5th Grade - Gateway 3

The Sessions within the Units distinguishes 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 3, Module 1, Session 3 “Multiplying and Dividing by Ten,” students are solving problems using money notation and considering the place value patterns that emerge when multiplying and dividing decimal numbers by 10. During the “Problems and Investigations” portion of this lesson students work with a scenario in which ten friends are planning an outing to the art museum and have some questions about the money they will spend. During this work time, students are asked clarifying questions to check understanding, then they share answers and sketches as a group. They also discuss their finding of patterns they noticed when using the number ten and the place value patterns that emerge within these findings. On the Student Book page, the students use a calculator to predict and record the results of multiplying and dividing by ten. They look for the patterns and record those observations.

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 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 partners to complete tasks that are based on the problems done 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, standard 5.NF.1 (add and subtract fractions with unlike denominators, include mixed numbers) is introduced in Grade 4, developed and mastered in Unit 2, and is reviewed/practiced/extended in Units 3 and 5. This standard continues to be developed in Number Corners from October through January, and in March through May. Another example is standard 5.NBT.2 (explain patterns in the number of zeros in the product when multiplying by powers of 10). This standard is developed in Unit 3, and mastered in Unit 7. It also is practiced during Number Corners from November through February.

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 considering ways to multiply a whole number by a unit fraction. They begin with a partner discussion as to how they might find the answer to the given combination. To further their understanding, the students are asked to work with a problem using a common rather than a unit fraction. Students pair up to share ideas before solving. Once they have their strategies in place, they play a game practicing and reinforcing the learning.

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 unit cubes to fill a rectangular prism, rulers to measure, geoboards, protractors, and graph paper to accurately represent problems.

For example, in Unit 6, Module 3, students are moving from using discrete counting to the continuous measurement definition of volume. Students are working to pack boxes of 1-inch marbles using paper cut-outs as boxes. They investigate how many marble boxes they can pack in each of the larger boxes for shipping. They predict, measure and review dimensions to solve the task. Once complete, they have a class forum (discussion) to share the results and check for accuracy.

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 1, Session 3 during a number string on fraction addition and subtraction, teachers are prompted to ask the following questions:

• "Which representation are you most likely to use?"
• "Is it easier to visualize 1/5 or 4/20?"
• "Which would you rather have in your pocket, 2 dimes or 20 pennies? Why?"

In Unit 3, Module 2, Session 2 - When discussing a fraction game, the teacher is prompted to ask the following questions:

• "How can you be sure you made the smallest or largest decimal possible?"
• "Is there ever a reason to not make the smallest or largest decimal possible?"
• "How do the wild cards change the game?"

In Unit 5, Module 2, Session 4, - As students are exploring multiplication with fractions, the teacher is prompted to ask the following questions:

• "How is multiplying with fractions different from multiplying with whole numbers?"
• "How is multiplying with fractions similar to multiplying with whole numbers?"
• "If you multiply a fraction less than 1 by another fraction less than 1, what kind of product will you get?"

In Unit 7, Module 4, Session 3, - students are working with decimal operations. The teacher is prompted to ask the following questions:

• "Given a situation that involves dividing $94.00 by 8, would an answer of $1.175 make sense? Why or why not? What about $11.75 or $117.50?"

In the December Number Corner Calendar Collector activity, the following questions are provided in the "Key Questions" section in the margin:

• "What does the ordered pair (x, y) represent in this situation?"
• "What does this line plot show about the heights of students in our class?"
• What is similar about these line plots? What is similar about them?"
• Imagine we added data about 10 more 5th grade classrooms to each graph. How do you think that would change the graphs? In what ways would the graphs remain similar?"

Materials contain adult-level explanations of the math 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 math concepts addressed in the unit. For example, Unit 3 states, "Understanding equivalence is critical to adding and subtracting fractions. We want students to have many meanings come to mind when they see a fraction. For 1/4, for example, a student might think of 1 quarter, 25 cents, $0.25, half of 1/2, double 1/8, 25%, dividing something by 4, 1/4 of an hour, 15 minutes out of 60 minutes, a distance 1/4 of a unit from 0, and so on. Then when students see 1/4 added to another fraction, they can use the meaning that is most helpful, given the denominator of the other fraction."

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 strategies, tools and representations within the sessions such as geoboards, protractors, mathematical language, and modeling. There are also explanations and samples of the various representations used within the unit such as area model, base ten pieces, ratio tables, 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 CCSS Focus for Grade 5 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 the Types of Assessments in Bridges sessions and Number corner.

The CCSS Where to Focus Grade 5 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 6 page, written baseline assessment that is designed to ascertain students' current levels of key number skills and concepts targeted for mastery in fourth grade – fluency with basic multiplication and division facts, solve story problems involving all four operations, compare and order fractions, add and subtract fractions and mixed numbers with like denominators, multiply fractions by whole numbers, and write and compare decimals to hundredths. The Comprehensive Growth Assessment contains 41 written items, addressing every Common Core standard for Grade 5. 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 3 Module 4 Session 2 “Support” section gives suggestions for struggling students. The materials suggest that the teacher should encourage students who want or need to build with the base ten area pieces before sketching to do so. An additional suggestion is that pairs have one partner model the problem with base ten area pieces while the other makes the sketch on the problem sheet.

