STAFFORD TOWNSHIP SCHOOL DISTRICT Manahawkin, NJ ...€¦ · The Hershey's Kisses Addition Book by...

STAFFORD TOWNSHIP SCHOOL DISTRICT Manahawkin, NJ

Mathematics Curriculum

3rd Grade

“Aligned to the 2010 Common Core Standards”

This Curriculum is reviewed and updated annually as needed

This Curriculum was approved at the Board of Education Meeting August 16, 2012, January 3, 2013, January 6, 2014, January 7, 2015

MISSION Stafford Township School District, together with parents and community, shall provide a secure, nurturing environment that promotes a positive self-image through solid educational achievements that promote attainment of the core curriculum content standards and promotes behavior enabling our students to become life-long learners in a technological society. PHILOSOPHY The approach is to intermittently introduce mathematics concepts and revisit them more formally at greater levels of sophistication before mastery is expected. This spiraling effect allows students’ understanding to strengthen and grow, enabling them to develop secure foundations for mathematics content. The Everyday Mathematics program is utilized by the district to teach the students mathematics. Teachers are encouraged to supplement this program as needed to ensure that the mathematics curriculum is met and mastered for each grade level.

NJ World Class Standards New Jersey Core Curriculum Content Standards and Common Core

Standards In 2009 New Jersey’s Core Curriculum Content Standards were revised for Visual & Performing Arts, Health & Physical Education, Science, Social Studies, World Languages, Technology, and 21st Century Life and Careers. In 2010 New Jersey adopted the Common Core Standards for English/Language Arts and Mathematics. The revised standards align with the knowledge and skills needed by all students for post-secondary opportunities, which may encompass: four-year college, community college, technical training, military service, direct entry into the workplace, and an array of future careers—including some that are just being envisioned and others that are currently unforeseen. In addition, the standards by necessity reflect a framework for teaching and learning that responds to the needs of 21st-century digital learners by incorporating the 'new literacies' required in an innovation economy: the ability to effectively access, evaluate, and synthesize vast amounts of information; to apply knowledge and skills to personal, workplace, and global challenges; to work collaboratively in cross-cultural settings; to solve problems creatively; and to act ethically as citizens of the world community. These 21st-century student outcomes require a deeper understanding of academic content at much higher levels than ever before. The revised standards facilitate this in-depth learning in all content areas through the systematic and transparent integration of 21st-century knowledge, skills, and themes; global perspectives; cross content connections; and technology. Consequently, successful implementation of the revised standards requires rethinking of traditional curricular and assessment approaches, as well as the creation of 21st-century learning environments in which teachers and students work across and beyond traditional disciplines and boundaries as engaged co-learners, critical and creative thinkers, and problem solvers. -Source: http://www.njcccs.org

About the Common Core Standards… In 2009, 48 states, 2 territories and the District of Columbia signed a memorandum of agreement with the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO), committing to a state-led process – the Common Core State Standards Initiative (CCSSI). On June 2, 2010, the Common Core State Standards for English Language Arts/Literacy and Mathematics (CCSS) were released. Building on the strength of current state standards, the CCSS are designed to be focused, coherent, clear and rigorous; internationally benchmarked; anchored in college and career readiness; and evidence and research based. Created by teachers, parents, education experts, and other from across the country, the Common Core Standards provide clear, consistent expectations for what students should be learning at each grade in order to be prepared for college and career.

Ocean County Mathematics Curriculum

Aligned to the 2010 Common Core Standards Ocean County School Districts have a Curriculum Council that is comprised of Curriculum Directors and Administrators that meet on a regular basis to discuss state mandates and initiates. This Council works together to understand the state’s new mandates and how best to implement these initiatives. When New Jersey adopted the Common Core Standards, school districts were faced with the expensive task of updating their current curriculum to the new Common Core Standards. The Ocean County Curriculum Council came up with the idea for all school districts in Ocean County to pull their resources and work together to look at the Common Core Standards and together write a county curriculum that addresses the new standards. Over the summer of 2010 & 2011, teachers and administrators from most of the school districts in Ocean County met over several weeks to develop the County Mathematics Curriculum document. After the County Curriculum team wrote the document, each school district took the document, added in the resources and materials that were specific to their own district, and then had the document approved by their board of education.

