community unit district 308: mathematics curriculum guide

Transcription

COMMUNITY UNIT DISTRICT
308: MATHEMATICS
CURRICULUM GUIDE
Third Grade
This document serves as the curriculum guide for District 308 mathematics. All components are fully
aligned to the Illinois Early Learning and Developmental Standards and Common Core State Standards.
2014-2015
EC-5 Math Curriculum Team Members:
Brokaw Early Childhood Center:


Susan Craig (EC)
Darlene Howell (EC)
Jean Rempala (K)
Jennifer Friel (K)
rd
Alexandra Wooden (3 )
nd
Brittany Morelli (2 )
nd
Christine Gamlin (2 )
th
Kelly Fleagle (5 )
st
Amber Denbo (1 )
th
Amy Huffman (5 )
Grande Park Elementary:


nd
Tamara Allen (2 )
th
Jeffrey Rainaldi (4 )


nd
Lisa Paluch (2 )
rd
Sara Studer (3 )



st
Anne Marie Simmons (1 )
th
Brendan Stephens (4 )
Jennifer Weaver (K)


nd
Tamarac Maddox (2 )
th
Susan Rost (4 )
Wolf’s Crossing Elementary:
nd
Jennifer Rusin (2 )
th
Julie Wilson (4 )
Hunt Club Elementary:


nd
Dana Miles (2 )
th
Deanne Todd (4 )
The Wheatlands Elementary:
Homestead Elementary:




Southbury Elementary:
Fox Chase Elementary:


rd
Kirsten Brandwein (3 )
th
Andrea Daleiden (5 )
nd
Jessica Richter (2 )
Prairie Point Elementary:
Churchill Elementary:





Old Post Elementary:
Boulder Hill Elementary:


st
Katelyn Hutchison (1 )
th
Ann Lutz (5 )
Long Beach Elementary:
East View Kindergarten Center:






st
Amanda Armitage (1 )
th
Joy Varney (5 )
Administrators:
rd
Rosa Brolley (3 -Dual)
st
Toni Morgan (1 -Dual)
Lakewood Creek Elementary:




Jodi Ancel
Tammie Harmon
Melissa McDowell
Lisa Smith
Community Unit District 308 Mathematics Units
Third Grade
Unit One: Addition and Subtraction within 1000
Approximate Time Frame: 4-6 Weeks August - September
Connections to Previous Learning:
In 2nd grade students solved one- and two-step situational problems of all three types (See Diagram page 3), involving both addition and subtraction within
100. They used number lines and other modalities to represent these problem-solving situations.
Focus on the Unit:
This unit focuses upon students deepening their understanding of place value through practice of rounding numbers to the nearest 10 or 100 using real world
situations that could involve money, time, and other daily life experiences. Students will reason and explain the answers they get when they round applying this
increased understanding of place value. Rounding is a life skill that will assist students in estimating in real-world situations. Students should have numerous
experiences using a number line and a hundreds chart as tools to support their work with rounding.
In addition, students will come to grasp that when moving from digit to digit within a multi-digit number that values increase or decrease, a skill that will
become more procedural in Grade 4.
Connections to Subsequent Learning:
Students will apply concepts of place value, recognizing that in a multi-digit whole number, the digit on the left represents ten times the value of the digit to its
right. They will be able to write multi-digit numbers in expanded form and compare two-digit numbers based on meaning and placement of the digits. They will
convey meaning of their comparisons using the symbols: <, >, =.
From the K-5, Number and Operations in Base Ten progression document, p. 11
At Grade 3, the major focus is multiplication, so students’ work with addition and subtraction is limited to maintenance of fluency within 1000 for some
students and building fluency to within 1000 for others.
3.NBT.2
Use place value understanding and properties of operations to perform multi-digit arithmetic Students continue adding and subtracting within 1000.
They achieve fluency with strategies and algorithms that are based on place value, properties of operations, and/or the relationship between addition and
subtraction. Such fluency can serve as preparation for learning standard algorithms in Grade 4, if the computational methods used can be connected with those
algorithms.
3.NBT.1
Students use their place value understanding to round numbers to the nearest 10 or 100.
They need to understand that when moving to the right across the
places in a number (e.g., 456), the digits represent smaller units. When rounding to the nearest 10 or 100, the goal is to approximate the number by the closest
number with no ones or no tens and ones (e.g., so 456 to the nearest ten is 460; and to the nearest hundred is 500). Rounding to the unit represented by the left
most place is typically the sort of estimate that is easiest for students. Rounding to the unit represented by a place in the middle of a number may be more
difficult for students (the surrounding digits are sometimes distracting). Rounding two numbers before computing can take as long as just computing their sum or
difference.
Priority Standards
Supporting Standards
Additional Standards
1
Third Grade
Taken from K-5 Number and Operations in Base Ten Progression Document, pages 8-11.
In grades K-2, students are introduced to such concepts as problem solving structures, unknowns in multiple locations and place value. (See table 2.) In grade
2, students learn to represent numbers up to 1,000 using manipulatives, pictures, symbols and real-life situations. They also learn to add and subtract within
100 fluently (moving from manipulatives and visual representations to symbols and real-life situations).
Students in grade 3 extend their understanding of place value and of 2-digit addition and subtraction to perform these operations on numbers within 1,000. In
addition, they use their understanding of place value to round numbers to the nearest 10 or nearest 100. They also apply their understanding of problem
solving structures (putting together, taking apart, adding to, taking from, and comparing) and unknowns in all 3 locations (start, change, result) to a new reallife application.
Taken from K-5 Counting and Cardinality Progression
Document, pg. 7.
In each type (shown as a row), any one of the three
quantities in the situation can be unknown, leading
to the subtypes shown in each cell of the table. The
table also shows some important language variants
which, while mathematically the same, require
separate attention. Other descriptions of the
situations may use somewhat different names.
Adapted from CCSS, p. 88, which is based on
Mathematics Learning in Early Childhood: Paths
Toward Excellence and Equity, National Research
Council, 2009, pp. 32–33.
Priority Standards
2
Third Grade
1 This can be used to show all decompositions of a given number, especially important for numbers within 10. Equations with totals on the left help
children understand that = does not always mean “makes” or “results in” but always means “is the same number as.” Such problems are not a problem
subtype with one unknown, as is the Addend Unknown subtype to the right. These problems are a productive variation with two unknowns that give
experience with ﬁnding all of the decompositions of a number and reﬂecting on the patterns involved.
2 Either addend can be unknown; both variations
should be included. 3 For the Bigger Unknown or Smaller Unknown situations, one version directs the correct
operation (the version using more for the bigger unknown and using less for the smaller unknown).
Transfer: Students will apply…
 Their understanding of place value to round numbers to the nearest 10 or 100
 Knowledge and skills to perform real world tasks such as estimating distance/mileage; calculating grocery bills by rounding prices and adding the
estimates; etc.
 Knowledge of estimation as a tool for checking their work.