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 4, Module 3, Session 4, the teacher leads students to apply what they already know about area models to using the standard algorithm to multiply multi-digit whole numbers (5.NBT.5). 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 November Problem Strings students revisit the previous month’s work with money and clock models in order to subtract unit and non-unit fractions with unlike denominators (5.NF.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 details the CCSS that is addressed.

For example, the Unit 2, Module 2, Session 6, Fraction Addition & Subtraction Checkpoint includes a Checkpoint Scoring Guide that lists each prompt, the correct answer, the standard, and the points possible. The Unit 2, Module 3, Session 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 Unit 6, Module 4, Session 4 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 January Number Corner Checkup 2 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 4, Session 1, students are working on the activity "Writing Division Story Problems." Support is offered: "...give students who are unable to finish during the math period extra time to complete the assignment before the end of the day." Challenge is offered: "Ask early finishers to prepare a second and even a third problems for their classmates."

In the Unit 2, Module 2, Session 1 "River Trail" activity, the following suggestions are provided:

• ELL: "Gather students and have them restate the task. Ensure that students understand the information they need to place at each landmark location.”
• Support: "Model how to create a 30-inch line with a measuring tape to simulate the trail and help students understand that it represents the full river length. Suggest that students attach their measuring tape to the butcher paper with tape.”
• Challenge: “Pair similar ability students together. Ask students to look at the list of landmarks, predict markers that will be placed at the same location, and then justify their thinking mathematically.”

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 4, Module 1, Session 4, students are sharing their strategies for the problems from the previous day’s activity about a fundraiser "Callie’s Cake Pops" in a math forum. Students worked in pairs to find the cost and profit for the fundraiser using any strategy and representation they wanted in multiplying decimals and whole numbers and then created a poster to present their solutions and strategies. In this session the teacher has pairs of students meet and compare their strategies with a particular focus on how they dealt with the decimals. Several pairs of students are asked to share their posters to highlight specific strategies they used to find the cost and profit for the fundraiser including: 1) splitting the dollars and cents, 2) changing the money to equivalent fractions, 3) using a ratio table to keep track of the money and cake pops, and finally 4) the doubling and halving strategy.

Another example is found in the Number Corner February Problem Strings. Students are solving a several strings of problems involving multiplication of whole number by fractions posed by the teacher using any strategy they want. Students’ strategies are modeled using various arrangements of counters and also various equations, which highlight the associative and distributive properties. The teacher solicits students’ strategies by prompting students to “Tell me how you thought about this one.” The teacher follows this question with, “ First, is the answer going to be greater than 16 or less than 16?”, in order to support students generalizations about multiplying fractions by whole numbers.

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 3 provides a ELL suggestion. The suggestion reads as follows: "Make sure students understand the problem. Write the problem in numbers as you say it in words. Draw the double number lines and the jumps as you talk about what is happening. Point to the jumps and the landing places as you write the corresponding numbers and fractions.”
• In Unit 5, Module 4, Session 3, students are working on solving problems where they divide a whole number by a fraction. The following Support suggestion is provided: "Encourage students who may be getting frustrated at this point to see if they can draw a picture that might help them understand the situation."
• In Unit 6, Module 2, Session 2, students are classifying quadrilaterals by their characteristics. The following ELL suggestions is provided as follows: “This is a particularly language-intensive assignment. You will almost certainly want to pair ELL students with partners who speak both languages. You might also have students record the name of each shape on the hierarchy in their own language as well as, or instead of, English.
• In the Number Corner April Computational Fluency, students are placing decimal on a number line using various strategies. The ELL suggestion is: "Make sure students understand the problem. Emphasize or review key vocabulary. You may want to have students work in pairs to discuss and solve the problems."

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 1, Session 1, students are solving problems that require addition of fractions with unlike denominators. The “Challenge” suggestions is as follows: “Encourage students to use the most efficient or sophisticated strategy they can think of. Then encourage them to look back at their work and see if they can think of an even more efficient strategy.”
• In Unit 4, Module 2, Session 4, students are solving equations that require multiplying whole numbers by fractions as part of the activity “Over & Under”. The "Challenge" suggestion is as follows: "Ask students to partner with another student and discuss the order in which they solved the problems. Ask them to verbalize a “path” they could take that minimizes the amount of work to solve one or both sets of problems.
• In the Number Corner January Calendar Grid, students are graphing patterns. The Challenge suggestions are as follows: "Invite students to make up their own number pattern and graph it instead of doing a pattern from the Calendar Grid. They can work independently, trade graphs with another student, and figure out what the rule was for generating the ordered pairs."

The materials provide a balanced portrayal of demographic and personal characteristics. Many of the contexts of problem solving involve objects and animals, such as baseballs, marbles, skateboards, money, apples, pet supplies, art supplies, and turtles, ducks. When people are shown, they are cartoons that appear to show a balance of demographic and personal characteristics.