OCEAN COUNTY MATHEMATICS

CURRICULUM

Content Area: Mathematics

Course Title: Elementary Grade Level: 3

Unit Plan 1: Number and Operations in Base Ten

Suggested: September/October

Unit Plan 2: Measurement and Data

(Represent and Interpret Data)

Suggested: October/November

Unit Plan 3: Operations and Algebraic

Thinking

Suggested: December/January

Unit Plan 4: Number and Operations-

Fractions

Suggested: February/March

Unit Plan 5: Geometry

Suggested: April

Unit Plan 6: Measurement and Data

(Volume, Mass and Area)

Suggested: May

Date Created: 2/08/12

L. Butler, Lacey Township School District

J. Carney, Long Beach Island Consolidated School District

D. Durning, Barnegat Township School District

D. Hewitt, Tuckerton Borough School District

C. McGettigan, Little Egg Harbor School District

M. Peck, Manchester Township School District

A. Phillips, Ocean Township School District

Board Approved on:

OCEAN COUNTY MATHEMATICS CURRICULUM

Unit Overview

Content Area: Mathematics Grade: 3

Domain: Number and Operations in Base Ten

Cluster:

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Cluster Summary

Prior to implementing rules for rounding, students need to have opportunities to investigate place value. A

strong understanding of place value is essential for the development of number sense and the subsequent

work that involves rounding numbers.

Building on previous understandings of the place value of digits in multi digit numbers, place value is used

to round whole numbers. Dependence on learning rules can be eliminated with strategies such as the use

of a number line to determine which multiple of 10 or of 100, a number is nearest (5 or more rounds up,

less than 5 rounds down). As students’ understanding of place value increases, the strategies for rounding

are valuable for estimating, justifying, and predicting the reasonableness of solutions in problem solving.

Strategies used to add and subtract two digit numbers are now applied to fluently add and subtract whole

numbers within 1000. These strategies should be discussed so that students can make comparisons and

move toward efficient methods.

Primary Interdisciplinary Connections:

Science, Social Studies, Language Arts, Technology, and 21st Century Life and Careers, For further

clarification see NJ Core Curriculum Content Standards at www.njcccs.org.

21st Century Themes:

Through instruction and life and career skills, all students acquire the knowledge and skills needed to

prepare for life as citizens and workers in the 21st Century. For further clarification see N.J. World Class

Standards at www.NJ.gov/education/aps/cccs/career/

Learning Targets

Content Standards

Number Common Core Standard for Mastery

3.NBT.1. Use place value understanding to round whole numbers to the nearest 10 or 100.

3.NBT 2 Fluently add and subtract within 1000 using strategies and algorithms based on place value,

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

3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 ×

60) using strategies based on place value and properties of operations.

Number Common Core Standard for Introduction

4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two

two-digit numbers, using strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area

models.

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-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.

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and

expanded form. Compare two multi-digit numbers based on meanings of the digits in each

place, using >, =, and < symbols to record the results of comparisons.

Unit Essential Questions

How do patterns in our place

value system assist in comparing

whole numbers?

How does understanding place

value help us add and subtract

large numbers?

How are the operations of

addition and subtraction related?

What are efficient methods for

finding multiples of numbers?

Unit Enduring Understandings

Students will understand that…

building and taking apart numbers provides a deep

understanding of the base 10 number system.

knowledge and use of place value for large numbers

provides context for distances.

addition and subtraction are related

Unit Objectives

Students will know…

place value and properties of operations

to add and subtract.

how to use a variety of estimation

strategies (e.g., rounding and mental

math) for estimating both quantities and

the result of computations to determine

if something is reasonable.

multiples of ten are based on place

value.

Unit Objectives

Students will be able to…

fluently add and subtract within 1000 using strategies and

algorithms based on place value, properties of operations,

and/or the relationship between addition and subtraction.

use place value to round whole numbers to the nearest 10 or

100.

multiply one digit whole numbers by multiples of 10.


Evidence of Learning

Formative Assessments

Teacher Observation

Performance Assessments

Exit Slips

Games

Anecdotal Records

Oral Assessments/Conferencing

Portfolio/Math Journals

Daily Classwork

Pre-assessments

Summative Assessments

Tests

Quizzes

National/State/District Wide Assessments

Modifications (ELLs, Special Education, Gifted and Talented)

Provide differentiated instruction as needed

Follow all IEP modifications/504 plan

Curriculum development Resources/Instructional Materials/Equipment Needed Teacher Resources:

Math Literature:

Place Value:

The King’s Commissioners by Aileen Freidman

Sir Cumference and the All the King's Tens by Cindy Neuschwander

Earth Day--Hooray! by Stuart Murphy

How Much is a Million? by David Schwartz

Addition and Subtraction:

The Mission of Addition by Brian P. Cleary

Addition Annie by David Gisler

The Hershey's Kisses Addition Book by Jerry Pallotta

Double Play: Monkeying Around with Addition by Betsy Franco

Subtraction:

The Action of Subtraction by Brian P. Cleary

Elevator Magic by Stuart J. Murphy

Subtraction Action by Loreen Leedy

Websites:

www.mathwire.com Offers various activities and lessons on addition and subtraction

http://www.funbrain.com/math/ Offers fun games for kids

Teacher Notes:

http://www.mathwire.com/

http://www.funbrain.com/math/


Unit Overview


Domain: Measurement and Data (Represent and Interpret Data)

Cluster:

Represent and interpret data

Cluster Summary

A clock is a common instrument for measuring time. Learning to tell time has much to do with learning to

read a dial-type instrument and little with time measurement. Building on previous understanding of

measuring time, students will tell and write time to the nearest minute and measure time intervals in

minutes.

