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MATHEMATICS

Grade 5: Unit 1

Understanding the Place Value System

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Course Philosophy/Description

In mathematics, students will learn to address a range of tasks focusing on the application of concepts, skills and understandings. Students will be

asked to solve problems involving the key knowledge and skills for their grade level as identified by the NJSLS; express mathematical reasoning and

construct a mathematical argument and apply concepts to solve model real world problems. The balanced math instructional model will be used as

the basis for all mathematics instruction.

Fifth grade Mathematics consists of the following domains: Operations and Algebraic Thinking (OA), Number and Operations in Base Ten (NBT),

Number and Operations-Fractions (NF), Measurement and Data (MD), and Geometry (G). In fifth grade, instructional time should focus on three

critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and

of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending

division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to

hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.

1) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike

denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and

make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between

multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is

limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations.

They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for

decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these

computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the

relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to

understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of

decimals to hundredths efficiently and accurately.

3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total

number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit

cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve

estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them

as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world

and mathematical problems

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ESL Framework

This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language programs

use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this document to

collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers may use the

appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this way, whether

it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the New Jersey Student Learning

Standards (NJSLS). The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA)

Consortium’s English Language Development (ELD) standards with the New Jersey Student Learning Standards (NJSLS). WIDA’s ELD standards

advance academic language development across content areas ultimately leading to academic achievement for English learners. As English learners

are progressing through the six developmental linguistic stages, this framework will assist all teachers who work with English learners to appropriately

identify the language needed to meet the requirements of the content standard. At the same time, the language objectives recognize the cognitive

demand required to complete educational tasks. Even though listening and reading (receptive) skills differ from speaking and writing (expressive) skills

across proficiency levels the cognitive function should not be diminished. For example, an Entering Level One student only has the linguistic ability

to respond in single words in English with significant support from their home language. However, they could complete a Venn diagram with single

words which demonstrates that they understand how the elements compare and contrast with each other or they could respond with the support of their

home language (L1) with assistance from a teacher, para-professional, peer or a technology program.

http://www.state.nj.us/education/modelcurriculum/ela/ELLOverview.pdf

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Pacing Chart – Unit 1

# Student Learning Objective NJSLS

Instruction: 8 weeks

Assessment: 1 week

1 Evaluate numerical expressions that contain parentheses, brackets and

braces. 5.OA.A.1

2 Write numerical expressions when given a verbal description or word

problem; interpret numerical expressions without evaluating them.

5.OA.A.2

3 Explain that a digit in one place represents 1/10 of what it would

represent in the place to its left and ten times what it would represent in

the place to its right.

5.NBT.A.1

4 Explain patterns in the number of zeros in the product when a whole

number is multiplied by a power of 10; represent powers of 10 using

whole-number exponents.

5.NBT.A.2*

5 Use the standard algorithm to multiply a whole number up to four digits

by a whole number up to two digits.

5.NBT.B.5*

6 Calculate whole number quotients of whole numbers with 4-digit

dividends and 2-digit divisors; explain and represent calculations with

equations, rectangular arrays, and area models.

5.NBT.B.6

7 Add, subtract, multiply, and divide decimals to hundredths using

concrete models or drawings and strategies based on place value,

properties of operations, and/or the relationship between addition and

subtraction; explain the reasoning used, relating the strategy to the

written method.

5.NBT.B.7*

8 Compare two decimals to thousandths using >, =, and < for numbers

presented as base ten numerals, number names, and/or in expanded

form.

5.NBT.A.3

9 Round decimals to any place value. 5.NBT.A.4

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Research about Teaching and Learning Mathematics Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997)

Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990)

Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt & Armour-Thomas, 1992)

Students interact in ways that support and challenge one another’s strategic thinking (Artzt, Armour-Thomas, & Curcio, 2008)

Activities structured in ways allowing students to explore, explain, extend, and evaluate their progress (National Research Council, 1999)

There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002):

Teaching for conceptual understanding

Developing children’s procedural literacy

Promoting strategic competence through meaningful problem-solving investigations

Teachers should be:

Demonstrating acceptance and recognition of students’ divergent ideas

Challenging students to think deeply about the problems they are solving, extending thinking beyond the solutions and algorithms

required to solve the problem

Influencing learning by asking challenging and interesting questions to accelerate students’ innate inquisitiveness and foster them to

examine concepts further

Projecting a positive attitude about mathematics and about students’ ability to “do” mathematics

Students should be:

Actively engaging in “doing” mathematics

Solving challenging problems

Investigating meaningful real-world problems

Making interdisciplinary connections

Developing an understanding of mathematical knowledge required to “do” mathematics and connect the language of mathematical

ideas with numerical representations

Sharing mathematical ideas, discussing mathematics with one another, refining and critiquing each other’s ideas and understandings

Communicating in pairs, small group, or whole group presentations

Using multiple representations to communicate mathematical ideas

Using connections between pictures, oral language, written symbols, manipulative models, and real-world situations

Using technological resources and other 21st century skills to support and enhance mathematical understanding

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Mathematics is not a stagnate field of textbook problems; rather, it is a dynamic way of constructing meaning about the world around

us, generating knowledge and understanding about the real world every day. Students should be metaphorically rolling up their

sleeves and “doing mathematics” themselves, not watching others do mathematics for them or in front of them. (Protheroe, 2007)

Balanced Mathematics Instructional Model

Balanced math consists of three different learning opportunities; guided math, shared math, and independent math. Ensuring a balance of all three

approaches will build conceptual understanding, problem solving, computational fluency, and procedural fluency. Building conceptual

understanding is the focal point of developing mathematical proficiency. Students should frequently work on rigorous tasks, talk about the math,

explain their thinking, justify their answer or process, build models with graphs or charts or manipulatives, and use technology.

When balanced math is used in the classroom it provides students opportunities to:

solve problems

make connections between math concepts and real-life situations

communicate mathematical ideas (orally, visually and in writing)

choose appropriate materials to solve problems

reflect and monitor their own understanding of the math concepts

practice strategies to build procedural and conceptual confidence

Teacher builds conceptual understanding by

modeling through demonstration, explicit

instruction, and think alouds, as well as guiding

students as they practice math strategies and apply

problem solving strategies. (whole group or small

group instruction)

Students practice math strategies independently to

build procedural and computational fluency. Teacher

assesses learning and reteaches as necessary. (whole

group instruction, small group instruction, or centers)

Teacher and students practice mathematics

processes together through interactive

activities, problem solving, and discussion.

