3rd 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 4, Module 3, Session 3 “Pattern Block Fractions,” students are investigating the fractions represented by several pattern blocks and then by combinations of pattern blocks when the hexagon is assigned a value of 1 in order to develop their understanding of a unit fraction and equivalency. During the Problem & Investigation section of the lesson students are given a problem, such as “if this (trapezoid) is ½ of the shape, what does the whole shape look like?” They work together to build what they believe the whole shape would be, then share their solutions and explain their reasoning, and discuss as a whole class. In the Student Book page, students are given a hexagon as the whole, or 1, and have to decide what the fraction of the whole is the shape given (trapezoid, rhombus, triangle). In the Work Place, “4D Hexagon Spin & Fill,” students play a game where they spin to determine a fractional amount, then take the correct pattern block or blocks and set them on the hexagon recording sheet until three hexagons are complete. They must always make trades to ensure that they have the fewest number of pattern blocks possible.

The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is 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 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 in whole group during the Problems & Investigations. The students then are given tasks that build on the session skills learned for the home connections.

For example, standard 3.OA.1 (interpret products of whole numbers) is introduced in Unit 2, developed in Units 2 and 5, mastered in Unit 5, and reviewed/practiced/extended in Unit 7. The standard continues to be developed in Number Corners from September through December and continues on in Grade 4. Another example is 3.NF.1 (develop and understanding of a unit fraction). This standard is introduced in Unit 4, developed in Units 4 and 7, and mastered in Units 7 and 8. It is also in Number Corners from October through the end of May, except for the month of March.

There is variety in what students are asked to produce. Throughout the grade, students are asked to respond and produce in various manners, often by 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 6, Module 2, Session 1, students are tasked with creating a series of polygons out of toothpicks by exploring the attributes of a progression of polygons. They are building their understanding of quadrilaterals, identifying shared attributes, and grouping them into different categories accordingly. They first build the polygons with given instructions. Then they record and label their polygons in their math journals, and they sketch the polygons and write their attributes. They, then have a discussion about the definition of a rhombus with the class. They further their understanding by identifying irregular polygons, using Word Resource Cards for definitions to make meaning, and then build their own pentagons and hexagons with irregular sides.

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 number lines, geoboards, pattern blocks, pan balance scales, and fraction tiles. When appropriate, they are connected to written representations.

For example, in Unit 4, Module 1, Session 5, students are learning about the difference between mass and weight by using pan balance scales to estimate, measure and solve problems about mass. They investigate and measure several items (paperclips, clay) using gram cubes with the balances and finally discuss how they can make each student’s clay have equal mass.

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 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 2, Module 2, Session 1, before students do a "count-around" with 9s after completing a "count-around" with 6s, teachers are prompted to ask the following questions:

• "Will there be more multiplies or fewer multiples of 9? Why?"
• "Will everyone get to call out a number? Why or why not?"
• "Can you estimate how many people will get to call out a number? Tell us more about your estimate."
• "What happens as the numbers we are counting get bigger?"

In Unit 4, Module 3, Session 2, "Comparing and Ordering Unit Fractions" teachers are prompted to ask the following questions:

• "If you like cookies, would you rather be in a group of 2 people sharing one cookie or 3 people sharing one cookie? Why?"
• "Would you rather be in a group of 2 people sharing 1 cookie or a group of 6 people sharing two cookies? Why?"
• "Would you rather be in a group of 50 people sharing or 25 people sharing? Why?"

In Unit 7, Module 3, Session 5, as students are playing a game called "Dozens of Eggs" which gives them practice building and recording egg carton (equivalent) fractions, the teacher is prompted to ask the following questions:

• "Can you tell how many twelfths there will be in the fraction on the card your classmate just drew for your team?"
• "How many more twelfths do you need to fill this egg carton? Is there a single fraction you could draw that would give you that many twelfths? How do you know?"
• "Which team is ahead, and by how much?"

In the November Number Corner Computational Fluency activity, "Array Race," the following questions are provided in the "Key Questions" section in the margin:

• "If you rolled an 8 and a 6, what would you shade in on the 10-by-10 array?"
• "Can you imagine how your array will look, and how many squares it will include before you shade it in?"
• "Now that your array is filled in, how many squares are there in each row? How many are there in each column?"
• "Can you find out how many squares are in this array without counting them one by one? How?"
• "How does the array model help you find the answers to multiplication problems?"
• "What is an efficient way to find your score?"

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 2 states, "Unit 2 focuses on helping students develop a conceptual understanding of multiplication. Investigations begin with contexts and problems that invite them to multiply, to think about equal groups and multiplicative comparisons. Story problems are not reserved as a culminating activity after students have explored multiplication. Instead, through the process of solving problems in context, students begin to make the transition from additive to multiplicative reasoning…”

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 give 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 3 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 MPs. 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 3 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 5-page, written baseline assessment that is designed to ascertain students' current levels of key number skills and concepts targeted for mastery in second grade – addition and subtraction story problems within 100; add single-digit numbers fluently; add and subtract 2- and 3-digit numbers; estimate, compare, and measure length in centimeters; and work with arrays with early fractions. The Comprehensive Growth Assessment contains 42 written items, addressing every Common Core standard for Grade 3. This can be administered as a baseline assessment as well as an end of the year summative or quarterly 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 as 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 remediating the misconceptions that students may have with the skill being taught.

