.

Units, modules, and sessions that connect two or more clusters in a domain or two or more domains include the following examples:

• Unit 1 Module 1, Session 1: "One Shoe" connects cluster K.CC.A to K.CC.B as students are counting the number of shoes up to 10, saying the number in the standard form and pairing each shoe with only one number name.
• Unit 1 Module 1, Session 2: "Two Shoes" connects clusters K.CC.A and K.CC.B to K.CC.C as students count the number of shoes to by ones, say the number in the standard form, pair each shoe with only one number name, and finally compare which group is greater than, less than, or equal.
• Unit 1 Module 1, Session 3: "Five Shoes" connects clusters K.CC.A and K.CC.B to K.CC.C as students count the number of shoes, saying the numbers in standard order and pairing each shoe with only one number name, and identifying whether the number of shoes in one group is greater than, less than, or equal to the number of shoes in the other group.
• Unit 1 Module 2, Session 1: "Shoes to Toes" connects cluster K.CC.B with K.OA.A as students are counting dots on five-frame cards, and then decomposing the sums in more than one way.
• Unit 1 Module 2, Session 2: "Fabulous Fives" connects cluster K.CC.B with K.OA.A as students are counting dots on five-frame cards, and then decomposing the sums in more than one way with Unifix cubes and five-frame cards.
• Unit 1 Module 2, Session 3: "Fives with Fingers" connects cluster K.CC.B with K.OA.A as students are counting dots on five-frame cards, and then decomposing the sums in more than one way with fingers.
• Unit 1 Module 2, Session 4: "Numerals 1 to 5" connects K.CC.A and K.CC.B with K.OA.A and K.MD.3 as students shake two-colored beans, count each color, find the sum and record numeral on sheet.
• Unit 1 Module 2, Session 5: "Filling Five-Frames" connects K.CC.B with K.OA.A as students are flashed five-frames they show the number of dots on one hand and the number of blank spaces on the other.
• Unit 1 Module 3, Session 4: "Beat You to Five" connects K.CC.A and K.CC.B with K.OA.A as students play the game of spinning a number and working in teams to see if they can cover five cubes on their frames before the teacher's side is covered. Students determine which group is greater than, less than, or equal to the number on the other side of the five-frame card.
• Unit 2 Module 1, Session 1: "Two Red, Three Blue" connects clusters K.CC.B to K.OA.A as students count, compare and answer how many more as they are looking at 5-frame cards.
• Unit 2 Module 1, Session 2: "Funny Five-Frame Flash" connects clusters K.CC.B to K.OA.A as students compare irregular 5-frame cards with regular 5-frame cards.
• Activities in Unit 5, Modules 1 and 2 connect K.G.4 with K.MD.3 when sorting pattern blocks, shapes and shape cards and then counting and recording the number of each.
• Activities in Unit 5, Module 3 connect K.G to K.CC.6 and K.MD.3 when sorting and recording the number in each category and then compare the quantities to find which is greater.
• Activities in Unit 6, Module 1 connect K.G to K.MD.3, K.CC.3, and K.CC.6 and K.CC.7 when sorting three-dimensional shapes into categories and counting and recording the number in each category and then comparing the numbers and quantities.
• Activities in Unit 6 Module 2 connect K.G to K.CC and K.OA.3 by playing a game “Make it Five” and recording combinations of five shapes.
• In Unit 3, Module 3, Session 3, students are counting objects (K.CC.2), comparing amounts (K.CC.6), describing the attributes of the objects (K.MD.1), and comparing more and less of the objects (K.MD.2).
• In Unit 4, Module 3, Session1 students work on counting and cardinality domain and all three clusters while comparing measurable attributes from the measurement and data domain.
• Activities in Unit 8, Module 1 connect K.CC.A to K.OA.A when counting objects and writing equations based on information from story problems.

### Rigor & Mathematical Practices

The materials reviewed for Kindergarten meet the expectations for Gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency and application. These three aspects are balanced within the lessons. The materials partially meet the expectations for the connections between the MP and the mathematical content. There are missed opportunities for attending to the full meaning of the MPs. More teacher guidance about how to support students in analyzing the arguments of others is needed.

##### Gateway 2
Meets Expectations

#### 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.

The materials reviewed for Kindergarten meet the expectations for this criterion by providing a balance of all three aspects of rigor throughout the lessons. To build conceptual understanding, the instructional materials include concrete materials, visual models, and open-ended questions. In the instructional materials students have many opportunities to build fluency with adding and subtracting within five. Application problems occur throughout the materials. The three aspects are balanced within the instructional materials.

##### 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.

The materials reviewed in Kindergarten for this indicator meet the expectations by attending to conceptual understanding within the instructional materials.

The instructional materials often develop a deeper understanding of clusters and standards by requiring students to use concrete materials and multiple visual models that correspond to the connections made between mathematical representations. The materials encourage students to communicate and support understanding through open-ended questions that require evidence to show their thinking and reasoning.

The following are examples of attention to conceptual understanding of K.CC.B:

• Unit 1, Module 1, Session 2: addresses and supports the developing understanding of cardinality and the conceptual understanding of K.CC.4 and K.CC.6 by sorting shoes in two lines then counting to identify which group is larger. The investigation uses concrete visual and verbal cues; there is correspondence across the mathematical representations as students are using verbal descriptions, concrete (actual shoes lined up on two different lines), and written value of each line.
• Unit 2, Module 1, Session 3: students reinforce their conceptual understanding of one-to-one correspondence (K.CC.4.A) as they are counting the number of boxes in the 10-frame and/or counting dots arranged on a ten-frame (K.CC.5). Students then use Unifix cubes to build a concrete representation of the 10-frame card, connecting the visual, verbal, and concrete representations.

The following are examples of attention to conceptual understanding of K.CC.6:

• Unit 2, Module 1, Sessions 4 and 5: conceptual understanding of comparing numbers is developed with a ten-frame. Strategies for determining which number has more or less are shared through discussion. In Session 5, the students play the game independently as the teacher observes and documents how students determine value of greater and less than.
• October Number Corner Calendar Collector: conceptual understanding is built with cubes and ten-frame representations. Discussion elicits evidence for which number is greater/less/equal by using multiple representations, including a simple array for comparison.

The following are examples of attention to conceptual understanding of K.OA.1:

• Unit 3, Module 2, Session 2: students develop conceptual understanding of addition and subtraction by acting out situations, using Unifix cubes, giving verbal explanations, and reading equations.
• Unit 6, Module 3, Session 3: students play the Work Place 6D Roll, Add & Compare game, roll 0-5 dice, build quantities to 10 with Unifix cubes, record the addition facts on a recording sheet, and then compare their total amount to their partners by snapping all their cubes together. Students are asked to justify their answer to who has more. Students connect the mathematical representations of dice, Unifix cubes/10-frames, written equations and Unifix trains to validate their comparison of who has more.
• April Number Corner Calendar Collector: writing addition equations is represented through direct modeling of frogs and represented as unit squares. Conceptual understanding of the addition equation sequence can be determined in multiple ways (example: 2 +1 + 1 + 1 is the same as 2 + 3).