(whole group or small group instruction)

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Effective Pedagogical Routines/Instructional Strategies

Collaborative Problem Solving

Connect Previous Knowledge to New Learning

Making Thinking Visible

Develop and Demonstrate Mathematical Practices

Inquiry-Oriented and Exploratory Approach

Multiple Solution Paths and Strategies

Use of Multiple Representations

Explain the Rationale of your Math Work

Quick Writes

Pair/Trio Sharing

Turn and Talk

Charting

Gallery Walks

Small Group and Whole Class Discussions

Student Modeling

Analyze Student Work

Identify Student’s Mathematical Understanding

Identify Student’s Mathematical Misunderstandings

Interviews

Role Playing

Diagrams, Charts, Tables, and Graphs

Anticipate Likely and Possible Student Responses

Collect Different Student Approaches

Multiple Response Strategies

Asking Assessing and Advancing Questions

Revoicing

Marking

Recapping

Challenging

Pressing for Accuracy and Reasoning

Maintain the Cognitive Demand

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Educational Technology

Standards

8.1.5.A.1, 8.1.5.A.3, 8.1.5.F.1, 8.2.5.C.4

Technology Operations and Concepts

Select and use the appropriate digital tools and resources to accomplish a variety of tasks including solving problems.

Example: Use this Order of Operations interactive game to improve math fluency.

http://learningwave.com/chapters/numbers/ordofops.html

Use a graphic organizer to organize information about problem or issue.

Example: Project the Bowling for Numbers task on an electronic board. Practice the activity with the whole class so that students in small

groups can demonstrate and record strategies in an organized manner. https://www.illustrativemathematics.org/content-standards/tasks/969

Critical Thinking, Problem Solving, and Decision making

Apply digital tools to collect, organize, and analyze data that support a scientific finding.

Example: Use a digital scale to model multi-step solutions. https://www.illustrativemathematics.org/content-standards/tasks/1562

Design

Collaborate and brainstorm with peers to solve a problem evaluating all solutions to provide the best results with supporting

sketches or models.

Example: Sumdog, and Moby Max. These sites allow students to work collaboratively and in competition in schools, at home and

between schools while promoting critical and computational thinking https://www.sumdog.com/, http://www.mobymax.com

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Career Ready Practices

Career Ready Practices describe the career-ready skills that all educators in all content areas should seek to develop in their students. They are

practices that have been linked to increase college, career, and life success. Career Ready Practices should be taught and reinforced in all career

exploration and preparation programs with increasingly higher levels of complexity and expectation as a student advances through a program of

study.

CRP2. Apply appropriate academic and technical skills. Career-ready individuals readily access and use the knowledge and skills acquired

through experience and education to be more productive. They make connections between abstract concepts with real-world applications, and

they make correct insights about when it is appropriate to apply the use of an academic skill in a workplace situation.

Example: Students will apply prior knowledge when solving real world problems. Students will make sound judgements about the use of specific

tools and use tools to explore and deepen understanding of place values.

CRP4. Communicate clearly and effectively and with reason. Career-ready individuals communicate thoughts, ideas, and action plans with

clarity, whether using written, verbal, and/or visual methods. They communicate in the workplace with clarity and purpose to make maximum

use of their own and others’ time. They are excellent writers; they master conventions, word choice, and organization, and use effective tone

and presentation skills to articulate ideas. They are skilled at interacting with others; they are active listeners and speak clearly and with

purpose. Career-ready individuals think about the audience for their communication and prepare accordingly to ensure the desired outcome.

Example: Students will communicate precisely using clear definitions and provide carefully formulated explanations when constructing

arguments. Students will communicate and defend mathematical reasoning using objects, drawings, diagrams, and/or actions. Students will ask

probing questions to clarify or improve arguments regarding place values, numerical expressions, patterns, multiplication, division, comparison of

decimals and rounding.

CRP8. Utilize critical thinking to make sense of problems and persevere in solving them. Career-ready individuals readily recognize

problems in the workplace, understand the nature of the problem, and devise effective plans to solve the problem. They are aware of problems

when they occur and take action quickly to address the problem; they thoughtfully investigate the root cause of the problem prior to introducing

solutions. They carefully consider the options to solve the problem. Once a solution is agreed upon, they follow through to ensure the problem

is solved, whether through their own actions or others.

Example: Students will understand the meaning of a problem and look for entry points to its solution. They will analyze information, make

conjectures, and plan a solution pathway. Students will monitor and evaluate progress and change course as necessary.

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CRP12. Work productively in teams while using cultural global competence.

Career-ready individuals positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to

avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They

plan and facilitate effective team meetings.

Example: Students will work collaboratively in groups to solve mathematical tasks. Students will listen to or read the arguments of others and

ask probing questions to clarify or improve arguments.

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WIDA Proficiency Levels

At the given level of English language proficiency, English language learners will process, understand, produce or use

6- Reaching

Specialized or technical language reflective of the content areas at grade level

A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse as

required by the specified grade level

Oral or written communication in English comparable to proficient English peers

5- Bridging

Specialized or technical language of the content areas

A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse,

including stories, essays or reports

Oral or written language approaching comparability to that of proficient English peers when presented with

grade level material

4- Expanding

Specific and some technical language of the content areas

A variety of sentence lengths of varying linguistic complexity in oral discourse or multiple, related

sentences or paragraphs

Oral or written language with minimal phonological, syntactic or semantic errors that may impede the

communication, but retain much of its meaning, when presented with oral or written connected discourse,

with sensory, graphic or interactive support

3- Developing

General and some specific language of the content areas

Expanded sentences in oral interaction or written paragraphs

Oral or written language with phonological, syntactic or semantic errors that may impede the

communication, but retain much of its meaning, when presented with oral or written, narrative or expository

descriptions with sensory, graphic or interactive support

2- Beginning

General language related to the content area

Phrases or short sentences

Oral or written language with phonological, syntactic, or semantic errors that often impede of the

communication when presented with one to multiple-step commands, directions, or a series of statements

with sensory, graphic or interactive support

1- Entering

Pictorial or graphic representation of the language of the content areas

Words, phrases or chunks of language when presented with one-step commands directions, WH-, choice or

yes/no questions, or statements with sensory, graphic or interactive support

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Culturally Relevant Pedagogy Examples

Bring in Guest Speakers: Invite guest speakers who can add context to your lesson and speak from a specific culture’s

general perspective.

Example: Ask a doctor to visit and speak to the students about how he uses operations with decimals when calculating

prescriptions. Or ask a banker to visit and discuss how operations with decimals assist with the accounting of money.

Everyone has a Voice: Create a classroom environment where students know that their contributions are expected

and valued.

Example: Norms for sharing are established that communicate a growth mindset for mathematics. All students are capable

of expressing mathematical thinking and contributing to the classroom community. Students learn new ways of looking at

problem solving by working with and listening to each other.

Use Learning Stations: Provide a range of material by setting up learning stations.

Example: Reinforce understanding of concepts and skills by promoting the learning through student interests and modalities,

experiences and/or prior knowledge. Encourage the students to make choices in content, based upon their strengths, needs,

values and experiences. Providing students with choice boards will give them a sense of ownership to their learning and

understanding.

Present New Concepts Using Student Vocabulary: Use student diction to capture attention and build understanding

before using academic terms.

