2nd Grade - Gateway 3

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 mathematics 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 1, Session 3, during "How Many More? How Many Fewer?," students are estimating, counting, and comparing quantities of beans in order to develop place value counting skills. During the Problems & Investigation section of the lesson, students are scooping kidney beans into containers in groups of 10. Students pair-share their estimates, revising as they work, and then discuss as a whole class. In the Student Book page, students are counting out beans, recording results, and calculating how many more beans or fewer beans did they have than 50. In the Work Place, "Scoop, Count, & Compare," students are playing a game where they get three chances to scoop as close to 125 beans as possible. They first create a benchmark by counting 10 or 25 beans. Then they scoop, count, and record the number of beans they actually scoop. Finally, students write an expression to show whether the number of beans they scooped was less than, greater than, or equal to 125; the player who is closest, wins.

The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is continued to be 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).

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 as a 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 Places activities are done in small groups or with 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, 2.MD.6 is Introduced in Unit 2, developed in Units 3 and 5, mastered in Unit 7, and is Reviewed/Practiced/Extended in Unit 8. The standard continues to be developed in Number Corners in Computational Fluency and Number Line sections in all months except March. The standard is continued in Grade 3 as a Reviewed/Practiced/Extended skill. Another example is 2.OA.2. This standard is Developed in Units 1, 2 and 3, and continues to be Developed in all months of Number Corners in Computational Fluency.

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 5, Module 1, Session 2, students are responding to a place-value task in a variety of ways by estimating, counting, recording, comparing, and adding total amounts. They are building on their place-value understanding to 1,000 as they count large amounts of craft sticks. They first estimate the total number and then work in groups to count and bundle the sticks into 10s and 100s. They then come together to record, compare, and order the number of sticks giving verbal explanations of their models and recording/comparing amounts using "greater than" and "less than" symbols. Finally, they find the total number of sticks and compare it to their original estimates.

Also, in the Number Corner December Number Line, students are responding to place value in a variety of ways: building, sharing, estimating, confirming as they work with craft sticks in singles and bundles of 10 and 100, the number line, and a greater than/less than chart. Students are first asked to represent a number (214) on the number line with the craft sticks and explain their thinking/observations of the final amount built on the place value chart. Next, they explore jumping on the number line, starting with jumps of 10 forward and backward then moving to jumps of 100 (214, 314, 414, 514...). They are asked to share with their neighbor on what number they think the next jump will land; then they confirm their thinking by actually jumping on the number line. Finally, students guess a secret number from the number line. With each guess, they are told "greater than or less than", and the numbers guessed are recorded on a greater than, less than chart. At the same time, students are moving clips on the number line to mark the numbers that have been guessed.

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 and craft sticks. When appropriate, they are connected to written representations.

For example, in Unit 4, Module 1, Sessions 1 and 2, students are exploring customary units of measure using traditional and invented tools (cut-outs of: student's foot - varies in size, teacher's foot - foot, giant's foot - using a yard stick 36"). They measure several things and distances around the room and outside. They compare measurements and discuss the differences in the tools chosen and the resulting measurements.

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 conduct discourse that allows 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 2, Module 1, Session 3, teachers are prompted to ask the following questions:

• "Who would like to share how many dots there are total on the domino I picked-the one with 4 and 3? "
• "What did you all get for the total number of dots on your domino-the one with 5 dots on one-half, and 4 on the other? "

In Unit 4, Module 2, Session 2, as students are working with a number line, the teacher removes numbers so that only the zero and 100 are showing with one blank space in between. Teachers are prompted to ask the following questions:

• "What number goes in the box?"
• "Yes, why would 50 go in the box? Can you say more about why it should be 50?"
• "If we replaced the 50 with a 5 would the number in the empty box change? Why? What would the new number be - how do you know?"

• "The emperor penguin is 45 inches tall. Is that more or less than 3 feet?"
• "Can someone tell us in your own words how you know?"
• "Does anyone see another way to do that?"
• "Would someone like to tell us why that's a good strategy?"

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

• "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 Corner April Days In School, the following questions are provided to help students think about numbers on the hundreds grid:

• How many squares are marked? How did you count them? Is there another way?
• What number comes next? How do you know?
• How many tens have we made so far?

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 3 states, "Unit 3 pushes students towards mastery of key number facts and fact strategies for single-digit addition and subtraction...The ability to subitize is central to a well-rounded sense of numbers and operational fluency in general. To subitize is to give up the need to count every object in a set in order to name the quantity...Unit 3 also emphasizes the concept of part-part-whole reasoning. A precursor to algebraic reasoning, knowledge of part-whole relationships is useful in problem contexts that involve either combining or separating numbers..."

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 line.

In the implementation section of the Online Resources, there is a "Math Coach" tab that provides the Implementation Guide, Scope & Sequence, Unpacked Content, and CCSSM Focus for Grade 2 Mathematics.

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 MPs 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 go into the Types of Assessments in Bridges Sessions and Number Corner.

The CCSSM Where to Focus Grade 2 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 math standards that are addressed in the previous grade as well as in the following grade. 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.

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

The September Number Corner Baseline Assessment is a 4-page, written assessment designed to ascertain students' current levels of skills targeted for mastery in Grade 1, such as story problems within 20, numbers to 120, place value, measurement, 2-digit addition/subtraction, and fractions. The Comprehensive Growth Assessment contains interview items and written items and addresses every Common Core standard for Grade 2. This can be administered as a baseline assessment, an end-of-the-year summative assessment or quarterly assessment to monitor students' progress.

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 when they work in Work Places.

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 those misconceptions that students may have with the skill being taught.