 Fluent addition and subtraction skills within 1000 to perform real world tasks.
Understandings: Students will understand that …
 Rounding is a method of approximating an answer.
 Addition and subtraction skills can be applied to a variety of situations with a variety of unknowns (ex. Comparison problem where the difference is
unknown. See chart on page 2 )
Essential Questions:
 How is rounding an efficient method of estimating?
 Why and when would you round?
 When do you apply addition and subtraction strategies to solve for an unknown variable?
Prerequisite Skills/Concepts: Students should already be able to…
 Demonstrate fluency in addition and subtraction skills within 100
 Demonstrate their experience with multi-digit addition and
subtraction skills to 1000.
 Solve one and two step situational problems
Knowledge: Students will know…
 When to round in a real-life situation.
 The relationship between addition and subtraction and their
application to real-life situations.
Priority Standards
Advanced Skills/Concepts: Some students may be ready to…
 Use place value understanding to round whole numbers to the
nearest 10 or 100. (3. NBT.1)
 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.2)
Skills: Students will be able to…
 Add and subtract within 1000.
 Model algorithms based upon place value, properties of operations
and/or the relationship between adding and subtracting.
3
Third Grade
WIDA Standard: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners will benefit from:
 Repetitive use of rounding/estimating terminology. The teacher modeling how to write the equation that represents the ten frame situation.
Desired Outcomes:
Standards:
Use place value understanding and properties of operations to perform multi-digit arithmetic.4
3. NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.
3. NBT.2 Fluently adds and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the
relationship between addition and subtraction.
Highlighted Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students will solve real-world problems involving the need for estimation and rounding and
make appropriate decision when rounding is required.
*2. Reason abstractly and quantitatively. Students make appropriate decisions when rounding is required. Students will also use rounding to estimate their
sums and differences when adding and subtracting.
*3. Construct viable arguments and critique the reasoning of others. Students will justify their process and reasoning for rounding numbers. They will listen
and comment upon others’ reasoning.
*4. Model with mathematics. Students will use modeling with various manipulatives, drawings, and words to represent estimation/rounding situation. Students
will model their addition and subtraction problem solving.
*5. Use appropriate tools strategically. Students will use manipulatives to model their rounding and/or their addition and subtraction solutions.
*6. Attend to precision. Students will justify their strategy for finding a solution.
*7. Look for and make use of structure. Students will model the rounding process and reasoning for rounding to represent the structure of the base-ten number
system.
*8. Look for and express regularity in repeated reasoning. Students notice the patterns of the number system and how it can be applied to the rounding
process.
Priority Standards
4
Third Grade
Academic Vocabulary:
Critical Terms
English:
Spanish:
English:
 Associative property of
 Propiedad asociativa de la
 Halfway
addition
suma
 Addition
 Commutative property of
 Propiedad conmutativa de
 Add
addition
la suma
 Addend
 Identity property of
 Propiedad de identidad de
 Subtraction
addition
la suma
 Ones
 Mental math
 Matemática mental
 Tens
 Parenthesis
 Paréntesis
 Hundreds
 Pattern
 Patrón
 Thousands
 Reasonable
 Razonable
 Decompose
 Regroup
 Reagrupar
 Unknown
 Incógnita
 Place value
 Valor posicional
 Round
 Redondear
 Digit
 Digito
 Expanded form
 Forma desarrollada
 Standard form
 Forma estándar
 Word form
 Forma verbal
 Bar diagram
 Diagrama de barra
 Estimate
 Estimación
 Inverse operations
 Operaciones inversas
 Sum
 Suma
 Subtract
 Restar
 Difference
 Diferencia
 Strategies
 Estrategias
 Properties
 Propiedades
 About (ELL)
 Casi (ELL)
 Close to (ELL)
 Cerca (ELL)
Supplemental Terms
Spanish:
 Intermedio
 Adición
 Sumar
 Sumandos
 Restar
 Unidades
 Decenas
 Centenas
 Mil
 Descomponer
Assessments: Utilize Pre-Assessments, Post-Assessments, Formative Assessments via My Math and Mastery Connect. Utilize District 308 Unit Summative
Assessments via Mastery Connect.
Priority Standards
5
Third Grade
Priority Standards
6
Third Grade
Unit Two: Multiplication and Division Concepts
Approximate Time Frame: 3-4 weeks October
Grade 2 students extend counting and modeling of quantities to equal groups and arrays. Students will begin to model multiplication using
rectangles partitioned into equivalent squares. They have begun extending the modeling of quantities to equal groups and arrays as a basis for
multiplication.
Focus of the Unit:
In this unit, Grade 3 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 and extend
students’ work with the distributive property. For example, in the picture the area of a 7 x 6 figure can be
determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums. Students will continue their
discovery of this concept and apply it to the composition and decomposition of shapes.
Students will utilize multiplication to find a total number of objects when there are a set number of groups with an equal number of objects in each group
or when an equal amount of objects are added or collected numerous times. There are two distinct models of division: partition models and measurement
(repeated subtraction) models.
 Partition models provide students with a total number and the number of groups. These models focus on the question, “How many objects are in
each group so that the groups are equal?” A context for partition models would be: There are 12 cookies on the counter. If you are sharing the
cookies equally among three bags, how many cookies will go in each bag?