Example: Teach math vocabulary in various modalities for students to remember. Use multi-modal activities, analogies, realia,

visual cues, graphic representations, gestures, pictures practice and cognates. Model to students that some vocabulary has

multiple meanings. Have students create the Word Wall with their definitions and examples to foster ownership. Work with

students to create a variety of sorting and match games of vocabulary words in this unit. Students can work in teams or

individually to play these games for approximately 10-15 minutes each week. This will give students a different way of

becoming familiar with the vocabulary rather than just looking up the words or writing the definition down.

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SEL Competency

Examples Content Specific Activity &

Approach to SEL

Self-Awareness Self-Management

Social-Awareness

Relationship Skills

Responsible Decision-Making

Example practices that address Self-Awareness:

• Clearly state classroom rules

• Provide students with specific feedback regarding

academics and behavior

• Offer different ways to demonstrate understanding

• Create opportunities for students to self-advocate

• Check for student understanding / feelings about

performance

• Check for emotional wellbeing

• Facilitate understanding of student strengths and

challenges

During the first week of school, establish

shared classroom rules and expectations and

consequences so that students can see the

impact of their own actions and behaviors

on outcomes.

Ask students to identify their own personal

interests, strengths, and weaknesses, in math

using a graphic organizer.

Encourage students to use mathematical

representations to elaborate their

understanding of decimals and the four

operations. (For example: Create a bar

graph on how they rate their ability to add,

subtract, multiply and divide decimals)

Self-Awareness

Self-Management Social-Awareness

Relationship Skills

Responsible Decision-Making

Example practices that address Self-Management:

• Encourage students to take pride/ownership in work

and behavior

• Encourage students to reflect and adapt to

classroom situations

• Assist students with being ready in the classroom

• Assist students with managing their own emotional

states

Teach self-management techniques such as

belly breathing, yoga positions, counting to

ten, self-talk, relaxation exercises or mental

rehearsal to help students develop concrete

techniques for managing their own stress or

anxiety.

Students will create goals based off of their

perceived math strengths and weaknesses.

They can be taught to self-assess progress

toward their learning goals, which is a

powerful strategy that promotes academic

growth. This should be an instructional

routine within the Independent phase of the

Balanced Instructional Math block.

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Have students create a graph to show

progress in SuccessMaker or Imagine Math

Facts.

Self-Awareness

Self-Management

Social-Awareness

Relationship Skills

Responsible Decision-Making

Example practices that address Social-Awareness:

• Encourage students to reflect on the perspective of

others

• Assign appropriate groups

• Help students to think about social strengths

• Provide specific feedback on social skills

• Model positive social awareness through

metacognition activities

Routinely ask students to talk about the

kinds of problems and puzzles they like to

solve and why. This will allow for students

to begin to see the ways in which other

students have similar or different

preferences and learn from each other about

why other concepts and problem-solving

approaches are interesting. Utilize games

that require math skills and promote

working together to solve them.

Model and routinely promote a rule or norm

of treating others the way you would want

to be treated.

Build respect for diversity in the classroom

by having students share their different

perspectives on situations or solution

strategies. (Teachers: They can engage

students in purposeful sharing of

mathematical ideas, reasoning and

approaches using varied representations.

Students: They can seek to understand the

approaches used by peers by asking

clarifying questions, trying out others’

strategies and describing the approaches

used by others.)

Self-Awareness

Self-Management Example practices that address Relationship

Skills:

Teach lessons on how to ask a peer or

teacher for help. Brainstorm with students

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Social-Awareness

Relationship Skills

Responsible Decision-Making

• Engage families and community members

• Model effective questioning and responding to

students

• Plan for project-based learning

• Assist students with discovering individual

strengths

• Model and promote respecting differences

• Model and promote active listening

• Help students develop communication skills

• Demonstrate value for a diversity of opinions

the most effective ways to request help.

Discuss and practice ways to say “thank

you.” Also teach students how to apologize

sincerely when frustrated, especially when

students express frustration inappropriately.

Develop speaking and listening skills (e.g.,

how to ask questions, how to listen well,

and how to effectively seek help when one

doesn’t understand academic content) and

the ability to collaborate to solve problems.

Self-Awareness

Self-Management

Social-Awareness

Relationship Skills

Responsible Decision-Making

Example practices that address Responsible

Decision-Making:

• Support collaborative decision making for

academics and behavior

• Foster student-centered discipline

• Assist students in step-by-step conflict resolution

process

• Foster student independence

• Model fair and appropriate decision making

• Teach good citizenship

Allow the students to select their own

strategy and/or tool to solve the problem.

(For example: Students can use a number

line, partial products, or area model to

multiply decimals.)

Teachers model and set the expectations for

the students to consistently assume

responsibility for following procedures for

independent and/or cooperative group work

and for the students to hold themselves

accountable for contributing productively to

their own learning.

Teacher models organization and homework

study skills for the students to be able to

independently make more positively

productive decisions. (For example: Show

students how to set up their binders,

creation of interactive notebooks, and study

skills.)

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Differentiated Instruction

Accommodate Based on Students Individual Needs: Strategies

Time/General

Extra time for assigned tasks

Adjust length of assignment

Timeline with due dates for

reports and projects

Communication system

between home and school

Provide lecture notes/outline

Processing

Extra Response time

Have students verbalize steps

Repeat, clarify or reword

directions

Mini-breaks between tasks

Provide a warning for

transitions

Partnering

Comprehension

Precise processes for balanced

math instruction model

Short manageable tasks

Brief and concrete directions

Provide immediate feedback

Small group instruction

Emphasize multi-sensory

learning

Recall

Teacher-made checklist

Use visual graphic organizers

Reference resources to

promote independence

Visual and verbal reminders

Graphic organizers

Assistive Technology

Computer/whiteboard

Tape recorder

Video tape

Tests/Quizzes/Grading

Extended time

Study guides

Shortened tests

Read directions aloud

Behavior/Attention

Consistent daily structured

routine

Simple and clear classroom

rules

Frequent feedback

Organization

Individual daily planner

Display a written agenda

Note-taking assistance

Color code materials

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Differentiated Instruction Accommodate Based on Content Specific Needs:

Teacher modeling

Use base ten manipulatives and color tiles to represent whole numbers, fractions and decimals while solving word problems.

Use centimeter, inch and unifix cubes to represent and solve real world problems.

Use base ten manipulatives to explore powers of ten.

Use interactive technology to improve fluency with multiplication.

Use rectangular arrays of objects to calculate whole number quotients.

Use concrete models to add, subtract, multiply and divide decimals to hundredths.

Use interactive technology to support steps in evaluating numerical expressions with parenthesis, brackets and braces.

Chart academic vocabulary with visual representations

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Interdisciplinary Connections

Model interdisciplinary thinking to expose students to other disciplines.