Materials provide sample dialogues to identify and address misconceptions. For example, the Unit 6 Module 2 Session 3 “Support” section gives suggestions for struggling students. The materials suggest that if students are still unsure that two triangles make up one rectangle, invite them to trace the figure on a sheet of geoboard paper, cut out the triangles, and slide them together to make a rectangle.

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 CCSSM 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. For example, in Unit 7, Module 3, Session 2, as students are working on toy store story problems, they are reviewing and practicing their ability to solve a two-step subtraction story problem with minuends to 100 (2.OA.1).

Ongoing review and practice is often provided through Number Corner routines. Each routine builds upon the previous month’s skills and concepts. For example, in the Number Corner December Calendar Grid, as students are working with 2-dimensional shapes to find patterns, they are getting review and practice with recognizing that shapes have specific attributes (2.G.1).

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 CCSSM is addressed.

For example, the Unit 2, Module 2, Session 1 Unit checkpoint includes a Checkpoint Scoring Guide that lists each prompt, the correct answer, each standard, and the points possible. The Unit 5, Module 3, Session 5 Post-Assessment includes a Post-Assessment Scoring Guide that lists all items, correct answers, standards and the possible points. The Number Corner Checkup 1 includes a Number Corner Checkup 1 Scoring Guide for the written part of the assessment that contains the item, the CCSSM, and the possible points.

Another example is Unit 6, Module 3, Session 5; this Unit 6 Assessment includes a Unit Scoring Guide that lists all items, correct answers, standards, and the possible points. Another example is Number Corner Checkup 4; the Interview Response Sheet has a CCSSM Correlation for each of the questions at the top of the Response Sheet as well as a Number Corner Checkup 4 Scoring Guide for the written part of the assessment.

Also, each item on the Comprehensive Growth Assessment lists the standards being emphasized on the Skills & Concepts Addressed sheet, the Interview Materials List and the Interview and Written Scoring Guides.

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, the 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 Assessment 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 during the school year to guide their decisions about the level of intervention required to ensure the success of 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.

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.

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.

For example, in Unit 5, Module 1, Session 4, students are playing the game, "Place Value Triple Roll." They are rolling dice and building various numbers with sticks and bundles. Support is offered: "...work with students to find the next smallest number on the list and record it on their sheets." Challenge is offered: "Ask students to complete the last section of the Record Sheet."

In the Unit 3, Module 3, Session 1 Work Place 3F "3-D Base Ten Triple Spin," the following suggestions are provided:

• Support: "If students are struggling to decide which denomination to take each spin in, have them take their three spins before they spin the Greater than/Less than spinner."
• ELL: "Review the idea of using sketches to record numbers. Draw the shorthand version for each base ten piece and, right next to the sketch, write the number it represents."
• Challenge: "If students are playing the game with confidence and ease and might benefit from a challenge, invite them to take six spins each, build two 3-digit numbers, and add the numbers to determine the winner."

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.

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 often are 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 2, Module 1, Session 4, students are working on showing "Tens & Ones" with cubes. The teacher begins with an intentional progression: 23 to 43; 43 to 73. As students share out their representations, the teacher continues to ask "Did someone have another way to make the change?" "How about a different way to show 45?" Students can represent the numbers by counting out single cubes, combinations of 10s, and combinations of 5s; some simply add two additional groups of ten to the original number (23-43). Any correct representation is accepted, and students are encouraged to solve using different representations.

Another example is found in the Number Corner February Computational Fluency. Students are working on their addition facts according to where they have demonstrated mastery. Each student has a different table where they are coloring in the facts that they've mastered, with non-mastered facts easily identified. The table also has a sidebar with various strategies that can be used to master facts to 20, for example: Add Zero Facts, Count on Facts, Doubles Facts, and Make Ten Facts. According to students' individual level of mastery and understanding of strategies, they identify the strategy that they will use in today's practice and the facts they will be working on. This fluency practice allows all students to enter the practice at their own point of understanding/mastery and allows for practice using different strategies.

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:

• Unit 3, Module 3, Session 6 provides an ELL and a support suggestion. The ELL suggestion reads as follows: "If students have trouble reading a problem, encourage them to examine the talk bubble for clues as to what they're supposed to do. If that doesn't work, suggest that they ask the student who wrote the problem to read it to them." The Support suggestion is to "be prepared to have two or three problems in mind to suggest to students who may have trouble choosing, or may have trouble solving some (perhaps most) of the problems posed by their classmates. You may even want to have one problem in mind to work with a smaller group of students as you send other students out to shop."
• In Unit 4, Module 2, Session 1, students are working on measuring feet and yards. The following "ELL" suggestion is provided: "Give students a visual reminder of the relationship between feet and yards to help them better understand that there are three feet in every yard. If possible, display a yardstick in the classroom with three inchworm rulers lined up beside it. Post the statement 1 yard = 3 feet near the display."
• In the Number Corner January Daily Rectangle, students are working on creating arrays in four different quadrants of a hundreds grid. The "Support" suggestion is: "Use two 5x8 cards to mask all but the quadrant in question each time. Also remind the students that they only need to worry about the gray squares in each quadrant, not all the squares."

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 3, Module 2, Session 2, the challenge part of this session encourages students to add on the number line.
• In Unit 4, Module 2, Session 2, in the Problems & Investigations section, students are measuring in yards. The "Challenge" suggestion is as follows: "Finding items that are exactly 2, 3 or 4 yards long may be difficult. Some pairs may enjoy competing to see who can get the closest to these measures. Discuss how you would judge who is the closest using a smaller unit of measure such as inches."
• In the Number Corner October Calendar Collector, students are collecting time in a game called" How Much Time Did We Collect." The Challenge suggestion is as follows: "Challenge students to figure out how many hours and minutes they collected over the month and record at the bottom of their Rolling for Minutes Record Sheet."

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, they are cartoons that appear to show a balance of demographic and personal characteristics.