Representation of a data set is extended from picture graphs and bar graphs with single-unit scales to

scaled picture graphs and scaled bar graphs.

Students are to measure lengths using rulers marked with halves and fourths of an inch and record the data

on a line plot.







Standards at www.NJ.gov/education/aps/cccs/career

Learning Targets

Content Standards


3.MD.1 Tell and write time to the nearest minute and measure time intervals

in minutes. Solve word problems involving addition and subtraction

of time intervals in minutes, e.g., by representing the problem on a

number line diagram.

3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a

data set with several categories. Solve one- and two-step ―how many

more‖ and ―how many less‖ problems using information presented in

scaled bar graphs. For example, draw a bar graph in which each square in

the bar graph might represent 5 pets.

3.MD.4 Generate measurement data by measuring lengths using rulers marked

with halves and fourths of an inch. Show the data by making a line

plot, where the horizontal scale is marked off in appropriate units—

whole numbers, halves, or quarters.



why is it important to be able to

tell time?

what can data tell you about your

class or school?

how do data displays help us

understand information?

what is the purpose of

measurement?



time measurement is a means to organize and structure each

day and our lives.

collection and use of data provides better understanding of

people and the world.

measurements can be used to describe, compare, and make

sense of phenomena.

Unit Objectives


time increments on analog and digital

clocks.

data can be displayed using various

types of graphs to organize and explain

information.

lengths can be measured to describe

countless objects.

Unit Objectives


tell and write time to the nearest minute and measure time

intervals. Solve word problems involving addition and

subtraction of time intervals in minutes.

interpret and represent data by solving 1 step and 2 step

word problems based on information presented in graphs.

measure lengths indirectly and by repeating length units.




Teacher Observation


Exit Slips

Games

Anecdotal Records



Daily Classwork

Pre-assessments


Tests

Quizzes






Math Literature:

Time:

Clocks and More Clocks by Pat Hutchings

Telling Time with Big Mama Cat by Dan Harper

Get Up and Go! By Stuart Murphy

Graphing:

The Best Vacation Ever! By Stuart Murphy

Lemonade for Sale by Stuart Murphy

Tiger Math: Learning to Graph from a Baby Tiger by Ann Whitehead Nagda

Websites:

www.mathwire.com Offers various activities and lessons on time and graphing concepts

http://www.teach-nology.com/themes/math/graphing/ Offers lessons, games, activities and worksheets

on graphing and time.

Teacher Notes:


Unit Overview


Domain: Operations and Algebraic Thinking

Cluster:

Represent and solve problems involving multiplication and division.

Understand properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Cluster Summary

Students develop an understanding of the meanings of multiplication and division of whole numbers

through activities and problems involving equal-sized groups, arrays, and area models; multiplication is

finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized

group situations, division can require finding the unknown number of groups or the unknown group size.

Students use properties of operations to calculate products of whole numbers, using increasingly

sophisticated strategies based on these properties to solve multiplication and division problems involving

single-digit factors. By comparing a variety of solution strategies, students learn the relationship between

multiplication and division.

Primary interdisciplinary connections:







Learning Targets

Content Standards


3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5

groups of 7 objects each. For example, describe a context in which a total number of objects

can be expressed as 5 × 7.

3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of

objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of

shares when 56 objects are partitioned into equal shares of 8 objects. For example, describe a

context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving

equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with

a symbol for the unknown number to represent the problem.

3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three

http://www.nj.gov/education/aps/cccs/career/

whole numbers. For example, determine the unknown number that makes the equation true in

each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24

is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2

can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative

property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 ×

(5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

3.OA.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the

number that makes 32 when multiplied by 8.

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between

multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or

properties of operations. By the end of Grade 3, know from memory all products of two one-

digit numbers.

3.OA.8 Solve two-step word problems using the four operations. Represent these problems using

equations with a letter standing for the unknown quantity. Assess the reasonableness of

answers using mental computation and estimation strategies including rounding.

3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table),

and explain them using properties of operations. For example, observe that 4 times a number

is always even, and explain why 4 times a number can be decomposed into two equal

addends.



How are multiplication and

division related?

Why do we use symbols to

represent missing numbers?

How can we predict the next

element in a pattern?



the four basic arithmetic operations are interrelated, and the

properties of each may be used to understand the others.

mathematical concepts can be understood using a variety of

models.

numbers are able to represent quantity, position, location,

and relationships, and symbols may be used to express

these relationships.

Unit Objectives


many situations in daily life can

be modeled with multiplication

and division.

problem solving in daily life may

include unknown variables that

impact outcomes.

patterns exist in the relationship

of multiplication and division.