Social Studies Connection: Millions and Billions of People Social Studies Standard 6.1.4.a

The purpose of this task is to help students understand the multiplicative relationship between commonly used large numbers (millions and

billions) by using their understanding of place value. The population estimates come from Historical Estimates of World Population from

the US Census Bureau. https://www.illustrativemathematics.org/content-standards/5/NBT/A/1/tasks/1931

English Language Arts: Hogwarts House Cup Language Arts Standard RL.5.4, RL.5.2

Students explore writing expressions and equations as well as simplifying expression in the context of points earned at Hogwarts. This task

should be carried over several class periods as these ideas are developed. This task could be introduced by reading short passages from one

of the Harry Potter books where points are given or deducted.

https://www.georgiastandards.org/Georgia-Standards/Frameworks/5th-Math-Unit-1.pdf

Science: Preparing a Prescription Science Standard 5-PS1-2

Students should understand how to use grid paper and partial products area models to determine multiplication products with numbers larger

than 10. Use this task or another one similar to it to help students make the transition from depending on manipulatives for determining

products of larger numbers to being able to determine these products through self-made diagrams.

https://www.georgiastandards.org/Georgia-Standards/Frameworks/5th-Math-Unit-1.pdf

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Enrichment

What is the purpose of Enrichment?

The purpose of enrichment is to provide extended learning opportunities and challenges to students who have already mastered, or can quickly master,

the basic curriculum. Enrichment gives the student more time to study concepts with greater depth, breadth, and complexity.

Enrichment also provides opportunities for students to pursue learning in their own areas of interest and strengths.

Enrichment keeps advanced students engaged and supports their accelerated academic needs.

Enrichment provides the most appropriate answer to the question, “What do you do when the student already knows it?”

Enrichment is…

Planned and purposeful

Different, or differentiated, work – not just more work

Responsive to students’ needs and situations

A promotion of high-level thinking skills and making connections

within content

The ability to apply different or multiple strategies to the content

The ability to synthesize concepts and make real world and cross-

curricular connections

Elevated contextual complexity

Sometimes independent activities, sometimes direct

instruction

Inquiry based or open ended assignments and projects

Using supplementary materials in addition to the normal range

of resources

Choices for students

Tiered/Multi-level activities with flexible groups (may change

daily or weekly)

Enrichment is not…

Just for gifted students (some gifted students may need

intervention in some areas just as some other students may need

frequent enrichment)

Worksheets that are more of the same (busywork)

Random assignments, games, or puzzles not connected to the

content areas or areas of student interest

Extra homework

A package that is the same for everyone

Thinking skills taught in isolation

Unstructured free time

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Assessments

Required District/State Assessments Unit Assessments

NJSLA

SGO Assessments

Suggested Formative/Summative Classroom Assessments Describe Learning Vertically

Identify Key Building Blocks

Make Connections (between and among key building blocks)

Short/Extended Constructed Response Items

Multiple-Choice Items (where multiple answer choices may be correct)

Drag and Drop Items

Use of Equation Editor

Quizzes

Journal Entries/Reflections/Quick-Writes

Accountable talk

Projects

Portfolio

Observation

Graphic Organizers/ Concept Mapping

Presentations

Role Playing

Teacher-Student and Student-Student Conferencing

Homework

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New Jersey Student Learning Standards

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as

18932 + 921, without having to calculate the indicated sum or product.

5. NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and

1/10 of what it represents in the place to its left.

5. NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the

placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

5. NBT.A.3 Read, write, and compare decimals to thousandths.

5.NBT.A.3a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100

+ 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

5.NBT.A.3b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the

results of comparisons.

5. NBT.A.4 Use place value understanding to round decimals to any place.

5. NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. (Benchmarked)

5. NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on

place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using

equations, rectangular arrays, and/or area models.

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value,

properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning

used.

25 | P a g e

Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

26 | P a g e

Grade: Five

Unit: 1 Topic: Understanding the Place Value

System

New Jersey Student Learning Standards (NJSLS):

5.OA.A.1, 5.OA.A.2, 5.NBT.A.1, 5.NBT.A.2, 5.NBT.A.3, 5.NBT.A 4, 5.NBT.B.5, 5.NBT.B.6, 5.NBT.B.7

Unit Focus:

Write and interpret numerical expressions

Understand the place value system

Perform operations with multi-digit whole numbers and with decimals to hundredths

New Jersey Student Learning Standard: 5.OA.A.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Student Learning Objective 1: Evaluate numerical expressions that contain parentheses, brackets and braces.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 8

5.OA.1

Expressions have depth

no greater than two,

e.g., 3[5 + (8 ÷ 2)] is

acceptable but 3[5 + (8

÷ {42})] is not.

Evaluate numerical expressions that

include grouping symbols (parentheses,

brackets or braces).

Evaluate numerical expressions that

include nested grouping symbols. For

example, 3 x [5 + 7 - 3)].

Create numerical expressions by using

cards, number cubes and grouping

symbols to calculate varying target

numbers.

Parenthesis, brackets or braces

can be used with expressions to

vary results.

Placing of parenthesis forces us

to complete the computations in

a different order than we would

according to the standard order

of operations.

Grouping symbols can reverse

the conventional practice of

performing

Bowling of Numbers

Expression Sets

Numerical Expressions

Order of Operations

Target Number Dash

Target Number

Trick Answers

27 | P a g e

SPED Strategies:

Provide foldables with sample

expressions.

Present information through different

modalities.

Provide students with ample

opportunities to explore numerical

expressions with mixed operations.

Review rules and provide a color coded

anchor chart.

ELL Strategies:

Review and provide a model for students

with illustrations and drawings:

Provide foldables with sample

expressions.

Review rules and provide color coded

anchor chart.

Provide students with:

Multilingual Math Glossary

Visuals and anchor charts

multiplication/division before

addition/subtraction.

You can remove parentheses,

brackets, and braces when they

do not change the order of

operations.

Why is order of operations

important?

How do grouping symbols

affect the order of operations?

Using Operations and

Parentheses

28 | P a g e

New Jersey Student Learning Standard: 5.OA.A.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as

18932 + 921, without having to calculate the indicated sum or product.

Student Learning Objective 2: Write numerical expressions when given a verbal description or word problem; interpret numerical

expressions without evaluating them.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 7

MP 8

5.OA.2-1

5.OA.2-2

Review and practice using multiplication

symbols (x, *, •).

Emphasize the difference between an

expression and an equation.

Provide explicit vocabulary instruction

for expression, equation grouping

symbols.

Translate verbal expressions to numerical

expressions.

Write simple numerical expressions from

verbal expressions without evaluating the

expression.

Translate numerical expressions to verbal

expressions.

The difference between an

equation and an expression

is that equations contain an

equal sign (=) and a result.

Equations and expressions

are needed to solve real

world situations.

How are numerical

expressions written and

interpreted?

Some mathematical phrases

can be represented using a

variable in an algebraic

expression.

What is the difference

between an expression and

an equation?