Number line (repeated subtraction) models provide students with a total number and the number of objects in each group. These models focus on
the question, “How many equal groups can you make?” A context for number line models would be: There are 12 cookies on the counter. If you
put 3 cookies in each bag, how many bags will you fill?
Students in Grade 3 will also work with the commutative property (rules about how numbers work) of multiplication. While students DO NOT need to use the
formal term commutative property, students must understand that properties are rules about how numbers work, and they need to apply each of them in varied
situations. Brief definitions for the commutative property are as follows:

The commutative property (order property) states that the order of numbers does not matter when you are adding or multiplying numbers.
For example, if a student knows that 5 x 4 = 20, then they also know that 4 x 5 = 20.
Priority Standards
7
Third Grade
Grade 4 students will multiply and divide with increasing difficulty. They will also extend multiplication and division concepts into factors and multiples.
Patterns that flow from these operations will be generated and analyzed. Finally, Grade 4 students will multiply a fraction by a whole number.
From the Operations & Algebraic Thinking progression document, pp. 22, 23, 24, 25- 26-27
Students focus on understanding the meaning and properties of multiplication and division and on finding products of single-digit multiplying and related
3.OA.1–7
quotients.
These skills and understandings are crucial; students will rely on them for years to come as they learn to multiply and divide with multi-digit
whole number and to add, subtract, multiply and divide with fractions and with decimals. Note that mastering this material, and reaching fluency in single-digit
3.OA.7
multiplications and related divisions with understanding
may be quite time consuming because there are no general strategies for multiplying or dividing
all single-digit numbers as there are for addition and subtraction. Instead, there are many patterns and strategies dependent upon specific numbers. So it is
imperative that extra time and support be provided if needed.
Table 3: Multiplication and division situations
AxB=
Equal Groups of Objects
A x  = C and C ÷ A = 
Unknown Product
There are A bags with B plums in each bag. How
many plums are in all?
Group Size Unknown
If C plums are shared equality into A bags, then
how many plums will be in each bag?
 x B = C and C ÷ B = 
Number of Groups Unknown
If C plums are to be packed B to a bag, then how
many bags are needed?
Equal groups language
Arrays of Objects
Unknown Product
There are A rows of apples with B apples in each
row. How many apples are there?
Unknown Factor
If C apples are arranged in A equal rows, how
many apples will be in each row?
Unknown Factor
If C apples are arranged into equal rows of B
apples, how many rows will there be?
Row and column language
Unknown Product
The apples in the grocery window are in A rows
and B columns. How many apples are there?
Unknown Factor
If C apples are arranged into an array with A rows,
how many columns of apples are there?
Unknown Factor
If apples are arranged into an array with B columns,
how many rows are there?
A>1
Compare
Larger Unknown
A blue hat costs $8. A red hat costs A times as
much as the blue hat. How much does the red hat
cost?
Smaller Unknown
A red hat costs $C and that is A times as much as
blue hat costs. How much does a blue hat cost?
Smaller Unknown
A blue hat costs $B. A red hat costs A as much as
the blue hat. How much does the red hat cost?
Larger Unknown
A red hat costs $C and that is A of the cost of a
blue hat. How much does a blue hat cost?
Multiplier Unknown
A red hat costs $C and a blue hat costs $B. How
many times as much does the red hat cost as the
blue hat?
A<1
Priority Standards
Multiplier Unknown
A red hat costs $C and a blue hat costs $B. What
fraction of the cost of the blue hat is the cost of
the red hat?
8
Third Grade
In equal groups, the roles of the factors differ. One factor is the number of objects in a group (like any quantity in addition and subtraction situations), and the