The following are examples of attention to conceptual understanding of K.OA.3:

• Unit 1, Module 3, Session 1: students move from the 5-frame to the 10-frame in Terrific Tens. The 10-frame model helps develop students' understanding of part-part-whole relationship of 10 (K.OA.3). As students explore the 10-frame, they use their fingers to show the amount on various 10-frames.
• Unit 8, Module 4, Session 1: students compose and decompose numbers less than or equal to 10 and explore how they might see equations in the ten-frame. Students record their way of seeing various quantities within 10: 5 = 4 + 1, 2 + 3, etc. Students are then asked to think about what subtraction equation they can write or the same 10-frames: 5 - 1 = 4, 5 - 3 = 2. Students are asked to "show where they see the equation on the 10-frame" and "who has a different equation?"

The following are examples of attention to conceptual understanding of K.NBT.1:

• Unit 7 Module 2 Session 1 and Session 2: conceptual understanding of teen numbers is elicited from building numbers on a double 10-frame to see the unit of 10 as a whole with some more (10 and 3 is 13). Number line representations are also used to guide the counting sequence of more than 10.
• Unit 8, Module 3, Session 1: students develop conceptual understanding using place value mats of ones/tens to build numbers in the 10-20 range in Place Value Build and Win. They build the quantity with cubes on the mat, compare the numbers, and write inequality statements using the greater than and less than symbols. Cubes are pre-grouped into trains of 10. As students build the numbers, they are asked to explain how they used their cubes to build the number. Emphasis is placed on the "10 and some more" concept.
• February Number Corner Number Line: conceptual understanding of “ten and some more” is reinforced through multiple and concrete representations (double 10-frame and a manipulative number line). Connections are made between the concrete visual representation of the teen number and the written numeral representation.

The following are examples of attention to conceptual understanding of K.G:

• Unit 2, Module 4, Session 3, Pattern Block Puzzles: students observe and explore pattern blocks, identify the shape using characteristics and correct mathematical name (K.G.2), describe the positions (above, below, beside) (K.G.1) and develop understanding that shapes are the same regardless of orientation or size. Students also compose simple shapes to form larger shapes (K.G.6), as they cover various shapes with smaller pattern blocks. There is correspondence across mathematical representations as student give verbal descriptions of a shape's characteristics, practice using the correct word for the shape, use concrete pattern blocks to compose a larger shape. Conceptual discussions with high level questions occur (students quietly observe the shapes of various pattern blocks and then are asked, "Can you tell me about these shapes?") Students pair-share ways to build designs, have the opportunity to build, and then are invited to share design and finally asked, "Can you show me a different way to cover the shapes?"
##### 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.

The Kindergarten materials meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.

• Throughout the materials, computational fluency is elicited with both addition and subtraction equations. Students are able to use counters, 10-frames or drawings to assist them. Evidence is gathered to note if a student has moved to procedural fluency and no longer needs concrete materials to add and subtract within five or within ten by using a unit of five. The expectation within the last two units is that students will be able to decompose five into parts fluently without the support of concrete materials to show procedural understanding.
• Students spend a significant amount of time and have a variety of opportunities to fluently add and subtract throughout number corners activities. K.OA.5 is addressed in two areas of Number Corner. Although the publisher does not list the K.OA.5 standard in any of their Computational Fluency workouts, instead most often listing K.OA.4 in relation to adding and subtracting within five, Computational Fluency workouts use finger patterns, 5-frames, and the number line to help students develop fluency with addition and subtraction facts to the number five. Calendar Collector workouts have students collecting various items to count throughout the month.
• In the March Calendar procedural fluency is guided by using subitizing images to state how many more to make a unit of 10. This builds from the conceptual understanding within an organized structure to see the parts of ten fluently without having to count (perceptual subitizing).
• In the May Computational Fluency workout fact fluency to 5 is investigated by using multiple representations (number cards, 10-frames). Routines focus on looking at decomposing 5 into 2 or even 3 addends to build number flexibility.
• Fluency is developed throughout the sessions of the Kindergarten instructional materials.
• In Unit 1, Module 2, Session 3 in “Fives with Fingers,” frames are flashed and students show number of dots with fingers of one hand and use their other hand to show how many empty boxes there are in the 5-frame and then add to find the total in all.
• In the Unit 6, Module 4, Session 1 Work Place “Shake Those Beans,” students count how many red and how many white beans and how many in all to determine all combinations of five.
• In Unit 7, Module 3, Session 5 in “Cubes in My Hand,” the teacher divides five cubes between her two hands. The teacher opens one hand to reveal cubes while keeping the other cubes hidden in other hand. Students determine how many cubes are hiding and then write the equations that represent the investigation.
• In Unit 8, Module 1 students fluently subtract with minuends to 5 by using spinners and drawings to represent minuends and subtrahends.
##### 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

Materials meet the expectations for having engaging applications of mathematics as they 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.

Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide single step contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. The majority of the application problems are done with guiding questions elicited from the teacher through whole group discussions that build conceptual understanding and show multiple representations of strategies. Materials could be supplemented to allow students more independent practice for application and real world contextual problems that are not teacher guided within discussions. This would provide students opportunities to show more evidence of their mathematical reasoning through common addition and subtraction situations as outlined in the CCSSM Glossary, Table 1.

The instructional materials include problems and activities aligned to K.OA.2 that provide multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. Examples of these applications include the following:

• In Unit 3, Module 3, Session 2, "Bicycle story problems," students are using their 10-frames to solve story problems given to them orally.
• In Unit 6, Module 4, students engage in application of addition skills to solve story problems.
• In Unit 7, Module 3, Sessions 1, 2 and 3, students solve frog addition/subtraction word problems using pictures. Students share out their strategies for solving. Students use Unifix cubes to model story problems and solve.
• In Unit 8, Module 1, Sessions 1-4, students use manipulatives, pictures, and 10-frame counting mats to demonstrate application of addition and subtraction skills for solving story problems.
• This module contains story problems set in the context of addition and subtraction. Student strategies are shared to elicit more sophisticated strategies over time within the unit. The unit also contains a checkpoint small group formative assessment to gather data to evaluate student strategies and misconceptions.
• In the Number Corner February Computational Fluency, students began to add to 10 in the context of themed story problems and application within the number corner computational fluency routine. Thinking within these contextual situations is extended toward building conceptual understanding of subtraction as a missing addend problem.
##### 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.

The materials reviewed in Kindergarten meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.

In the Kindergarten materials all there aspects of rigor are present in the instructional materials. All three aspects of rigor are used both in combination and individually throughout the Unit Sessions and in Number Corner activities. Application problems are seen to utilize procedural skills and require fluency of numbers. Conceptual understanding is enhanced through application of previously explored clusters. Procedural skills and fluency learned in early units are applied in later concepts to improve understanding and conceptual understanding.