Comparing Products

Expression Puzzle

Hogwarts House Cup

The Beanbag Dartboard

Video Games Scores

29 | P a g e

SPED Strategies:

Listen to and demonstrate understanding

by writing the numerical expressions of a

given word problem or scenario, which

uses key technical vocabulary in a series

of simple sentences.

Use words to interpret the numerical

expression.

Allow students to use calculators to

determine the value of given expressions. Include mnemonics to assist students

with remembering the order that

expressions should be solved.

ELL Strategies:

Listen to and demonstrate understanding

by writing the numerical expressions of a

given word problem or scenario which

uses key technical vocabulary in a series

of simple sentences.

Use words to interpret the numerical

expression.

Allow students to use calculators to

determine the value of given

expressions.

Include mnemonics to assist students

with remembering the order that

expressions should be solved.

How can I write an

expression that

demonstrates a situation or

context?

In what kinds of real world

situations might we use

equations and expressions?

30 | P a g e

New Jersey Student Learning Standard: 5.NBT.A.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and

1/10 of what it represents in the place to its left.

Student Learning Objective 3: Explain that a digit in one place represents 1/10 of what it would represent in the place to its left and ten

times what it would represent in the place to its right.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

MP 7

5.NBT.1

Tasks have “thin

context” or no context.

Tasks involve the

decimal point in a

substantial way (e.g.,

by involving a

comparison of a tenths

digit to a thousandths

digit or a tenths digit to

a tens digit).

In fourth grade, students examined the

relationships of the digits in numbers for

whole numbers only. In Grade 5, the

students extend this understanding to the

relationship of decimal fractions.

Students use base ten blocks, pictures of

base ten blocks, and interactive images of

base ten blocks to manipulate and

investigate the place value relationships.

They use their understanding of unit

fractions to compare decimal places and

fractional language to describe those

comparisons.

Quantitative relationships exist between

the digits in place value positions of a

multi-digit number.

Use manipulatives, drawings or

equations to represent how many of a

certain decimal unit will comprise one

unit.

Unit decimal fractions are

named according to the

number of same-sized items

needed to compose a value

of one. Non-unit decimal

fractions are named

according to:

a) the number of same-

sized items needed to

compose a value of one

b) how many of those items

are being considered (ten

0.1 are needed to make

one).

Each place value to the left

of another is ten times

greater than the one to the

right

IFL PBA:

Decimal Place Value

Additional Tasks:

Kipton’s Scale

Millions and Billions of

People

Tenths and Hundredths

Value of a Digit

Which number is it?

31 | P a g e

Students model each situation with

diagrams and or numbers.

Have students reason about the

magnitude of numbers. The tens place is

ten times as much as the ones place, and

the ones place is 1/10 the size of the tens

place.

Extensive modeling and practice with

whole numbers and decimals is needed to

solidify this concept.

Define a number in one place as 1/10 of

its value in the place to its left.

Define a number in one place as 10 times

its value in the place to its right.

SPED Strategies:

Number cards, number cubes, spinners

and other manipulatives can be used to

generate decimal numbers. For example,

have students roll three number cubes,

then create the largest and smallest

number to the thousandths place. Ask

Each place value to the right

is 1/10 the place value to the

left.

What changes the value of a

digit?

32 | P a g e

students to represent the number with

numerals and words.

Present students with a millions through

thousandths place value chart to utilize as

a reference.

Use base ten blocks to distinguish

between place value labels such as

hundredths and thousandths.

ELL Strategies:

Number cards, number cubes, spinners

and other manipulatives can be used to

generate decimal numbers.

Proportional materials such as base ten

blocks can help English language

learners distinguish between place value

labels like hundredths and thousandths

more easily by offering clues to their

relative sizes.

Provide mental imagery for mathematical

idea.

Provide students with:

Multilingual Math Glossary

Visuals and anchor charts

33 | P a g e

New Jersey Student Learning Standard: 5.NBT.A.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement

of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Student Learning Objective 4: Explain patterns in the number of zeros in the product when a whole number is multiplied by a power of 10;

represent powers of 10 using whole-number exponents.

Modified Student Learning Objectives/Standards: M.EE.5.NBT.A.2: Use the number of zeros in numbers that are powers of 10 to determine which values are equal, greater than, or less than.

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

MP 7

5.NBT.2-2

For the explain aspect of

5.NBT.2 tasks do not

involve reasoning about

place value in service of

some other goal (e.g., to

multiply multi-digit

numbers). Rather, tasks

involve reasoning directly

about the place value

system, in ways consistent

with the indicated content

scope.

Students reason that not just the

decimal point is moving but that you

are multiplying or dividing to make

the number 10 times greater or less.

Since we are multiplying by a power

of 10, the decimal point moves to the

right.

A pattern is created when a number is

multiplied by a power of 10 and

students show their solutions through

multiple representations.

Sets of ten, 100, and so forth must be

perceived as single entities when

interpreting numbers using place

value. Write whole number exponents to

denote powers of 10.

Multiplying a whole number by power

of 10 will result in a product with as

many 0s at the end as were in the

power of 10.

Sets of ten, one hundred, and so forth

must be perceived as single entities

when interpreting numbers using place

value (e.g., 1 hundred is one group, it is

10 tens or 100 ones).

When multiplying a number by a

power of ten, the exponent does not

indicate the number of zeroes in the

product should be emphasized. For

example: 30 x 102 = 3,000. The

exponent indicates the number of zeros

added to the number.

What pattern is our number system

based on?

Distance from the Sun

What Comes Next?

Multiplying a Whole

Number by a Power of 10

Multiplying a Decimal by

a Power of 10

34 | P a g e

Illustrate and explain a pattern for

how multiplying or dividing any

decimal by a power of 10 relates to

the placement of the decimal point.

SPED Strategies:

Consider allowing students to

research and present to classmates the

origin of number names like googol

and googolplex.

Allow students to explore with a

calculator.

Provide place value chart as a visual.

ELL Strategies:

Explain orally and in writing the

patterns of the number of zeros and

the placement of the decimal point in

a product or quotient in L1(student’s

native language) and/or use gestures,

pictures and selected words.

Provide:

Multilingual Math Glossary

Interactive Word/Picture Wall

Visuals and anchor charts

Patterns are created when we multiply

a number by powers of ten.

What happens when we multiply a

number by powers of ten?

How does multiplying a whole number

by a power of ten affect the product?

35 | P a g e

New Jersey Student Learning Standard: 5.NBT.B.5: Fluently multiply multi-digit whole numbers using the standard algorithm.

Student Learning Objective 5: Use the standard algorithm to multiply a whole number of up to a four digits by a whole number of up to

two digits.

Modified Student Learning Objectives/Standards: M.EE.5.NBT.5 Multiply whole numbers up to 5 x 5.

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

MP 7

MP 8

5.NBT.5

Tasks assess accuracy.

The given factors are

such as to require an

efficient/standard

algorithm (e.g., 26

4871).