other is a multiplier that indicates the number of groups. So, for example, 4 groups of 3 objects is arranged differently than 3 groups of 4 objects. Thus there
are two kinds of division situations depending on which factor is the unknown (the number of objects in each group or the number of groups). In the array
situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the
situation. But rows and columns depend on the orientation of the array. If an array is rotated 90°, the rows become columns and the columns become rows.
This is useful for seeing the commutative property for multiplication 3.OA.5 in rectangular arrays and areas. This property can be seen to extend to equal group
situations when equal group situations are related to arrays by arranging each group in a row and putting the groups under each other to form an array. Array
situations can be seen as equal group situations if each row or column is considered as a group. Relating equal group situations to arrays, and indicating rows
or columns within arrays, can help students see that a corner object in an array (or a corner square in an area model) is not double counted: at a given time, it
is counted as part of a row or as a part of a column but not both.
As noted in Table 3, row and column language can be difficult. The array problems given in the table are of the simplest form in which a row is a group and
Equal Groups language is used (“with 6 apples in each row”). Such problems are a good transition between the Equal Groups and array situations and can
support the generalization of the commutative property discussed above. Problems in terms of “rows” and “columns,” e.g., “The apples in the grocery window
are in 3 rows and 6 columns,” are difficult because of the distinction between the number of things in a row and the number of rows. There are 3 rows but the
number of columns (6) tells how many are in each row. There are 6 columns but the number of rows (3) tells how many are in each column. Students do need
to be able to use and understand these words, but this understanding can grow over time while students also learn and use the language in the other
3.MD.5–7
multiplication and division situations. Grade 3 standards focus on area measurement.
Area problems where regions are partitioned by unit squares are
3.MD
foundational for Grade 3 standards because area is used as a model for single-digit multiplication and division strategies.
Throughout multiplication and division learning, students gain fluency and begin to know certain products and unknown factors. All of the understandings of
multiplication and division situations, of the levels of representation and solving, and of patterns need to culminate by the end of Grade 3 in fluent multiplying
3.OA.7
and dividing of all single-digit numbers by 10.
Such fluency may be reached by becoming fluent for each number (e.g., the 2s, the 5s, etc.) and then
extending the fluency to several, then all numbers mixed together. Organizing practice so that it focuses most heavily on understood but not yet fluent products
and unknown factors can speed learning. To achieve this by the end of Grade 3, students must begin working toward fluency for the easy numbers as early as
possible. Because an unknown factor (a division) can be found from the related multiplication, the emphasis at the end of the year is on knowing from memory
all products of two one-digit numbers. As should be clear from the foregoing, this isn’t a matter of instilling facts divorced from their meanings, but rather the
outcome of a carefully designed learning process that heavily involves the interplay of practice and reasoning. All of the work on how different numbers fit with
the base-ten numbers culminates in these “just know” products and is necessary for learning products. Fluent dividing for all single-digit numbers, which will
combine just knows, knowing from a multiplication, patterns, and best strategy, is also part of this vital standard.
 The use of equal sized groups, arrays, and area models to multiplication and division situations.
 Problem-Solving strategies to multiply and divide to solve real-world problem situations.
Priority Standards
9
Third Grade