Unit Objectives


interpret products of whole numbers.

interpret whole number quotients.

use multiplication and division to solve word

problems.

determine the unknown whole number in an equation

of three whole numbers.

apply properties of operations to multiply and divide

memorize all products of two single-digit numbers.

solve two step word problems using four operations

and solving for the unknown.

identify patterns in arithmetic.




Teacher Observation


Exit Slips

Games

Anecdotal Records



Daily Classwork

Pre-assessments


Tests

Quizzes






Math Literature:

Multiplication:

Hershey’s Kisses by Jerry Pollatta

365 Penguins by Jean Luc Fromental

The Doorbell Rang by Pat Hutchings

Division:

Safari Park by Stuart Murphy

The Doorbell Rang by Pat Hutchings

Websites:

http://www.factmonster.com Offers games and activities on multiplication and division concepts.

http://harcourtschool.com/search/search.html Offers games and activities on multiplication and

division concepts.

www.mathwire.com Offers lessons and activities on multiplication and division concepts.

Teacher Notes:


Unit Overview


Domain: Number and Operations-Fractions

Cluster:

Develop understanding of fractions as numbers

Cluster Summary

Students develop an understanding of fractions, beginning with the representation of parts compared to a

whole. Students understand that the size of a fractional part is relative to the size of the whole.

Students are able to use fractions to represent numbers equal to, less than, and greater than one.

They solve problems that involve comparing fractions by using visual fraction models and strategies based

on noticing equal numerators or denominators.








Learning Targets

Content Standards


3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a

whole is partitioned into b equal parts; understand a fraction a/b as

the quantity formed by a parts of size 1/b.

3.NF.2 Understand a fraction as a number on the number line; represent

fractions on a number line diagram.

a. Represent a fraction 1/b on a number line diagram by defining the

interval from 0 to 1 as the whole and partitioning it into b equal

parts. Recognize that each part has size 1/b and that the endpoint

of the part based at 0 locates the number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by marking off

a lengths 1/b from 0. Recognize that the resulting interval has size

a/b and that its endpoint locates the number a/b on the number line.

3.NF.3 Explain equivalence of fractions in special cases, and compare

fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the

same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 =

2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by

using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that

are equivalent to whole numbers. Examples: Express 3 in the form

3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point

of a number line diagram.

d. Compare two fractions with the same numerator or the same

denominator by reasoning about their size. Recognize that

comparisons are valid only when the two fractions refer to the

same whole. Record the results of comparisons with the symbols

>, =, or <, and justify the conclusions, e.g., by using a visual

fraction model.


4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as

62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that

comparisons are valid only when the two decimals refer to the same whole. Record the

results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using

a visual model.


How many ways can a whole

number be represented?

How do we show part of a unit?

How can a fraction be represented

in different equivalent forms?



fractions represent equal parts of a whole unit

fractions are represented on a number line

fractions with different numerators and denominators can

be compared by reasoning about their size

Unit Objectives


fractions represent equal parts of

a whole unit.

fractions are represented on a

number line.

fractions can still be equivalent

even though they appear to be

different.

Unit Objectives


construct a fraction based on an object partitioned into

equal parts.

compare fractions by using visual fraction models and

number lines to understand equivalent fractions.

compare two fractions with the same numerator or the same

denominator by reasoning about their size.




Teacher Observation


Exit Slips

Games

Anecdotal Records



Daily Classwork

Pre-assessments


Tests

Quizzes






Math Literature:

Fractions:

Fraction Fun by David Adler

Give Me Half! By Stuart Murphy

Clean Sweep Campers by Lucille Recht Penner

Hershey Fractions Book by Jerry Pollatta

Websites:

Mathwire.com Offers various activities, lessons and games for fraction concepts

http://www.aaamath.com/fra.html Offers various math games

Teacher Notes:

http://www.aaamath.com/fra.html


Unit Overview


Domain: Geometry

Cluster:

Reason with shapes and their attributes.

Cluster Summary

Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify

shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their

fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.






prepare for life as citizens and workers in the 21st Century. For further clarification see N.J. World Class.


Learning Targets

Content Standard


3.G.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may

share attributes (e.g., having four sides), and that the shared attributes can define a larger

category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of

quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these

subcategories.

3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of

the whole. For example, partition a shape into 4 parts with equal area, and describe the area of

each part as 1/4 of the area of the shape.


4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping

parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve

addition and subtraction problems to find unknown angles on a diagram in real world and

mathematical problems, e.g., by using an equation with a symbol for the unknown angle

measure.

4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and

parallel lines. Identify these in two-dimensional figures.

4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular

lines, or the presence or absence of angles of a specified size.

4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that

the figure can be folded along the line into matching parts. Identify line-symmetric figures and

draw lines of symmetry.