Factors in the task do

not suggest any obvious

ad hoc or mental

strategy (as would be

present for example in a

case such as 7250 40).

Tasks do not have a

context.

For purposes of

assessment, the

possibilities are 1-digit

x 2-digit, 1-digit x 3-

digit, 2-digit x 3-digit,

or 2-digit x 4-digit.

Tasks are not timed.

Being able to estimate and mentally

multiply a 2- or 3- digit number by a1-digit

number to determine reasonable answers.

Students often overlook the place value of

digits, or forget to use zeros as place

holders, resulting in an incorrect partial

product and ultimately the wrong answer.

Students should use multiple strategies:

Area model:

225 x 12

SPED Strategies:

Allow students to use graphing paper to

assist with the lining up of the number.

What are different models or

strategies for multiplication?

make equal sets/groups

create fair shares

represent with objects,

diagrams, arrays, area

models

identify multiplication

patterns

What are efficient methods for

finding products?

use identity and zero

properties of multiplication

apply doubling/halving

concepts to multiplication

(ex: 16x5 is half of 16 x10)

demonstrate fluency with

multiplication facts of

factors 0-12

identify factors/divisors of a

number and multiples of a

number

IFL Tasks:

“Decimal

Operations:

Multiplication and

Division”

Additional Tasks:

Multiplication Three

in a Row

Preparing a

Prescription

Field Trip Funds

Elmer’s

Multiplication Error

36 | P a g e

Allow students to use other strategies to

assist with multiplying.

Allow students to explore with a calculator.

Provide visuals and anchor charts for

students to reference.

ELL Strategies:

Allow students to use other strategies to

assist with multiplying.

Provide:

Multilingual Math Glossary

Interactive Word/Picture Wall

Visuals and anchor charts

multiply any whole number

by a two-digit factor

use and examine algorithms:

partial product and

traditional

New Jersey Student Learning Standard: 5.NBT.B.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place

value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using

equations, rectangular arrays, and/or area models.

Student Learning Objective 6: Calculate patterns in the number of quotients of whole numbers with 4-digit dividends and 2-digit divisors;

explain and represent calculations with equations, rectangular arrays, and area models.

Modified Student Learning Objectives/Standards: M.EE.5.NBT.6-7: Illustrate the concept of division using fair and equal shares.

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 3

5.NBT.6

Tasks do not have a

context.

Division can mean equal sharing or

partitioning of equal groups or arrays and

is the same as repeated subtraction. The

If there is a whole number of

groups, repeated addition can be

used because the size of each group

(i.e., the unit or non-unit decimal

Are These All 364

Division Four in a

Row

37 | P a g e

MP 4

MP 5

MP 7

Tasks involve 3- or 4-

digit dividends and

one- or two-digit

divisors.

quotient can be thought of as a missing

factor.

Use problems where the divisor is the

number of groups and where the dividend

is the size of the groups.

Reinforce the difference between the

divisor and dividend.

Ensure students are using rectangular

arrays and area models to represent their

calculations.

Example:

There are1,716 students participating in

Field Day. They are put into teams of 16.

How many teams get created? If you

have left over students, what do you do

with them?

fraction) can be added repeatedly

(e.g., 14 groups of 0.1).

If there is a whole number of

groups, the smaller the amount in

each group, the smaller the

product. The larger the amount in

each group, the larger the product.

How can estimating help us when

solving division problems?

What strategies can we use to

effectively solve division

problems?

Lion Hunt

38 | P a g e

SPED Strategies: Fluency- Include fluency practice to allow

students opportunities to retain past

number understandings and to sharpen

39 | P a g e

those understandings needed for

subsequent work.

Consider including body movements to

accompany skip-counting exercises (e.g.,

jumping jacks, toe touches, arm stretches,

or dance movements like the Macarena).

Interactive math journals

Allow students to use various models and

strategies to divide.

Provide students with

multiplication reference sheet.

EL ELL Strategies:

Illustrations/diagrams/drawings and

selected words

Provide color coded examples of

equations, rectangular arrays, and area

models.

Anchor charts

Interactive math journals

Provide students with

multiplication reference sheet.

40 | P a g e

New Jersey Student Learning Standard: 5.NBT.B.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value,

properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning

used.

Student Learning Objective 7: Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawings and

strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; explain the reasoning used,

relating the strategy to the written method.

Modified Student Learning Objectives/Standards: M.EE.5.NBT.6-7: Illustrate the concept of division using fair and equal shares.

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 3

MP 4

MP 5

MP 7

5.NBT.7-1

Tasks do not have a

context.

Only the sum is

required. Explanations

are not assessed here.

Prompts may include

visual models, but

prompts must also

present the addends as

numbers, and the answer

sought is a number, not a

picture.

Each addend is greater

than or equal to 0.01 and

less than or equal to

99.99.

20% of cases involve a

whole number—either

the sum is a whole

number, or else one of

Draw diagrams or use number

reasoning for each situation.

Model each word problem with

diagrams and/or numbers.

Use knowledge of repeated addition of

decimal composites along with

decomposition of decimal composites

to solve problems.

Use manipulatives, drawings, or

equations to represent how many of a

certain decimal unit comprise one

whole.

Example:

4 – 0.3

The wholes must be divided into

tenths.

If there is less than a whole number

of groups, (e.g., 0.5 groups or 0.25

groups), the product will be less than

the amount in the group and the

number of groups because less than

one whole group is being utilized.

If there is more than a whole number

of groups, the product will be greater

than the amount in the group

because more than one group of the

decimal fraction/composite unit is

being utilized.

Either or both factors can be

decomposed to form equivalent

values (e.g., 2.3 x 6 is the same as 2

x 6) + (0.3 x 6).

Any representation (e.g., area

model/arrays, number lines, set, or

IFL Tasks:

“Decimal Operations:

Multiplication and

Division”

Additional Tasks:

Ten is the Winner

The Value of

Education

Hanging by a Hair

Field Trip

Base Ten Activity

Clay Boxes

Road Trip

41 | P a g e

the addends is a whole

number presented

without a decimal point.

(The addends cannot

both be whole numbers.)

5.NBT.7-2

Tasks do not have a

context.

Only the difference is

required.

Prompts may include

visual models, but

prompts must also

present the subtrahend

and minuend as

numbers, and the answer

sought is a number, not a

picture.

The subtrahend and

minuend are each

greater than or equal to

0.01 and less than or

equal to 99.99. Positive

differences only. (Every

included subtraction

problem is an unknown-

addend problem

included in 5.NBT.7-1.)

20% of cases involve a

whole number—either

the difference is a whole

number, or the

subtrahend is a whole

0.22 x 5

2.4 ÷ 4

1.6 ÷ 0.2

Estimate decimal computation before

computing with pencil and paper.

When answering a division problem

involving a whole quotient, it is

important for students to be able to

decide whether the context requires the

result to be reported as a whole

number with remainder or a mixed

number/decimal.