Knowledge of equal groups in multiplication and division to solve real-world problems.
Problem-solving skills to find the unknown in both multiplication and division problems.
Knowledge of the commutative property of multiplication to solve problems.
Use of arrays to solve unknown factor problems. Example: Twenty stickers have been arranged on a sheet into 5 rows. How many columns will
there be? 5 x? = 20
 Use of arrays to solve unknown product problems. Example: There are 7 bags with 3 apples in each bag for the field trip. How many apples in
all? 7 x 3 = ?
 Visual images and numerical patterns of multiplication and division will assist in solving problems.
 The commutative property of multiplication will help in performing computation as well as in problem-solving situations.
 Modeling multiplication and division problems based upon their problem-solving structure can help in finding solutions.
 How do modeling multiplication and division problems help in finding solutions?
 How can you use multiplication to solve division problems?
 How can modeling multiplication and division problems help in finding their solutions?
 How is the commutative property useful in solving multiplication problems?
 Model multiplication and division with equal groups.
 Identify and work with factors and multiples.
 Solve equations for the unknown.
 Multiply and divide multi-digit whole numbers.
 Solve basic problem-solving structures.
 Solve multi-step problems.
 That multiplication/division is an extension of repeated
 Interpret products of whole numbers as the total number of objects in
addition/subtraction
“so many” groups of “so many” objects each. (3.OA.1)
 How to apply the commutative property to multiplication
 Interpret whole-number quotients of whole numbers as the number of
objects in each share or as a number of equal shares. (3.OA.2)
 Use multiplication and division within 100 to solve word problems
in situations involving equal groups, arrays, and measurement
quantities. (3.OA.3)
 Use drawings and equations with a symbol for the unknown number to
represent the problem. (3.OA.3)
 Fluently multiply and divide within 100, using various strategies.
(3.OA.7)
 Measure and estimate liquid volumes and masses of objects using
Fluently multiply and divide within 100, using strategies such as the
relationship between multiplication and division (e.g., knowing that 8
Priority Standards
10
Third Grade
× 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. (3.OA.7)
 Concrete models of multiplication and division processes.
 Repeated verbalization of processes along with an anchor chart highlighting terms and steps.
 Anchor charts highlighting mathematical vocabulary specific to unit.
 Repeated practice verbalizing solution pathways.
Desired Outcomes:
Standards:
Represent and solve problems involving multiplication and division.
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 x 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 each. 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.
Multiply and divide within 100.
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.
1.
Make
sense
of
problems
and persevere in solving them. Students demonstrate their ability to persevere and identify appropriate strategies to solve
*
multiplication and division problems embedded within sophisticated problem-solving situations.
*2. Reason abstractly and quantitatively. Students demonstrate reasoning by justifying and explaining products of whole numbers as groups of objects (equal
groups/equal sharing). Students will make the connection between quantity and area models of multiplication and division.
*3. Construct viable arguments and critique the reasoning of others. Students will explain why specific multiplication and division strategies work. They will
also listen to each other and explain what their peers have said.
*4. Model with mathematics. In this unit, students are asked to use various modalities to show multiplication and division situations (drawings, arrays,
objects, etc.). Students are asked to communicate how their visuals represent these situations.
*5. Use appropriate tools strategically. Students will use concrete models to represent multiplication and division situations.
*6. Attend to precision. Students represent and describe the process of computations using inverse operations to justify their work.
* 7. Look for and make use of structure. Students will recognize and identify patterns existing within and between multiplication and division. Students will
Priority Standards
11
Third Grade
utilize parentheses to display the structure of these problems, i.e., 2(3 x 4) or 15 – (2 x 3). Students use this knowledge when applying strategies to evaluate
real-world situations of multiplication and division embedded within various problem-solving structures.
*8. Look for express regularity in repeated reasoning. Students will observe commonalities within and between multiplication and division.
Critical Terms
Supplemental Terms
English:
Spanish:
English:
Spanish:
 Multiply/multiplication
 Multiplicar/multiplicación
 Distributive property
 Propiedad distributiva
 Array
 Arreglo
 Zero property
 Propiedad del cero de la
multiplicación
 Product
 Producto