#### Criterion 2.2: Math Practices

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

The materials reviewed for Kindergarten partially meet this criterion. The MPs are often identified and often used to enrich mathematics content. There are, however, several sessions that are aligned to MPs with no alignment to Standards of Mathematical Content. The materials often attend to the full meaning of each practice. However, there are instances where the standards are superficially attended to. The materials reviewed for Kindergarten attend to the standards' emphasis on mathematical reasoning. Students are prompted to explain their thinking, listen to and verify the thinking of others, and justify their own reasoning. Although the materials often assist teachers in engaging students in constructing viable arguments, more guidance about how to guide students in analyzing the arguments of others is needed.

##### Indicator {{'2e' | indicatorName}}
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Kindergarten meet the expectations for identifying the MPs and using them to enrich the mathematical content. Although a few entire sessions are aligned to MPs without alignment to grade-level standards, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.

The Kindergarten Assessment Guide provides teachers with a Math Practices Observation Chart to record notes about students' use of MPs during Sessions. The Chart is broken down into four categories: Habits of Mind, Reasoning and Explaining, Modeling and Using Tools, and Seeing Structure and Generalizing. The publishers also provide a detailed, "What Do the Math Practices Look Like in Kindergarten?" guide for teachers (AG, page 16).

Each Session clearly identifies the MPs used in the Skills & Concept section of the Session. Some Sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher that shows the connection between the indicated MP and the content standard. Examples of the MPs in the instructional materials include the following:

• In Unit 1, Module 3 each of the six sessions list the same two Math Practices: MP6 and MP7. There is a "Math Practices In Action" reference in two of the six Sessions.
• In Unit 2, Module 3 in the Skills & Concepts section, four sessions (1, 2, 3, 6) list MP6, four sessions (1, 2, 5, 6) list MP7, three sessions list MP8 (3, 4, 5) and one session lists MP3 (4).
• In Unit 2, Module 3, Sessions 1, 4 and 6 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
• In Unit 4, Module 2 all five sessions list in the Skills & Concepts section two MPs: MP6 and MP7.
• In Unit 7, Module 1, sessions 2 and 5 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
• In the September Number Corner MP2 is referenced in the Calendar Collector; MP4 is referenced in Days in School; MP7 is addressed in Calendar Grid, Computational Fluency, and Number Line; and MP8 is addressed in Calendar Grid, Computational Fluency, Number Line, Days in School and Calendar Collector.

Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These lessons occur at the beginning and the end of the year. These sessions that focus entirely on MPs include the following:

• Unit 1, Module 4, Session 1
• Unit 1, Module 4, Session 2
• Unit 1, Module 4, Session 3
• Unit 1, Module 4, Session 4
• Unit 8, Module 4, Session 4
• Unit 8, Module 4, Session 5
##### Indicator {{'2f' | indicatorName}}
Materials carefully attend to the full meaning of each practice standard

The materials partially meet the expectations for attending to the full meaning of each practice standard. Although the instructional materials often attend to the full meaning of each practice standard, there are instances where the MPs are only attended to superficially. There is limited discussion or practice standards within Sessions, Number Corner, and Assessments.

Each Session clearly identifies the MPs used in the Skills & Concept section of the Session. Typically there are two MPs listed for each session, so there is not an overabundance of identification. Some Sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. Although the MPs are listed at the session level, they are not discussed or listed in unit overviews or introductions (Major Skills/Concepts Addressed); however, they are listed in Section 3 of the Assessment Overview. With limited reference in these sections, overarching connections were not explicitly addressed.

In Number Corners, the MPs are listed in the Introduction in the Target Skills section with specific reference to which area of Number Corner in which the MP is addressed (Calendar Grid, Calendar Collector, Days in School, Computational Fluency, Number Line). The MP are also listed in the Assessment section of the Introduction as well. Although the MPs are listed in these sections, there is no further reference to or discussion of them within Number Corner.

At times, the instructional materials fully attend to a specific MP. The following are examples:

• In Unit 1, Module 3, Session 2, the Skills & Concepts section lists MP6 and MP7. The session also references the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action." This section states that "(w)hen students pair the numerals and quantities and then arrange them in order, they are looking for and making use of structure..." Students were provided with several opportunities to "communicate precisely to others" their counting strategies as they are asked to explain how they counted their 10-frame cards, asked to explain other ways to count, and asked if there an easy way to count the dots? Students paired numerals with 10 frame cards and then arranged them in order; they are using the structure of the 10-frame cards to recognize patterns and describe the structure through repeated reasoning.
• In Unit 1, Module 3, Session 4, the Skills & Concepts section lists MP6 and MP7. The session also references the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action." This sections states that "(w)hen you help young students keep track of their counting, you are helping them attend to precision..." Students are playing the game, "Beat you to Five." They are counting accurately attending to precision using strategies so that they include each object once without losing track. As students spin, they are using the Unifix cubes to show the number needed to reach five.
• Unit 4, Module 2, Session 5 attends to MP7. In the "Beat You to Twenty" Work Place, grouping the cubes by 10 and having students count on from 10 helps them recognize the structure of our number system.

At times, the instructional materials only attend superficially to MPs. The following are examples:

• Standard MP3 is addressed in Unit 2, Module 3, Session 4. Students play the game, "Which Bug Will Win?" by spinning a spinner with two different bugs. Students mark an "x" on the column according to which bug the spinner landed on. The first student to fill a column wins. Students are asked, "Who won?" and "Why?" This session does not attend to the full meaning of constructing mathematical arguments and/or critiquing the reasoning of others.
• MP8 is addressed in Unit 2, Module 3, Session 3. As students play the game, "Which Bug Will Win?," they make predictions about the two different spinners (one has an equal amount of two different bugs and the other spinner has four of one bug and two of the other). Students then play the game multiple times using the two different spinners and then adjust their predictions based on their outcomes. Students are looking for regularities as they spin multiple times during the game. There is a missed opportunity to revisit students' predictions during the final discussion. Students could identify the differences in the spinners and then describe why one spinner results in a different outcome than the other spinner.
• The materials partially attend to the meaning of MP4. The intent of this practice standard is to apply mathematics to contextual situations in which the math arises in everyday life. Often when MP4 is labeled within the instructional materials students are simply selecting a model to represent a situation. For example, in Unit 3, Module 1, Session 1, MP 4 is indicated, but students are simply representing a number on a ten frame. The Math Practices in Action note states that "Students will use drawings, numbers, expressions, and equations to model with mathematics."
##### 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.

The materials reviewed for Kindergarten meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

Students are asked to explain their thinking, listen to and verify other's thinking, and justify their reasoning. This is done in interviews, whole group teacher lead conversations, and in student pairs. For the most part, MP3 is addressed in classroom activities and not in Home Connection activities.

• In Unit 2, Module 1, Session 2, within the Problems and Investigations portion, students are introduced to the "think-pair-share" routine. They are asked to listen and explain their partners' thinking.
• In Unit 5, Module 4, Session 2 students are introduced to "There's a Shape in My Pocket." In this activity, students present arguments and critique the reasoning of their classmates to come to an agreement about which cards to remove.
• In Unit 7, Module 4, Session 1 students engage in a "think-pair-share" routine. As in other Sessions in the instructional materials, this activity allows students to share their thoughts, listen to the thoughts of classmates, and justify their own reasoning.
• In the March Calendar Grid and Number Line students share their thinking and justify their reasoning in developing their combinations of numbers to construct a ten.
##### 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.