SPED Strategies:

Reduce length of assignment and

provide a different instructional mode

of delivery.

equations) of repeated

addition/multiplication of a number

by a number illustrates the number

of groups, the size of each group

(i.e., the unit or non-unit decimal

fraction), and the resulting product

or partial products.

When dividing by a decimal number

less than one, the quotient will be

more than the dividend because

either; you are making groups of an

amount less than one; or you are

making less than one group.

Any representation of the division of

a number divided by a number (area

model, number line, or set model)

highlights the starting amount, the

final amount, and the impact of the

division.

What are some ways you can add,

subtract, multiply and divide

decimals?

Hit the Target

Competitive Eating

Records

Rolling Around with

Decimals

Watch Out for

Parenthesis

42 | P a g e

number presented

without a decimal point,

or the minuend is a

whole number presented

without a decimal point.

(The subtrahend and

minuend cannot both be

whole numbers.)

5.NBT.7-3

Tasks do not have a

context.

Only the product is

required.

Prompts may include

visual models, but

prompts must also

present the factors as

numbers, and the

answer sought is a

number, not a picture.

Each factor is greater

than or equal to 0.01

and less than or equal

to 99.99.The product

must not have any

non-zero digits

beyond the

thousandths place.

(For example, 1.67 x

0.34 = 0.5678 is

excluded because the

product has an 8

beyond the

Increase one-on-one time.

Utilize working contact between you

and student at-risk.

ELL Strategies:

Model the process. Talk aloud while

solving problems on the overhead or

chalkboard to show the thinking

process and common errors.

Have students explain their thinking

process aloud to a classmate while

solving a problem. Integrate reading and writing through

the use of journals, learning logs,

poems, literature, etc.

43 | P a g e

thousandths place; cf.

5.NBT.3, and see p.

17 of the Number and

Operations in Base

Ten Progression

document.)

Problems are 2-digit x

2-digit or 1-digit by 3-

or 4-digit. (For

example, 7.8 x 5.3 or

0.3 x 18.24.)

20% of cases involve

a whole number—

either the product is a

whole number, or else

one factor is a whole

number presented

without a decimal

point. (Both factors

cannot both be whole

numbers.)

5.NBT.7-4

Tasks do not have a

context.

Only the quotient is

required.

Prompts may include

visual models, but

prompts must also

present the dividend and

divisor as numbers, and

the answer sought is a

number, not a picture.

44 | P a g e

Divisors are of the form

XY, X0, X, X.Y, 0.XY,

0.X, or 0.0X (cf.

5.NBT.6), where X and

Y represent non-zero

digits. Dividends are of

the form XY, X0, X,

XYZ.W, XY0.Z, X00.Y,

XY.Z, X0.Y, X.YZ,

X.Y, X.0Y, 0.XY, or

0.0X, where X, Y, Z,

and W represent non-

zero digits.

Quotients are either

whole numbers or else

decimals terminating at

the tenths or hundredths

place. (Every included

division problem is an

unknown-factor problem

included in 5.NBT.7-3.)

20% of cases involve a

whole number—either

the quotient is a whole

number, or the dividend

is a whole number

presented without a

decimal point, or the

divisor is a whole

number presented

without a decimal point.

(If the quotient is a

whole number, then

neither the divisor nor

45 | P a g e

the dividend can be a

whole number)

New Jersey Student Learning Standard: 5.NBT.A.3: Read, write, and compare decimals to thousandths.

Student Learning Objective 8: Compare two decimals to thousandths using >, =, and < for numbers presented as base ten numerals, number names, and/or in expanded form.

Modified Student Learning Objectives/Standards: M.EE.5.NBT.3: Compare whole numbers up to 100 using symbols (<, >, =).

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP2

MP 5

MP 6

MP 7

5.NBT.3a

Tasks assess conceptual

understanding, e.g., by

including a mixture of

expanded form, number

names, and base ten

numerals.

Tasks have “thin context”

or no context.

5.NBT.3b

Tasks have “thin context”

or no context.

Tasks assess conceptual

understanding, e.g., by

including a mixture (both

within and between

items) of expanded form,

Read and write decimals to the

thousandths using base 10

numerals.

Read and write decimals to the

thousandths using expanded form

(with fractions of 1/10, 1/100 and

1/1000 to denote decimal places).

Use concrete models,

representations, and number lines

to extend understanding of

decimals to the thousandths.

Comparing decimals is simplified

when students use their

understanding of fractions to

compare decimals.

Some students may believe that a

longer number is a larger number.

Like whole numbers, the location of a

digit in decimal numbers determines the

value of the digit.

The longer the number does not

necessarily indicate a greater number.

How do we compare decimals?

How do we round decimals?

PBA:

Decimal Place Value

Additional Tasks:

Are These Equivalent

to 9.52?

Decimal Designs

Decimal Garden

Decimal Lineup

High Roller Revisited

46 | P a g e

number names, and base

ten numerals.

Use several examples with

different representations.

SPED Strategies: Present information through

different modalities.

Provide mental imagery for the

mathematical idea.

ELL Strategies: Review prerequisite skills and

concepts.

Provide opportunities for project

based assignments.

New Jersey Student Learning Standard: 5.NBT.A.4: Use place value understanding to round decimals to any place.

Student Learning Objective 9: Round decimals to any place value.

Modified Student Learning Objectives/Standards: M.EE.5.NBT.4: Round two-digit whole numbers to the nearest 10 from 0 – 90

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

MP 7

5.NBT.4

Tasks have “thin context”

or no context.

Use horizontal and vertical number

lines showing the placement of

decimals and determine the relative

values of decimal numbers.

Use the position of a number on a

number line to round the number

Rounding decimals is dependent upon

the accuracy and level of precision

needed for decision making.

Estimating by rounding can be used to

determine a reasonable solution.

Batter Up

Check This

Decimals

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with and without knowing its exact

value. Use benchmark numbers to

support this work.

Justify the reasonableness of a

solution using estimation and

benchmarks.

SPED Strategies: Read, listen to, and understand a

given word problem or math

question dealing with rounding

decimals to any place that includes

key technical vocabulary in a series

of simple sentences.

Charts/Posters

Present information through

illustrations/diagrams/drawings

Visual models and anchor charts

should be presented.

Provide mental imagery for the

mathematical idea

ELL Strategies: Consider showing both a horizontal

and vertical line and comparing

their features so that students can

see the parallels and gain comfort

in the use of the vertical line.

How can you round whole numbers

and decimals to any place value

position?

Reasonable Rounding

Round to Tenths and

Hundredths

The Right Cut

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Integrated Evidence Statements 5.NBT.A.Int.1 Demonstrate understanding of the place value system by combining or synthesizing knowledge and skills articulated in

5.NBT.A

5.NBT.Int.1 Perform exact or approximate multiplications and/or divisions that are best done mentally by applying concepts of place

value, rather than by applying multi-digit algorithms or written strategies.