Identity
property

Propiedad de identidad de la
 Factor
 Factor
suma
 Property
 Divisor
 Divisor

Propiedad

Identity
property
 Dividend
 Dividendo
 Razonabilidad
 Reasonableness
 Quotient
 Cociente
 Computación mental

Mental
computation
 Combination
 Combinación
 Suma repetida
 Repeated addition
 Commutative property of
 Propiedad conmutativa de la
 Múltiple
multiplication
multiplicación
 Multiple
 Estimación
 Equal groups
 Grupos iguales
 Estimation
 Incógnita
 Multiplication sentence
 Enunciado de multiplicación
 Unknown
 Patrones
 Tree diagram
 Diagrama de árbol
 Patterns
 Division sentence
 Enunciado de división
 Fact family
 Familia de operación
 Inverse operations
 Operación inversas
 Partition
 Partición
 Related facts
 Operaciones relacionadas
 Repeated subtraction
 Resta repetida
 Decomposing
 Descomponer
 Divide/division
 Dividir/división
Priority Standards
12
Third Grade
Unit Three: Multiplication and Division Fluency and Application
Approximate Time Frame: 9-10 weeks November – January
Grade 2 students extend counting and modeling of quantities to equal groups and arrays. Students will begin to model multiplication using
rectangles partitioned into equivalent squares. They have begun extending the modeling of quantities to equal groups and arrays as a basis for
multiplication.
Focus of the Unit:
In this unit, Grade 3 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 and extend
students’ work with the distributive property. For example, in the picture the area of a 7 x 6 figure can be
determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums. Students will continue their
discovery of this concept and apply it to the composition and decomposition of shapes.
This unit will further address various problem-solving structures that students are expected to use while solving word problems involving multiplication and
division. Students should use a variety of representations for creating and solving one-step word problems. They explain and apply properties of operations as
strategies for finding their solutions to problems. In addition, students will solve two-step problems involving all four of the operations. Students will
determine the unknown in a multiplication and division equation and understand division as an unknown-factor problem. Patterns within multiplication and
division will be identified and students will explain them in more depth. This application unit will focus upon the multiplication and division situations of Equal
Groups and Arrays (See Table 3)
Fluency is also a focus of this unit. By studying patterns and relationships in multiplication facts and relating the operations of multiplication and division,
students will build a foundation for fluency with multiplication and division facts. Students will demonstrate fluency with multiplication facts through 10 and
the related division facts. Multiplying and dividing fluently refers to the skill of performing these operations accurately (using a reasonable amount of steps and
time), flexibility (using strategies such as the distributive property), and efficiently. NOTE: By the end of Grade 3, students will know from memory all products
of two one-digit numbers.
They will continue to work with products and quotients of whole numbers and apply properties of operations to problem-solving situations. Students will
uncover patterns within multiplication and division and use these to build fluency.
Students will utilize multiplication to find a total number of objects when there are a set number of groups with an equal number of objects in each group
or when an equal amount of objects are added or collected numerous times. There are two distinct models of division: partition models and number line
(repeated subtraction) models.
 Partition models provide students with a total number and the number of groups. These models focus on the question, “How many objects are in
each group so that the groups are equal?” A context for partition models would be: There are 12 cookies on the counter. If you are sharing the
cookies equally among three bags, how many cookies will go in each bag?
Ex. ? ? ? = 12
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Third Grade