The instructional materials reviewed for Kindergarten 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. Although the instructional materials often assist teachers in engaging students in constructing viable arguments, there is minimal assistance to teachers in how to guide their students in analyzing the arguments of others.

There are Sessions containing the "Math Practice In Action" sidebars that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. A few of the sessions contain direction to the teacher for prompts and sample questions and problems to pose to students.

Many lessons give examples of teacher/student discourse by providing teachers a snapshot of what questions could be used to generate conjectures and possible student thinking samples. The following are examples of sample discourse:

• Unit 4, Module 1, Session 2
• Unit 7, Module 2, Session 5
• Unit 8, Module 1, Session 5

Although teachers are provided guidance to help students construct arguments and students are provided many opportunities to share their arguments, more guidance is need to support teachers in guiding their students through the analysis of arguments once they are shared. For example, in Unit 5, Module 4, Session 2, students are asked to think-pair-share about their observations of a Shape Card pocket chart. Students are invited to report to the group what they heard their partner say. Students continue to engage as they turn and talk to their partner about what problem they are trying to solve and asking questions about the shape cards, and finally coming to an agreement about which cards should be removed. When students come to an agreement about which cards to remove, they are presenting arguments and critiquing the reasoning of their classmates, engaging in logical reasoning. Although this activity allows students to analyze the arguments of classmates, the teacher is not provided enough support to help students with this analysis.

##### Indicator {{'2g.iii' | indicatorName}}
Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Kindergarten meet the expectations for explicitly attending 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 is present throughout the materials.

The instructional materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers and arguments in small group, whole group teacher directed, and teacher one-to-one settings.

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples of this include using geometry terminology such as rhombus, hexagon, and trapezoid and using operations and algebraic thinking terminology such as equation, difference, and ten-frame.

• Many sessions include a list of mathematical vocabulary that will be utilized by students in the session.
• The online Teacher Materials component of Bridges provides teachers with "Word Resources Cards" which are also included in the Number Corner Kit. The Word Resources Cards document includes directions to teachers regarding the use of the mathematics word cards. This includes research and suggestions on how to place the cards in the room. There is also a "Developing Understanding of Mathematics Terminology" included within this document which provides guidance on the following: providing time for students to solve problems and ask students to communicate verbally about how they solved, modeling how students can express their ideas using mathematically precise language, providing adequate explanation of words and symbols in context, and using graphic organizers to illustrate relationships among vocabulary words
• At the beginning of each section of Number Corner, teachers are provided with "Vocabulary Lists" which lists the vocabulary words for each section.
• In Unit 6, Module 1, Section 5, in the Problems & Investigation section, the teacher is reminded to use the vocabulary for three-dimensional shapes, such as edge, face, vertex, surface.

### Usability

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Use & Design

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

Materials are well-designed, and lessons are intentionally sequenced. Typically students learn new mathematics in the Problems & Investigations portion of Sessions while they apply the mathematics and work towards mastery during the Work Station portion of Sessions and during Number Corner. Students produce a variety of types of answers including both verbal and written answers. Manipulatives such as 10-frames, craft sticks and tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

##### 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.

The sessions within the units distinguish the problems and exercises clearly. In general, students are learning new mathematics in the Problems & Investigations portion of each session. Students are provided the opportunity to apply the math and work toward mastery during the Work Station portion of the session as well as in daily Number Corners.

For example, in Unit 2, Module 2 of Session 4, students are learning the new mathematics and in Session 5, students are applying that learning in the Work Station. In the Problem & Investigations, students are learning the new mathematics concept of identifying if the number of objects in one group is greater than, less than, or equal to the number of objects in another group. They initially observe the Count and Compare game board and discuss the meaning of the words "greater than," "less than," and "equal to." Students are shown and then demonstrate hand movements that represent greater than, less than, and equal to. Students and teacher each choose a 10-frame dot card and share strategies for determining the amount. Students then spin the greater than/less than spinner to determine who wins the two cards. In another Problems & Investigations, students get another opportunity to play the game Count and Compare Dots. The teacher observes students at play, checking for understanding of the greater than/less than concept as well as the directions of the game, and clarifies any questions. In the Work Place, students engage in the game Count and Compare Dots where they apply their understanding of identifying if the number of objects in one group is greater than, less than, or equal to the number of objects in another group (K.CC.6).

In the October Calendar Collector, students are given another opportunity to apply their understanding of identifying if the number of objects in one group is greater than, less than, or equal to the number of objects in another group. Students are observing the weekly Pattern Blocks Data Collection Graph. Students use the Word Resource Cards in the pocket chart; the cards show greater than, less than, most, least, and equal. Each card contains a base ten model that represents the word on the card. Students share their observations of the graph and are encouraged to use the mathematical terms on the Word Resource Cards.

##### Indicator {{'3b' | indicatorName}}
Design of assignments is not haphazard: exercises are given in intentional sequences.

The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is continuously reviewed, practiced and extended throughout the year.

The "Skills Across Grade Level" table is present at the beginning of each Unit. This table shows the major skills and concepts addressed in the Unit. The table also provides information about how these skills are addressed elsewhere in the Grade, including Number Corner, and in the grade that follows. Finally, the table indicates if the skill is introduced (I), developed (D), expected to be mastered (M), or reviewed, practiced or extended to higher levels (R/E).

For example, K.CC.6 is found in Units 1, 2, 3, 4, 5, 6, 7, 8 and in Number Corner in October, December, January, February, March, April, and May. In Unit 1 this standard is introduced. In Units 2, 3, 4 and 5 it is developed, and in Unit 6 the standard is mastered. The standard is again Reviewed/Practiced/Extended in Units 7 and 8. Another example is K.CC.4.B found in Units 1, 2, 3, 4, 6 and in all Number Corners. In Unit 1 this standard is introduced. It is developed in Units 2 and 3, and it is mastered in Unit 4. The standard is once again Reviewed/Practiced/Extended in Unit 6.

Concepts are developed and investigated in daily lessons and are reinforced through independent and guided activities in Work Places. Number Corner, which incorporates the same daily routines each month (not all on the same day) has a spiraling component that reinforces and builds on previous learning. Assignments, both in class and for homework, directly correlate to the lesson being investigated within the unit.

The sequence of the assignments is placed in an intentional manner. First, students complete tasks whole group in a teacher directed setting. Then students are given opportunities to share their strategies used in the tasks completed in the Problems & Investigations. The Work Place activities are done in small groups or partners to complete tasks that are based on the problems done as a whole group in the Problems & Investigations. The students then are given tasks that build on the session skills learned for the home connections. For example, in Unit 7, Module 2, Session 1 the focus in the Problems & Investigations is using double 10-frames to identify numbers between 10 and 20 with sight and equations. Then, students use a number line to determine how far from 20 the number is so they can determine the winner. Then, in the Work Place, students are given the same tasks of identifying numbers between 10 and 20 with sight and equations and determining, on a number line, how far from 20 the number line is with partners.