What words in geometry are

also used in daily life?

Why can different geometric

terms be used to name the same

shape?



geometric figures are described by their attributes

attributes of objects can be measured with appropriate tools

Unit Objectives


how spatial relationships can be

described by careful use of geometric

language.

how geometric relationships help to

solve problems and/or make sense of

phenomena.

Unit Objectives


use properties of standard 2-D shapes to identify, classify, and

describe (vertex, side, edge, face, and angle).

recognize rhombus, rectangles, and squares as examples of

quadrilaterals and determine examples of quadrilaterals that

do not belong.

partition shapes (unit fractions) into sections to determine

parts of a whole.




Teacher Observation


Exit Slips

Games

Anecdotal Records



Daily Classwork

Pre-assessments


Tests

Quizzes






Math Literature:

Shapes:

When a Line Bends . . . A Shape Begins by Rhonda Gowler Greene

Shapes, Shapes, Shapes by Tanya Hoban

Cubes, Cones, Cylinders, & Spheres by Tanya Hoban (Introduction)

Lines, Segments, Rays, and Angles by Claire Piddick

Websites:

http://www.factmonster.com Offers games and activities on measurement concepts

http://harcourtschool.com/search/search.html Offers games and activities on measurement

www.mathwire.com Offers lessons and activities on area, volume and perimeter concepts

Teacher Notes:


Unit Overview


Domain: Measurement and Data (Represent and Interpret Data)

Cluster:

Geometric Measurement: Understand concepts of liquid volume, mass, perimeter, area and relate area to

multiplication and to addition.

Cluster Summary

Students will recognize perimeter and area as an attribute of two-dimensional regions. They measure the area of a

shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a

square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays

can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays

of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.


Science, Social Studies, Language Arts, Technology, and 21st Century Life and Careers, For further clarification see

NJ Core Curriculum Content Standards at www.njcccs.org.


Through instruction and life and career skills, all students acquire the knowledge and skills needed to prepare for life

as citizens and workers in the 21st Century. For further clarification see N.J. World Class Standards at

www.NJ.gov/education/aps/cccs/career/

Learning Targets

Content Standards


3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams

(g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word

problems involving masses or volumes that are given in the same units, e.g., by using

drawings (such as a beaker with a measurement scale) to represent the problem.

3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.

A square with side length 1 unit, called ―a unit square,‖ is said to have ―one square unit‖ of

area, and can be used to measure area. A plane figure which can be covered without gaps or

overlaps by n unit squares is said to have an area of n square units.

3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and

improvised units).

3.MD.7 Relate area to the operations of multiplication and addition. Find the area of a rectangle with

whole-number side lengths by tiling it, and show that the area is the same as would be found

by multiplying the side lengths.

Multiply side lengths to find areas of rectangles with whole-number side lengths in the

context of solving real world and mathematical problems, and represent whole-number

products as rectangular areas in mathematical reasoning.

Use tiling to show in a concrete case that the area of a rectangle with whole-number side

lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive

property in mathematical reasoning.

Recognize area as additive. Find areas of rectangular figures by decomposing them into non-

overlapping rectangles and adding the areas of the non-overlapping parts, applying this

technique to solve real world problems

3.MD.8 Solve real world and mathematical problems involving perimeters of polygons, including

finding the perimeter given the side lengths, finding an unknown side length, and exhibiting

rectangles with the same perimeter and different areas or with the same area and different

perimeters.


4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common

endpoint, and understand concepts of angle measurement:

4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified

measure.

4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm;

kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express

measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in

a two-column table.


How can measurements be used to solve real

world problems?



everyday objects have a variety of attributes, each of which

can be measured in many ways.

what we measure affects how we measure it.

measurements can be used to describe, compare, and make

sense of phenomena.

Unit Objectives


how to measure and estimate liquid volumes and

masses of objects.

how to measure area of given shapes and objects.

how to relate area to the operations of multiplication

and addition.

how to identify the perimeter of a shape.

how to solve word problems using perimeter and

area.

Unit Objectives


measure and estimate liquid volumes and masses using

specific units such as grams, kilograms, and liters.

measure the area of a given shape by counting unit squares

(square cm, square m, square in, etc.).

measure the area of a rectangle by multiplying side lengths

or using repeated addition.

recognize that the perimeter is found along the outside of a

given shape and can problem solve to find an unknown side

length.

comprehend that perimeter and area are used to solve real

world problems.