Materials provide sample dialogues to identify and address misconceptions. For example, the Unit 4 Module 3 Session 3 “Support” section gives suggestions for struggling students. The materials suggest that although most students will likely have had several years of experience with the pattern blocks, the urge to play with these manipulatives never quite goes away. The materials suggest that teachers may find that students are more focused on the problems they are about to pose if you give them a couple of minutes to make designs or structures with their small collection of blocks.

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 5, Module 1, Session 2, students are applying what they already know about multiplication and division in order to solve story problems with products and dividends to 100 involving situations of measurement quantities (3.OA.3). 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 February Number Line as students create their own number lines marked with fractions (halves, fourths, eights, thirds, and sixths), students review and practice using the 0-1 number lines from the previous month (3.NF.2).

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 details the CCSS that is addressed.

For example, the Unit 3, Module 2, Session 1, Rounding & Multi-Digit Addition Checkpoint includes a Checkpoint Scoring Guide that lists each prompt, the correct answer, the standard, and the points possible. The Unit 5, Module 4, Session 6 Post-Assessment includes a Post-Assessment Scoring Guide that lists all items, correct answers, standards and the possible points, as well as a Student Reflection Sheet. The Unit 6, Module 4, Session 4, Unit 6 Post-Assessment includes a Scoring Guide that lists all items, correct answers, standards, and the possible points, as well as a Student Reflection Sheet. The October Number Corner Checkup 1 includes a Scoring Guide that contains the item, the CCSSM, and the possible points. The May Number Corner Checkup 4 includes a Scoring Guide that contains the item, the CCSSM, and the possible points.

Also, each item on the Comprehensive Growth Assessment lists the standard emphasized in the Skills & Concepts Addressed chart as well as on the Comprehensive Growth Assessment Scoring Guide.

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 of each student. There are cut scores and designations assigned to each range to help teacher 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 3, Module 2, Session 4, students are works on "Book & Books & Books." Support is offered: "...give them additional time." Challenge is offered: "ask them to write and solve a story problem with more than one step and more than one operation."

In the Unit 1, Module 3, Session 2 "Adding Lengths" activity, the following suggestions are provided:

• Support: "Help students model their thinking. Sometimes students can think of a strategy but do not know how to put their thinking on paper. Help them by listening carefully and modeling their work."
• ELL: "Help the student understand the meaning of the questions. Use objects or pictures of objects to represent the objects in the problems. Use gestures or line up the actual objects."
• Challenge: "Encourage students to use he most efficient and sophisticated strategies. Ask them if they can make bigger jumps. Ask students to describe their strategies."

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 1, students are working on "The Pet Store." This session is about seeing, identifying and marking groups to lay the foundation for multiplication. As students share out their representations, the teacher continues to ask "So how many chew toys can you see?" "So there are 4 groups of 3?" "Two dogs. I wonder how many feet they have?" "How can I write that?" During the session, 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 multiplying by 3, 4, and 8. Each student has a different multiplication 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 facts for example: Zero Facts, Doubles Facts, and Ten Facts.

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 2, Module 2, Session 2 provides a Support/ELL suggestion. The suggestion reads as follows: "As you work through the rest of the session, point to the groups of cubes on the train. Use the physical models to help clarify student comments and your questions."
• In Unit 5, Module 1, Session 5, students are working on Game Store Problems. The following ELL suggestion is provided: "Encourage students to write problems in their own language, using illustrations to make the problems more accessible to classmates who don't speak that language." The following Support suggestion is provided: "Tape copies of Game Store Problems 1, 2, 3 and 4 to the whiteboard, and let students know that these are examples that might give them ideas for their own problems."
• In the Number Corner April Calendar Collector, students are working with fractions of an hour. The ELL suggestion is: "As always, take time to emphasize important vocabulary through labels, sketches, and examples. Help ELL students connect terms in English to terms in their native language. Try to figure out what students already know in their native language and how that can help them understand the material in this workout."

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 1, Module 2, Session 2, the challenge part of this session provides extra questions and asks how each problem relates to the half facts on the Subtraction Table.
• In Unit 3, Module 1, Session 3, in the Problem String section, students are solving 2-digit and 3-digit addition problems. The "Challenge" suggestion is as follows: "Invite students who understand the give & take strategy to come up with problems that would work well with it. Have them also consider what numbers do not lend themselves to this strategy."
• In the Number Corner April Number line, students are working with fractions in a game called" Put It on the Line." The Challenge suggestions are as follows: "For student who can solve problems efficiently and easily, you may want to have a few challenge questions posted for them to work on as they finish the problems for the game," and "Challenge students to consider what the total sum must be for all of the answers (the sum of their score and your score)."

The materials provide a balanced portrayal of demographic and personal characteristics. Many of the contexts of problem solving involve objects and animals, such as cats and dogs. When people are shown, they are cartoons that appear to show a balance of demographic and personal characteristics.