##### 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.

There is variety in what students are asked to produce. Throughout the grade, students are asked to respond and produce in various manners. Often, working with concrete and moving to more abstract models as well as verbally explaining their strategies. Students are asked to produce written evidence using drawings, representations of tools or equations along with a verbal explanation to defend and make their thinking visible.

For example, in Unit 2, Module 2, Session 5 in the Problems & Investigation section of the lesson, students are working with three different models to show combinations of 5: 5-frames, finger patterns, and number racks. First, students are flashed 5-frame cards and asked to show the number of red dots with their fingers on one hand and the number of blue dots with their fingers on their other hand. Students are asked to determine the total number of dots. Various 5-frame cards are flashed as students are working to support their development of cardinality. Next, students transition to the number racks, moving their beads to represent the 5-frame cards and are guided to verbally explain the process: "I pushed 3 red beads and then I added 2 white beads. Now I have 5 beads in all." Students continue to represent the amounts on various five-frame cards and turn and talk to their partners to describe what they did using the sentence frame. The lesson is wrapped up with students using the think-pair-share routine to discuss the various ways they built combinations of five.

##### Indicator {{'3d' | indicatorName}}
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

Manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Manipulatives are used and provided to represent mathematical representations and provide opportunities to build conceptual understanding. Some examples are the 10-frames, number lines, Unifix cubes, number racks, coins, craft sticks and tiles. When appropriate, they are connected to written representations.

For example, in Unit 8, Module 2, Session 4, students are working with Unifix cubes to measure various items around the room. After recording their estimates and actual measurements, they write actual measurements in expanded form. Also, in the Number Corner February Number Line, students are playing a game called Roll & Count On From Ten. They roll a die to determine how many hops forward the frog will make on the number line. They connect the number line to an equation to represent the frog's hops forward. Another example is Unit 4, Module 2, Session 4. Students take turns rolling the numbered 0-5 die and covering the indicated number of pictured cubes with Unifix cubes. Students work together to see if they can be the first to collect 20 Unifix cubes on their side of the game board.

Also, in Unit 7, Module 4, Session 1, students use double 10-frames and craft stick bundles to demonstrate counting by 10's and 1's and organizing the counting process. The 10-frame numbers are compared to the task of bundling the sticks into groups of 10's and 1's.

##### 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.

The material is not distracting and does support the students in engaging thoughtfully with the mathematical concepts presented. The visual design of the materials is organized and enables students to make sense of the task at hand. The font, size of print, amount of written directions and language used on student pages is appropriate for kindergarten. The visual design is used to enhance the math problems and skills demonstrated on each page. The pictures match the concepts addressed such as having the characters that are in the story problems placed in picture format on the page as well. Some problems may even require students to use the pictures to solve the story problems.

For example, in the Number Corner March Calendar Grid, the grid is visually appealing, with easy to read/interpret diagrams of a sequence of numbers that represent the specific day of the month. There is also a corresponding 20-frame that represents the number as well. Kindergarteners would easily be able to see that the diagrams and 20-frame representations are increasing by one each day.

#### Criterion 3.2: Teacher Planning

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

The instructional materials support teacher learning and understanding of the standards. The instructional materials provide questions and discourse that support teachers in providing quality instruction. The teacher's edition is easy to use and consistently organized and annotated. The teacher's edition explains the mathematics in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.

##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Lessons provide teachers with guiding questions to elicit student understanding and opportunities for discourse to allow student thinking to be visible. Discussion questions provide a context for students to communicate generalizations, find patterns, and draw conclusions.

Each unit has a sessions page, which is the daily lesson plan. The materials have quality questions throughout most lessons. Most questions are open-ended and prompt students to higher level thinking.

In Unit 1, Module 1, Session 2, teachers are prompted to ask the following questions:

• What did you notice is the same about these two shoes?
• How are these two shoes alike?
• How do you know there are two?

In Unit 1, Module 4, Session 4, students are working with patterns, and teachers are prompted to ask the following questions:

• So, you're saying that these cards all show a pattern? How do you know?
• Is that true for all the cards?
• Can you show us what you mean with cubes?
• What should come next? How do you know?

In Unit 7, Module 4, Session 2, students are working to count dots on a double 10-frame, and teachers are prompted to ask the following questions:

• What do you think is the same and what is different about these cards?
• What else do you notice?
• What do you mean they look different?
• Can you tell me a bit more? How do they look different?

In Number Corners, there are are sidebars labeled "Key Questions" throughout the sections. For example, in the Number Corner December Calendar Grid, the"Key Questions" sidebar includes the following examples:

• Where do you think the teddy bear will be on the next marker? Why?
• Where is the bear on the 3rd marker?
• I see a teddy bear behind a box, which marker am I looking at?
• Can you use the patterns we've discovered to predict what the marker for the day after tomorrow will look like?
##### 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.

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; however, additional teacher guidance for the use of embedded technology to support and enhance student learning is needed.

There is ample support within the Bridges material to assist teachers in presenting the materials. Teacher editions provide directions and sample scripts to guide conversations. Annotations in the margins offer connections to the mathematics practices and additional information to build teacher understanding of the mathematical relevance of the lesson.

Each of the eight units also have an Introductory section that describes the mathematical content of the unit and includes charts for teacher planning. Teachers are given an overview of mathematical background, instructional sequence, and the ways that the materials relate to what the students have already learned and what they will learn in the future units and grade levels. There is a Unit Planner, Skills Across the Grade Levels Chart, Assessment Chart, Differentiation Chart, Module Planner, Materials Preparation Chart. Each unit has a Sessions page, which is the Daily Lesson Plan.

The Sessions contain:

• Sample Teacher/Student dialogue;
• Math Practices In Action icons as a sidebar within the sessions - These sidebars provide information on what MP is connected to the activity;
• A Literature Connection sidebar - These sidebars list suggested read-alouds that go with each session;
• ELL/Challenge/Support notations where applicable throughout the sessions; and
• A Vocabulary section within each session - This section contains vocabulary that is pertinent to the lesson and indicators showing which words have available vocabulary cards online.

Technology is referenced in the margin notes within lessons and suggests teachers go to the online resource. Although there are no embedded technology links within the lessons, there are technology resources available on the Bridges Online Resource page such as videos, whiteboard files, apps, blogs, and online resource links (virtual manipulatives, images, teacher tip articles, games, references). However, teacher guidance on how to incorporate these resource is lacking within the materials. It would be very beneficial if the technology links were embedded within each session, where applicable, instead of only in the online teacher resource. For instance, the teacher materials would be enhanced if a teacher could click on the embedded link, (if using the online teacher manual) and get to the Whiteboard flipchart and/or the virtual manipulatives.

##### 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.

Materials contain adult-level explanations of the mathematics concepts contained in each unit. The introduction to each unit provides the mathematical background for the unit concepts, the relevance of the models and representations within the unit, and teaching tips. When applicable to the unit content, the introduction will describe the algebra connection within the unit.