Teacher Observation


Exit Slips

Games

Anecdotal Records



Daily Classwork

Pre-assessments


Tests

Quizzes






Math Literature:

Area and Perimeter:

Perimeter and Area at the Amusement Park by Diane Irving

Perimeter, Area and Volume by David Adler

Volume and Mass:

Perimeter, Area and Volume by David Adler

Mr. Archimedes Bath by Pamela Allen

Websites:

http://www.factmonster.com Offers games and activities on measurement concepts

http://harcourtschool.com/search/search.html Offers games and activities on measurement

www.mathwire.com Offers lessons and activities on measurement concepts

Teacher Notes:

Stafford Township School District Grade Level Expectations and Pacing Guides

After our district participated in creating the Ocean County Mathematics Curriculum we shared the document with our staff. The County Curriculum addresses the Common Core Standards and had an enormous amount of resources for our staff to use, but still did not give teachers a clear understanding of how to implement the standards in their classrooms. Over the 2011-12 School Year, we looked at the materials and resources that we have available in Stafford, and worked with teachers to create grade level expectations and pacing guides. These documents outline what teachers need to cover in specific grade levels and timeframes. The purpose is to create consistency within a grade level to ensure that all students at a specific grade level will be exposed to specific skills, strategies, expectations, and assignments.

Stafford Township School District Mathematics Pacing Guide

MATHEMATICS Unit Assessment Schedule

Grade 3 2012~2013

Unit 1 ~ Test by September 28, 2012 Unit 2 ~ Test by October 16, 2012 Unit 3 ~ Test by November 2, 2012 Unit 4 ~ Test by November 30, 2012 Unit 5 ~ Test by December 21, 2012 Unit 6 ~ Test by January 25, 2013 Unit 7 ~ Test by February 14, 2013 Unit 8 ~ Test by March 5, 2013 Unit 9 ~ Test by April 5, 2013 Unit 10 ~ Test by April 26, 2013 *NJ ASK May 13th-May 16th, 2013 Unit 11 ~ Test by May 31, 2013

Stafford Township School District Mathematics Resources

Everyday Math Everyday Math online White Marker Boards Math manipulative kit Calculators Study Island online

Stafford Township School District Accommodations for Special Education Students as Specified in Their IEP

& for English Language Learners Access to Special Equipment and Instructional Materials

• Calculator. • Tape recorder/CD player. • Slant board. • Special pens, pencils and paper. • Computer. • FM systems. • Communication board /augmentative communication devices.

Class Work Curricular Procedures • Allow additional time to complete work. • Assist in organizing materials. • Cue students’ attention. • Functional or practical emphasis (specify in IEP). • Limit oral reading unless student volunteers. • Modify reading assignments. • Modify writing requirements. • Ongoing monitoring of work produced during class. • Peer tutoring or partner learning. • Preferential seating. • Provide alternate content objectives (specify in IEP). • Provide copies of class notes. • Provide organizational aids (folders, schedules). • Provide repetition/practice. • Provide student a daily schedule/established routine. • Provide study guides. • Reading assistance in content area. • Reduce content objectives. • Reduce length/number of written assignments. • Review, restate and repeat directions. • Simplify verbal/.written directions. • State expectations clearly. • Teach to learner’s auditory style. • Teach to learner’s multi sensory style. • Teach to learner’s visual style.

Facilitating Appropriate Behavior • Assign activities which require some movement. • Avoid the use of confrontational techniques. • Consistently enforce school rules. • Designate a “time-out” location within the classroom. • Encourage student to accept responsibility for behavior. • Ignore minor annoying behaviors. • Involve parents/guardians to coordinate approach. • Minimize situations that the student may perceive as embarrassing. • Monitor levels of tolerance and be mindful of signs of frustration. • Provide clear and concise classroom expectation and consequences. • Provide student with alternative. • Reinforce student for appropriate behaviors. • Speak privately, without the audience of peers, to student about inappropriate

behavior. • Use praise generously. • Check for student behavior plan.

Homework

• Homework assignment book checked by special education mainstream. teacher(s) and parent/guardian daily.

• Reduce length of assignments. • Reduced alternative reading work. • Reduced alternative written work. • Time in school to complete.

Modifying Test

• Allow open book tests/open note tests (circle). • Allow student to edit a “first draft” on essay questions and grade final edited

copy. • Allow student to give answers orally. • Allow student to type. • Allow student to make corrections on returned tests for additional credit. • Allow test to be given by special education teacher. • Allow use of instructional aids (calculator, computer, etc.). • Improve font/spatial organization of test. • Math tools for tests (fact chart, number line calculator). • Minimize essay questions. • Modify test format. • Modify the content of the test. • Retake failed tests using alternative strategies. • Test only on key concepts. • Time of test determined by instructor. • Use word banks.

Modifying the Environment • Frequently check the organization of the student’s notebook. • Help keep the student’s work area free of unnecessary materials. • Monitor the student’s use of his/her assignment sheet. • Provide a specific place for turning in completed assignments. • Provide opportunities for movement/activity change. • Seat student in an area free of distractions. • Use checklists to help the student get organized. • Use preferential seating. • Use study carrels.

Modifying the Grading

• Allow for spelling errors. • Grade determined through collaboration of general/special education teacher. • Modify homework/class work expectations.

