Kindergarten - 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 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.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

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."

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).

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.

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.”

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.

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.