Tasks do not have a context.

5.Int.1 Solve one-step word problems involving multiplying multi-digit whole numbers.

The given factors are such as to require an efficient/standard algorithm (e.g., 726 4871). Factors in the task do not suggest any obvious

ad hoc or mental strategy (as would be present for example in a case such as 7250 400).

For purposes of assessment, the possibilities for multiplication are 1-digit x 2- digit, 1-digit x 3-digit, 2-digit x 3-digit, 2-digit x 4-digit, or

3-digit x 3-digit.

Word problems shall include a variety of grade-level appropriate applications and contexts.

5.Int.2 Solve word problems involving multiplication of three two-digit numbers.

The given factors are such as to require an efficient/standard algorithm (e.g., 76 48 39). Factors in the task do not suggest any obvious

ad hoc or mental strategy y (as would be present for example in a case such as 50 20 15).

Word problems shall include a variety of grade-level appropriate applications and contexts.

5.C.1-1 Base explanations/reasoning on place value and/or understanding 6521of operations.

Tasks do not have a context.

5.C.2-1 Base explanations/reasoning on the relationship between multiplication and division. Content Scope: Knowledge and skills

articulated in 5.NBT.6

5.C.2-2 Base explanations/reasoning on the relationship between addition and subtraction or the relationship between multiplication

and division. Content Scope: Knowledge and skills articulated in 5.NBT.7

5.C.1.2 Base explanations/reasoning on the properties of operations. Content Scope: Knowledge and skills articulated in 5.NBT.7

Tasks do not have a context.

Students need not use technical terms such as commutative, associative, distributive, or property. Unneeded parentheses should not be used. For example, use 4 + 3 x 2 rather than 4 + (3 x 2).

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Integrated Evidence Statements 5.C.3 Reason about the place value system itself.

Tasks do not involve reasoning about place value in service of some other goal (e.g., to multiply multi-digit numbers). Rather, tasks

involve reasoning directly about the place value system, in ways consistent with the indicated content scope.

5.C.4-3 Base arithmetic explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed

by the student in her response), connecting the diagrams to a written (symbolic) method. Content Scope: Knowledge and skills articulated

in 5.NBT.7

5.C.5-3 Base explanations/reasoning on a number line diagram (whether provided in the prompt or constructed by the student in her

response).

5.C.7-4 Distinguish correct explanation/reasoning from that which is flawed, and – if there is a flaw in the argument – present corrected

reasoning. (For example, some flawed ‘student’ reasoning is presented and the task is to correct and improve it.)

Tasks may have scaffolding 1, if necessary, in order to yield a degree of difficulty appropriate to Grade.

5.D.2 Solve multi-step contextual problems with degree of difficulty appropriate to Grade 5, requiring application of knowledge and

skills articulated in 4.OA, 4.NBT, 4.NF, 4.MD

Tasks may have scaffolding, if necessary, in order to yield a degree of difficulty appropriate to Grade 5. ii) Multi-step problems must have at

least 3 steps.

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Unit 1 Vocabulary

Algorithm

Area Model

Base-ten Numerals

Benchmark Numbers

Braces

Brackets

Decimal

Decimal Point

Diagrams

Divide

Dividend

Divisor

Equation

Expanded Form

Exponents

Expression

Estimating

Models

Manipulatives

Expression

Estimating

Hundredths

Models

Manipulatives

Multiplicand

Multiplier

Multiply

Numerical Expressions

Number Lines

Number Name

Order of Operations

Partial Product

Partial Quotient

Place Value

Precision

Product

Properties of Operations

Quotient

Repeated Subtraction

Representations

Rectilinear Array

Remainder

Repeated addition

Rounding

Tenths

Thousandths

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References & Suggested Instructional Websites

Imagine Math Facts: https://www.imaginelearning.com/programs/math-facts

SuccessMaker: https://paterson1991.smhost.net/lms/sm.view

National Council of Teachers of Mathematics - This website contains activities and lessons, and virtual manipulatives organized by strand.

http://illuminations.nctm.org

Internet for Classrooms – This site is a list of math sites for lessons and teacher tools. www.internet4classrooms.com

The National Library of Virtual Manipulatives has tutorials and virtual manipulatives for the classroom.

http://nlvm.usu.edu/en/nav/index.html

Georgia Standards contain exceptional tasks and curriculum support. www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx

Illustrative Mathematics is a library of tasks linked to Common Core State Standards. www.illustrativemathematics.org/

Inside Mathematics site contains tools for teachers, classroom videos, common core resources, rubric scored student samples, problems of the day

and performance tasks. http://www.insidemathematics.org

K-5 Math Teaching Resources site contains free math teaching resources, games, activities and journal tasks.

http://www.k-5mathteachingresources.com.

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Field Trip Ideas MoMath/Museum of Mathematics: Mathematics illuminates the patterns and structures all around us. The dynamic exhibits gallery, and

programs will stimulate inquiry, spark curiosity, and reveal the wonders of mathematics. MoMath has innovative exhibits that will engage

folks from 105 to 5 years old (and sometimes younger), but with a special emphasis on activities for 4th through 8th graders.

http://momath.org/

Liberty Science Center: Student mastery of STEM (Science, Technology, Engineering, and Mathematics) has never been more important.

Under the newest national standards, educators are required to instruct students in science and technology with active question-and-answer

pedagogy and hands-on investigation. Liberty Science Center understands educators’ needs and offers a full portfolio of age-appropriate,

curriculum-linked STEM programs suitable for preschoolers through technical school students, including pupils with special needs.

http://lsc.org/for-educators/

Discovery Times Square: New York City’s first large-scale exhibition center presenting visitors with limited-run, educational and immersive

exhibit experiences while exploring the world’s defining cultures, art, history, mathematics and events. http://discoverymuseum.org/

Great Falls National Park: A Revolutionary Idea: Cotton & silk for clothing; locomotives for travel; paper for books & writing letters;

airplanes, & and the mathematics needed for manufacturing. What do they have in common? They all came from the same place -

Paterson, NJ. http://www.nps.gov/pagr/index.htm

Passaic County Historical Society Lambert Castle Museum: The museum consists of mostly self-guided exhibits. Tours of the first floor will

be offered every half hour (as interest permits) or as visitors arrive on weekdays. If you are interested in a tour of the first floor, please let

our docent know when you arrive. From the financial cost to Carolina Lambert, area, distance and value (cost and loss) of artwork,

provides any aspiring historian with a deeper understanding and perspective of life, prosperity and loss throughout the rich history of

Lambert‘s Castle. http://www.lambertcastle.org/museum.html

The Paterson Museum: Founded in 1925, it is owned and run by the city of Paterson and its mission is to preserve and display the industrial

history Paterson. Manufacturing, finance and daily operation of this museum build a deeper understanding of the mathematics involved in

the building and decline of this great city. http://www.patersonnj.gov/department/index.php?structureid=16