At the beginning of each Unit, the teacher's edition contains a "Mathematical Background" section. This includes the mathematics concepts addressed in the unit. For example, Unit 1 states, "This unit addresses three major concepts... First, students must master the number word sequence, that is they must be able to say the number words in the correct order... Students must also understand one-to-one correspondence, the idea that when counting to find the total number of objects in a collection, they must count each object once and only once... Finally, students must have a full grasp of cardinality, that is, that the last number they say when counting a group of objects indicates the total number in the collection."

The Mathematical Background also includes sample models with diagrams and explanations, strategies, and algebra connections. There is also a Teaching Tips section following the Mathematical Background that gives explanations of routines within the sessions such as think-pair-share, craft sticks, and choral counting. There are also explanations and samples of the various models used within the unit such as frames, number racks, tallies/bundles/sticks, and number lines.

In the Implementation section of the Online Resources, there is a "Math Coach" tab that provides the Implementation Guide, Scope & Sequence, Unpacked Content, and CCSS Focus for Kindergarten Mathematics.

##### 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.

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.

In the Unit 1 binder there is a section called "Introducing Bridges in Mathematics." In this section there is an overview of the components in a day (Problems & Investigations, Work Places, Assessments, Number Corner). Then there is an explanation of the Mathematical Emphasis in the program. Content, Practices, and Models are explained with pictures, examples and explanations. There is a chart that breaks down the mathematical practices and the characteristics of children in that grade level for each of the math practices. There is an explanation of the skills across the grade levels chart, the assessments chart, and the differentiation chart to assist teachers with the use of these resources. The same explanations are available on the website. There are explanations in the Assessment Guide that goes into they Types of Assessments in Bridges sessions and Number Corner.

The CCSS Where to Focus Kindergarten Mathematics document is provided in the Implementation section of the Online Resources. This document lists the progression of the major work in grades K-8.

Each unit introduction outlines the standards within the unit. A “Skills Across the Grade Level” table provides information about the coherence of the mathematics standards that are addressed in other units in Kindergarten and in Grade 1. The "Skills Across the Grade Level" document at the beginning of each Unit is a table that shows the major skills and concepts addressed in the Unit and where that skill and concept is addressed in the curriculum in the previous grade as well as in the following grade.

##### 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).

The materials provide a list of lessons in the teacher's edition cross-referencing the standards covered and providing an estimated instructional time for each lesson and unit. The "Scope and Sequence" chart lists all modules and units, the CCSSM standards covered in each unit, and a time frame for each unit. There is a separate "Scope and Sequence" chart for Number Corners.

##### 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.

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

Home connection materials and games sometimes include a “Note to Families” to inform them of the mathematics being learned within the unit of study.

Additional Family Resources are found at the Bridges Educator's Site.

• Grade K Family Welcome letter in English and Spanish- This letter introduces families to Bridges in Mathematics, welcomes them back to school, and contains a broad overview of the year's mathematical study.
• Grade K Unit Overviews for Units 1-8, in English and Spanish.
##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials contain explanations of the instructional approaches of the program. In the beginning of the Unit 1 binder, there is an overview of the philosophy of this curriculum and the components included in the curriculum. There is a correlation of the CCSSM and MPs as the core of the curriculum in the Mathematical Emphasis section. The assessment philosophy is given in the beginning of the assessment binder. The types of assessments and their purpose is laid out for teachers. For example, informal observation is explained as "one of the best but perhaps undervalued methods of assessing students...Teachers develop intuitive understandings of students through careful observation, but not the sort where they carry a clipboard and sticky notes. These understandings develop over a period of months and involve many layers of relaxed attention and interaction."

#### Criterion 3.3: Assessment

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

The instructional materials offer teachers resources and tools to collect ongoing data about student progress. The September Number Corner Baseline Assessment allows teachers to gather information on student's prior knowledge, and the Comprehensive Growth Assessment can be used as a baseline, quarterly, and summative assessment. Checkpoint interviews and informal observation are included throughout the instructional materials. Throughout the materials, Support sections provide common misconceptions and strategies for addressing common errors and misconceptions. Opportunities to review and practice are provided in both the Sessions and Number Corner routines. Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments clearly indicate the standards being assessed and include rubrics and scoring guidelines. There are, however, limited opportunities for students to monitor their own progress.

##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The September Number Corner Baseline Assessment is designed to gauge incoming students' numeracy skills. Also, the Comprehensive Growth Assessment contains 22 interview items and 8 written items and addresses every Common Core standard for Kindergarten. This can be administered as a baseline assessment as well as an end of the year summative or quarterly to monitor students' progress. Each unit contains at least two interview checkpoints within small groups to gather data for progress monitoring within the unit.

Informal observation is used to gather information. Many of the sessions and Number Corner workouts open with a question prompt: a chart, visual display, a problem, or even a new game board. Students are asked to share comments and observations, first in pairs and then as a whole class. This gives the teacher an opportunity to check for prior knowledge, address misconceptions, as well as review and practice with teacher feedback. There are daily opportunities for observation of students during whole group and small group work as well as independent work as they work in Work Places.

##### Indicator {{'3n' | indicatorName}}
Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Most Sessions have a Support section and ELL section that suggests common misconceptions and strategies for re-mediating these misconceptions that students may have with the skill being taught.

Materials provide sample dialogues to identify and address misconceptions. For example, the Unit 2 Module 2 Session 5 “Support” section gives suggestions for students struggling with one-to-one correspondence and cardinality. Each unit assessment also lists reteaching suggestions for students who did not master the learning targets for the unit.

##### Indicator {{'3o' | indicatorName}}
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review and practice, with feedback for students in learning both concepts and skills.

The scope and sequence document identifies the CCSS that will be addressed in the Sessions and in the Number Corner activities. Sessions build toward practicing the concepts and skills within independent Work Places. Opportunities to review and practice are provided throughout the materials. Ongoing review and practice is often provided through Number Corner routines. Each routine builds upon the previous month’s skills and concepts. For example, K.CC.2 is reviewed and practiced in Bridges Units 4, 6 and 8, and this standard is reviewed and practiced in all Number Corner months.

##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.

All assessments, both formative and summative, clearly outline the standards that are being assessed. In the assessment guide binder, the assessment map denotes the standards that are emphasized in each assessment throughout the year. Each assessment chart notes which CCSS is addressed.

For example, in Unit 1, Module 2, Session 5, the “Elements of Early Number Sense Checkpoint” includes four prompts targeting standard K.CC.4.B. There is a Checkpoint Scoring Guide that lists each prompt and each standard. Another example is Number Corner Checkup 2; the Interview Response Sheet has a CCSS Correlation for each of the questions at the top of the Response Sheet as well as a Number Corner Checkup 2 Scoring Guide. Also, each item on the Comprehensive Growth Assessment lists the standard being emphasized listed on the Skills & Concepts Addressed sheet as well as on the Interview Materials List and the Interview and Written Scoring Guides.

##### 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.

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting students' performance and suggestions for follow-up.

All Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments are accompanied by a detailed rubric and scoring guideline that provide sufficient guidance to teachers for interpreting student performance. There is a percentage breakdown to indicate Meeting, Approaching, Strategic, and Intensive scores. Section 5 of the Assessments Guide is titled "Using the Results of Assessments to Inform Differentiation and Intervention.” This section provides detailed information on how Bridges supports RTI through teachers' continual use of assessments throughout the school year to guide their decisions about the level of intervention required to ensure success for each student. There are cut scores and designations assigned to each range to help teachers identify students in need of Tier 2 and Tier 3 instruction. There is also a breakdown of Tier 1, 2 and 3 instruction suggestions.

##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

There is limited evidence in the instructional materials that students are self-monitoring their own progress.

Section 4 of the Assessment Guide is titled, "Assessment as a Learning Opportunity." This section provides information to teachers guiding them in: setting learning targets, communicating learning targets, encouraging student reflection, exit cards and comparing work samples from earlier and later in the school year.

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

Session and Number Corner activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The Bridges and Number Corner materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.

##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Units and modules are sequenced to support student understanding. Sessions build conceptual understanding with multiple representations that are connected. Procedural skills and fluency are grounded in reasoning that was introduced conceptually, when appropriate. An overview of each unit defines the progression of the four modules within each unit and how they are scaffolded and connected to a big idea. For example, in Unit 2 “Numbers to Ten” (K.CC) Module 1 compares five with ten frames, Module 2 compares five and ten units using the number rack (rekenrek), and Module 3 compares numbers within 10 using multiple visual models (10-frames, number rack, tally cards, craft sticks).

In the Sessions and Number Corner activities there are ELL strategies, support strategies, and challenge strategies to assist with scaffolding lessons and making content accessible to all learners.

The Assessment & Differentiation portion of Unit 1, Session 2, Module 4 in the “Spill 5 Beans” Work Place Guide provides suggestions for teachers on how to scaffold the Work Place. Guidance includes “(i)f you see that...(a) student is struggling with one-to-one correspondence then... support the student by pairing the student with someone with solid one-to-one correspondence. Together they can pull off and count beans as they organize them on 5-frame counting mat.”

In the Unit 7, Module 3, Session 1 Problem & Investigation, students are solving word problems by counting the number of eyes on the frogs. The following is "Support" and "ELL" suggestions are provided:

• "Support" - Some students may be completely stumped and not know how to start. Have them look at the picture and ask again, "How many frogs could there be?" Continue with, "Can you think of something to use to help you?"
• "ELL"- As you discuss and read the problem, be sure to point to the parts (eyes, log, pond) as you say them, circle all the eyes as you say, "8 eyes," point to the eyes as they're counted.

In the January Number Corners Number Line, as students are working on the number line to determine which number is greater and less than another number, the following "Support" suggestion is provided:

• If students are having a difficult time telling which number is greater than the other using numeral cards, show your class two small groups of cubes or other small objects, count the items into 10-frames, and ask which group has the greater number - reminding them that the word "greater" in mathematics means "more."
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials provide teachers with strategies for meeting the needs of a range of learners.

A chart at the beginning of each unit indicates which sessions contain explicit suggestions for differentiating instruction to support or challenge students. Suggestions to make instruction accessible to ELL students is also included in the chart. The same information is included within each session as it occurs within the teacher guided part of the lesson. Each Work Place Guide offers suggestions for differentiating the game or activity. The majority of activities are open-ended to allow opportunities for differentiation. Support and intervention materials are provided online and include practice pages, small-group activities and partner games.

In Unit 2, Module 2, Session 1, as students are working with two-color 10-frames, the teacher is provided with ELL, Support, and Challenge strategies to meet the needs of a range of learners.

• ELL - "When asking students about the top row and bottom row be sure to point to that row on the card, and when asking about how many there are in all, sweep your hand in a circular motion to indicate what you mean."
• Support - "Seat students who are not yet solid with one-to-one correspondence and numeral counting sequences to ten, close to or right in front of you. Once you have flashed the 10-frame to the rest of the students, continue to show the 10-frame to these students to view while they build what they see. Show the 10-frame again and give students the opportunity to check their work as you leave the card displayed."
• Challenge - "Provide students who are already facile with subitizing and building quantities to ten with a student whiteboard and dry erase marker and ask them to record an equation that describes the 10-frame.”

In Unit 6, Module 3, Session 2, as students are working on guessing and writing the mystery numbers, the teacher is provided with Support and Challenge strategies to meet the needs of a range of learners.

• Support - "If students have difficulty writing the numerals, say aloud what you are doing as you write them. For example, when writing 17, say, "For the 1, I'll start at the top and make a straight line down. For the 7, I'll start at the top and make a short line straight across and then make another slanted line down to the bottom."
• Challenge - "Ask students to explain the "10 and some more" property of each number (14 means there are 10 and 4 more).
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations. Tasks are typically open ended and allow for multiple entry-points in which students are representing their thinking with various strategies and representations (concrete tools as well as equations).

In the Problems and Investigations section, students are often given the opportunities to share strategies they used in solving problems that were presented by the teacher. Students are given multiple strategies for solving problems throughout a module. They are then given opportunities to use the strategies they are successful with to solve problems in Work Places, Number Corner and homework.

For example, in Unit 1, Module 3, Session 2, students are using the 10-frame, counting how many dots they see, and discussing various ways they counted. As students share their strategies, the following sample dialogue is provided: T - "How did you count the dots?" S - "I knew that there were 5 on the top and then I said 6, 7, 8." T - "Does someone have a different way that they counted?" S - "I just counted them all 1, 2, 3, 4, 5, 6, 7, 8." S - "There's 10 boxes and 2 are empty, so that makes 8 with dots.”

Another example is found in Unit 2, Module 2, Session 5, as students are working with the number rack, 5-frames, fingers, and equations to build combinations of 5. Students practice building these combinations and, then, share out the various ways they can build combinations of 5 using the various models.

In Unit 6, Module 4, Session 1, students are identifying the attributes of the group of students selected by the teacher. Students can enter the discussion with a wide variety of attributes they noticed and then provide various strategies to represent the attributes. For example, they can draw a picture of the short-sleeve shirts and the long-sleeve shirts, write 3 short-sleeve and 2 long-sleeve, write 3 + 2, write 2 + 3, or write 2 + 3 = 5. Students are able to respond with sketches, words and numbers, just numbers, expressions and equations.

##### 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).

The instructional materials suggest supports, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Online materials support students whose primary language is Spanish. The student book, home connections and component masters are all available online in Spanish. Materials have built in support in some of the lessons in which suggestions are given to make the content accessible to ELL students of any language.