Modifying the Presentation of Materials

• Assign tasks at an appropriate reading level. • Break assignment into segments of shorter tasks. • Cue student by calling his/her name before asking questions. • Familiarize student with any new vocabulary before beginning the lesson. • Frequently check on progress of independent work. • Give additional presentations by varying the methods using repetition, simpler

explanations, more examples and modeling. • Give written directions to supplement verbal directions. • Highlight important concepts to be learned in text of material. • Monitor the student’s comprehension of language used during instruction. • Provide a checklist for long, detailed tasks. • Provide clear and well defined worksheets. • Provide clear, concise directions and concrete examples for homework/class

work assignments. • Provide due date on written assignments. • Provide graph paper for math computation. • Reduce the number of concepts presented at one time. • Student qualifies for Books on Tape. • Use concrete examples of concepts before teaching the abstract. • Utilize manipulative, hands-on activities. • Utilize visual aids such as charts and graphs.

Instructional Strategies and Techniques that Address Learning Style • Multi-sensory approach. • Stress visual presentation. • Stress auditory presentation. • Modify written expectations. • Provide positive reinforcement. • Provide support/encouragement. • Provide structured environment. • Set clear limits and consequences. • Use concrete examples. • Simplify directions. • Provide repetition and practice. • Allow frequent activity changes. • Provide small group instruction. • Reduce work increments. • Set short-term goals. • Preferential seating. • Use study carrels to increase concentration. • Seat in area of minimal distraction. • Model skills/techniques to be mastered. • Have student repeat directions. • Utilize peer buddy as needed.

Common Core State StandardS for matHematICSG

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Mathematics|Grade3

InGrade3,instructionaltimeshouldfocusonfourcriticalareas:(1)

developingunderstandingofmultiplicationanddivisionandstrategies

formultiplicationanddivisionwithin100;(2)developingunderstanding

offractions,especiallyunitfractions(fractionswithnumerator1);(3)

developingunderstandingofthestructureofrectangulararraysandof

area;and(4)describingandanalyzingtwo-dimensionalshapes.

(1)Studentsdevelopanunderstandingofthemeaningsofmultiplication

anddivisionofwholenumbersthroughactivitiesandproblemsinvolving

equal-sizedgroups,arrays,andareamodels;multiplicationisfinding

anunknownproduct,anddivisionisfindinganunknownfactorinthese

situations.Forequal-sizedgroupsituations,divisioncanrequirefinding

theunknownnumberofgroupsortheunknowngroupsize.Studentsuse

propertiesofoperationstocalculateproductsofwholenumbers,using

increasinglysophisticatedstrategiesbasedonthesepropertiestosolve

multiplicationanddivisionproblemsinvolvingsingle-digitfactors.By

comparingavarietyofsolutionstrategies,studentslearntherelationship

betweenmultiplicationanddivision.

(2)Studentsdevelopanunderstandingoffractions,beginningwith

unitfractions.Studentsviewfractionsingeneralasbeingbuiltoutof

unitfractions,andtheyusefractionsalongwithvisualfractionmodels

torepresentpartsofawhole.Studentsunderstandthatthesizeofa

fractionalpartisrelativetothesizeofthewhole.Forexample,1/2ofthe

paintinasmallbucketcouldbelesspaintthan1/3ofthepaintinalarger

bucket,but1/3ofaribbonislongerthan1/5ofthesameribbonbecause

whentheribbonisdividedinto3equalparts,thepartsarelongerthan

whentheribbonisdividedinto5equalparts.Studentsareabletouse

fractionstorepresentnumbersequalto,lessthan,andgreaterthanone.

Theysolveproblemsthatinvolvecomparingfractionsbyusingvisual

fractionmodelsandstrategiesbasedonnoticingequalnumeratorsor

denominators.

(3)Studentsrecognizeareaasanattributeoftwo-dimensionalregions.

Theymeasuretheareaofashapebyfindingthetotalnumberofsame-

sizeunitsofarearequiredtocovertheshapewithoutgapsoroverlaps,

asquarewithsidesofunitlengthbeingthestandardunitformeasuring

area.Studentsunderstandthatrectangulararrayscanbedecomposedinto

identicalrowsorintoidenticalcolumns.Bydecomposingrectanglesinto

rectangulararraysofsquares,studentsconnectareatomultiplication,and

justifyusingmultiplicationtodeterminetheareaofarectangle.

(4)Studentsdescribe,analyze,andcomparepropertiesoftwo-

dimensionalshapes.Theycompareandclassifyshapesbytheirsidesand

angles,andconnectthesewithdefinitionsofshapes.Studentsalsorelate

theirfractionworktogeometrybyexpressingtheareaofpartofashape

asaunitfractionofthewhole.


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operations and algebraic thinking

• represent and solve problems involving multiplication and division.