There are ELL, Support, and Challenge accommodations throughout the Sessions and Number Corner activities to assist teachers with scaffolding instructions. Examples of these supports, accommodations and modifications include the following:

• In Unit 6, Module 4, Session 3, students are introduced to a new game called "Fill It Up Five +." Students are working on filling a 10-frame with 5 and some more. The ELL support provided for this session suggests "When using the terms, ‘top row’ and ‘bottom row’ be sure to point to that row on the display card, and run your hand in a circular motion around the card when you say ‘in all.’”
• For ELL support, in Unit 7, Module 2, Session 2, the materials suggest stressing the "-teen" ending of the numbers to differentiate from numbers ending in "-ty."
• For ELL and Support, Unit 7, Module 4, Session 2 suggests that teachers “(r)emind students what less means by demonstrating with a large pile of cubes and a small pile of cubes.”
• The Number Corner December Computational Fluency "Five and More" page in Activity 3 suggests “asking students to work together at the same pace while you read each prompt aloud. Students who are able to work ahead may do so, but providing this kind of scaffolding may help the students who are still learning to read and write numbers.”
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The instructional materials provide opportunities for advanced students to investigate mathematics content at greater depth. The Sessions, Work Places, and Number Corners include "Challenge" activities for students who are ready to engage deeper in the content.

Challenge activities found throughout the instructional materials include the following:

• In Unit 2, Module 2, Session 3, the challenge part of this session encourages students to write equations related to the number rack investigation of counting two sets of beads.
• In Unit 4, Module 2, Session 5, as students are working in the Work Place, "Beat You to Twenty." the Work Place Guide offers the following differentiation to challenge students: “Have students record an equation to describe their turn. Invite students to play Game Variation A or B.”
• In Unit 5, Module 4, Session 4, students are working on sorting 2-D shapes using Shape Sorting Cards. Students use characteristics from the cards such as "curved sides"and "three corners to eliminate all shapes that do not have the card's characteristic. The "Challenge" suggestion is to have students explain why there are squares in the "blue group" instead of in the "square group.”
• In the May Number Line Number Corners, students are playing "Cross out Fifty," a game requiring naming and crossing out all of the numbers to 50 on the One Hundred Grid. There are three "Challenge" suggestions: 1) Each time a team rolls, before the numbers are crossed out, ask students to figure out the last number that will be crossed out on that turn and explain their thinking. 2) Ask students to figure out how many more squares need to be crossed out to reach 50. How do they know? Can they prove it? 3) With input from the class, write an inequality statement about the two color amounts or write an addition equation about the two color amounts and the total number of squares.
##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

The materials provide a balanced portrayal of demographic and personal characteristics. Most of the contexts of problem solving involve objects and animals, such as frogs and penguins. When students are shown performing tasks, there are cartoons that appear to show a balance of demographic and personal characteristics.

##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials offer flexible grouping and pairing options. Throughout the Units, Work Places, and Number Corners there are opportunities to group students in various ways such as whole group on the carpet, partners during pair-share, and small groups during Problem & Investigations and Work Places.

In Unit 1, Mocule 2, Session 4, students are playing the game "Spill Five Beans" with a partner. In Unit 4, Module 2, Session 5, students begin the session sitting whole group in a discussion circle. In this Session, they move into Work Places where they get to choose what Work Place they would like to participate in. Grouping can be individual, pairs, or small groups, depending on the Work Place chosen.

##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

There is limited evidence of the instructional materials encouraging teachers to draw upon home language and culture to facilitate learning. The materials provide parent welcome letters and unit overview letters that are available in English and Spanish.

#### 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.

All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide additional, technology-based assessment opportunities, and the digital resources are not easily customized for individual learners.

##### 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.

The digital materials are web-based and compatible with multiple internet browsers. They appear to be platform neutral and can be accessed on tablets and mobile devices.

All grade level Teacher Editions are available online at bridges.mathlearningcenter.org. Within the Resources link (bridges.mathlearningcenter.org/resources), there is a sidebar that links teachers to the MLC, Math Learning Center Virtual Manipulatives. These include games, Geoboards, Number Line, Number Pieces, Number Rack, Number Frames and Math Vocabulary. The resources are all free and available in platform neutral formats: Apple iOS, Microsoft and Apps from Apple App Store, Window Store, and Chrome Store. The apps can be used on iPhones and iPads. The Interactive Whiteboard files come in two different formats: SMART Notebook Files and IWB-Common Format. From the Resource page there are also many links to external sites such as ABCYA, Sheppard Software, Illuminations, Topmarks, and Youtube.

##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials do not include opportunities to assess student mathematical understanding 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.

The instructional materials are not easily customizable for individual learners or users. Suggestions and methods of customization are not provided.

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.).

The instructional materials provide opportunities for teachers to collaborate with other teachers and with students, but opportunities for students to collaborate with each other are not provided. For example, a Bridges Blog offers teacher resources and tools to develop and facilitate classroom implementation.

##### 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.

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the MPs.

Each session within a module offers online resources that are in alignment with the session learning goals. Online materials offer an interactive whiteboard file as a tool for group discussion to facilitate discourse in the MPs. Resources online also include virtual manipulatives and games to reinforce skills that can be used at school and home. In the Bridges Online Resources there are links to the following:

• Virtual Manipulatives - a link to virtual manipulatives such as number lines, geoboard, number pieces, number racks, number frames, and math vocabulary
• Interactive Whiteboard Files - Whiteboard files that go with each Bridges Session and Number Corner
• Online Games- online games such as 100 Hunt using the hundreds grid, 2-D Shape Pictures, Interactive math dictionary, Addition With Manipulatives, and Balloon Pop Comparisons (greater than/less than)
• Images - for example, 1,000 M&M candies arranged on hundred grids by students

Within the Teacher's Edition, there is no direct reference to online resources. If embedded within the Teacher's Edition, the resources would be more explicit and readily available to the teacher.

## Report Overview

### Summary of Alignment & Usability for Bridges In Mathematics | Math

#### Math K-2

The instructional materials reviewed for K-2 meet the expectations for alignment and usability in each grade. The materials spend the majority of the time on the major work of the grade, and the assessments are focused on grade-level standards. Content is aligned to the standards and progresses coherently through the grades. There is also coherence within modules of each grade. The lessons include conceptual understanding, fluency and procedures, and application. There is a balance of these aspects of rigor. The Standards for Mathematical Practice (MPs) are used to enrich the learning, but the materials do not always attend to the full meaning of each MP and additional teacher assistance is needed in engaging students in constructing viable arguments and analyzing the arguments of others.  The K-2 materials also meet the criterion for usability which includes the following areas: use and design, planning support for teachers, assessment, differentiation, and technology.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

#### Math 3-5

The instructional materials reviewed for grades 3-5 meet the expectations for alignment and usability in each grade. The materials spend the majority of the time on the major work of the grade, and the assessments are focused on grade-level standards. Content is aligned to the standards and progresses coherently through the grades. There is also coherence within modules of each grade. The lessons include conceptual understanding, fluency and procedures, and application. There is a balance of these aspects of rigor. The Standards for Mathematical Practice (MPs) are used to enrich the learning, but the materials do not always attend to the full meaning of each MP and additional teacher assistance is needed in engaging students in constructing viable arguments and analyzing the arguments of others.  The 3-5 materials also meet the criterion for usability which includes the following areas: use and design, planning support for teachers, assessment, differentiation, and technology.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

## Report for {{ report.grade.shortname }}

### Overall Summary

###### Alignment
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###### Usability
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