• Understand properties of multiplication and the relationship between multiplication and division.

• multiply and divide within 100.

• Solve problems involving the four operations, and identify and explain patterns in arithmetic.

number and operations in Base ten

• Use place value understanding and properties of operations to perform multi-digit arithmetic.

number and operations—fractions

• develop understanding of fractions as numbers.

measurement and data

• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

• represent and interpret data.

• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

• Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometry

• reason with shapes and their attributes.

mathematical Practices

1. Makesenseofproblemsandpersevereinsolvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

Grade 3 overviewmathematical Practices

1. Makesenseofproblemsandperseverein

solvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritique

thereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeated

reasoning.


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operations and algebraic thinking 3.oa

Represent and solve problems involving multiplication and division.

1. Interpretproductsofwholenumbers,e.g.,interpret5×7asthetotalnumberofobjectsin5groupsof7objectseach.For example, describe a context in which a total number of objects can be expressed as 5 × 7.

2. Interpretwhole-numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1

4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

5. Applypropertiesofoperationsasstrategiestomultiplyanddivide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

6. Understanddivisionasanunknown-factorproblem.For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Multiply and divide within 100.

7. Fluentlymultiplyanddividewithin100,usingstrategiessuchastherelationshipbetweenmultiplicationanddivision(e.g.,knowingthat8×5=40,oneknows40÷5=8)orpropertiesofoperations.BytheendofGrade3,knowfrommemoryallproductsoftwoone-digitnumbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

8. Solvetwo-stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3

9. Identifyarithmeticpatterns(includingpatternsintheadditiontableormultiplicationtable),andexplainthemusingpropertiesofoperations.For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

1SeeGlossary,Table2.2Studentsneednotuseformaltermsfortheseproperties.3Thisstandardislimitedtoproblemsposedwithwholenumbersandhavingwhole-numberanswers;studentsshouldknowhowtoperformoperationsintheconven-tionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).

Common Core State StandardS for matHematICSg

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Number and Operations in Base Ten 3.NBT

Use place value understanding and properties of operations to perform multi-digit arithmetic.4

1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.

2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.

3. Multiplyone-digitwholenumbersbymultiplesof10intherange10–90(e.g.,9×80,5×60)usingstrategiesbasedonplacevalueandpropertiesofoperations.

Number and Operations—Fractions5 3.NF

Develop understanding of fractions as numbers.1. Understandafraction1/basthequantityformedby1partwhena

wholeispartitionedintob equalparts;understandafractiona/basthequantityformedbyapartsofsize1/b.

2. Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.

a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.

b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.

3. Explainequivalenceoffractionsinspecialcases,andcomparefractionsbyreasoningabouttheirsize.

a. Understandtwofractionsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.

b. Recognizeandgeneratesimpleequivalentfractions,e.g.,1/2=2/4,4/6=2/3.Explainwhythefractionsareequivalent,e.g.,byusingavisualfractionmodel.

c. Expresswholenumbersasfractions,andrecognizefractionsthatareequivalenttowholenumbers.Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

d. Comparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

Measurement and Data 3.MD

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.

4Arangeofalgorithmsmaybeused.5Grade3expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,6,and8.


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2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).6Add,subtract,multiply,ordividetosolveone-stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.7

Represent and interpret data.

3. Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveone-andtwo-step“howmanymore”and“howmanyless”problemsusinginformationpresentedinscaledbargraphs.For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

4. Generatemeasurementdatabymeasuringlengthsusingrulersmarkedwithhalvesandfourthsofaninch.Showthedatabymakingalineplot,wherethehorizontalscaleismarkedoffinappropriateunits—wholenumbers,halves,orquarters.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

5. Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.

a. Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“onesquareunit”ofarea,andcanbeusedtomeasurearea.

b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.

6. Measureareasbycountingunitsquares(squarecm,squarem,squarein,squareft,andimprovisedunits).

7. Relateareatotheoperationsofmultiplicationandaddition.

a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.

b. Multiplysidelengthstofindareasofrectangleswithwhole-numbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.

c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.

d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

8. Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.

6Excludescompoundunitssuchascm3andfindingthegeometricvolumeofacontainer.7Excludesmultiplicativecomparisonproblems(problemsinvolvingnotionsof“timesasmuch”;seeGlossary,Table2).


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Geometry 3.G

Reason with shapes and their attributes.

1. Understandthatshapesindifferentcategories(e.g.,rhombuses,rectangles,andothers)mayshareattributes(e.g.,havingfoursides),andthatthesharedattributescandefinealargercategory(e.g.,quadrilaterals).Recognizerhombuses,rectangles,andsquaresasexamplesofquadrilaterals,anddrawexamplesofquadrilateralsthatdonotbelongtoanyofthesesubcategories.

2. Partitionshapesintopartswithequalareas.Expresstheareaofeachpartasaunitfractionofthewhole.For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

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