Number line (repeated subtraction) models provide students with a total number and the number of objects in each group. These models focus on
the question, “How many equal groups can you make?” A context for number line models would be: There are 12 cookies on the counter. If you
put 3 cookies in each bag, how many bags will you fill?
Students in Grade 3 will also work with properties (rules about how numbers work) of multiplication. This grade will take their understanding beyond previous
expectations, in which students were asked to simply identify properties. While students DO NOT need to use the formal terms of these properties, student must
understand that properties are rules about how numbers work, and they need to apply each of them in varied situations. A brief definition for each of the
properties follows:
 The associative* property states that the sum or product stays the same when the order of addends or factors is changed. For example, when a
student multiplies 7 x 5 x 2, a student could rearrange the numbers to first multiply 5 x 2 = 10 and then multiply 10 x 7 = 70.
 The commutative property (order property) states that the order of numbers does not matter when you are adding or multiplying numbers.
For example, if a student knows that 5 x 4 = 20, then they also know that 4 x 5 = 20.
 The distributive* property of multiplication over addition is a strategy for using products students know to solve products they don’t know. Students
would be using mental math to determine a product. Here are ways that students could use the distributive property to determine the product of 7 x
6. Again, students should use the distributive property, but can refer to this in informal language such as “breaking numbers apart”.
7x6
7x6
7x6
7
x
3
=
21
5
x
6 =30
7 x 5 = 35
7 x 3 = 21
2 x 6 = 12
7x1=7
21 + 21 =42
30 + 12 = 42
35 + 7 = 42
Another example of the distributive property is the area model of multiplication. This helps students determine the products and factors of problems by
breaking numbers apart and visualizing their products. For example, for the problem 7 x 8 =?, students can decompose the 7 into a 5 and 2, and reach the
answer by multiplying 5 x 8 = 40 and 2 x 8 =16 and adding the two products (40 +16 = 56).
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*The use of parentheses are important in modeling the associative and distributive properties.
With respect to fluency, the standards use the word fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility
(using strategies such as the distributive property). By studying patterns in multiplication facts and relating multiplication to division, students will build a
foundation for fluency with these facts. Some strategies students may use to attain fluency are listed below:
• Multiplication by zeroes and ones
• Doubles (2s facts), Doubling twice (4s), Doubling three times (8s)
• Tens facts (relating to place value, 5 x 10 is 5 tens or 50)
• Five facts (half of tens)
• Skip counting (counting groups of and knowing how many groups have been counted)
• Square numbers (ex: 3 x 3)
• Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3)
• Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6)
• Turn-around facts (Commutative Property)
• Fact families (Ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24)
• Missing factors
NOTE: Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms.
Grade 4 students will multiply and divide with increasingly difficulty. They will also extend multiplication and division concepts into factors and
multiples. Patterns that flow from these operations will be generated and analyzed. Finally, Grade 4 students will multiply a fraction by a whole
number.
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Progression Citation: From the K-5 Operations and Algebraic progression document pp. 19, 23, 24, 26, 27.
Table 3: Multiplication and division situations
AxB=
Equal Groups of Objects
A x  = C and C ÷ A = 
Unknown Product
There are A bags with B plums in each bag. How
many plums are in all?
Group Size Unknown
If C plums are shared equality into A bags, then
how many plums will be in each bag?
 x B = C and C ÷ B = 
Number of Groups Unknown
If C plums are to be packed B to a bag, then how
many bags are needed?
Equal groups language
Arrays of Objects
Unknown Product
There are A rows of apples with B apples in each
row. How many apples are there?
Unknown Factor
If C apples are arranged in A equal rows, how
many apples will be in each row?
Unknown Factor
If C apples are arranged into equal rows of B
apples, how many rows will there be?
Row and column language
Unknown Product
The apples in the grocery window are in A rows
and B columns. How many apples are there?
Unknown Factor
If C apples are arranged into an array with A rows,
how many columns of apples are there?
Larger Unknown
A blue hat costs $8. A red hat costs A times as
much as the blue hat. How much does the red hat
cost?
Smaller Unknown
A red hat costs $C and that is A times as much as
blue hat costs. How much does a blue hat cost?
Unknown Factor
If apples are arranged into an array with B columns,
how many rows are there?
A>1
Compare
Multiplier Unknown
A red hat costs $C and a blue hat costs $B. How
many times as much does the red hat cost as the
blue hat?
A<1
Smaller Unknown
A blue hat costs $B. A red hat costs A as much as
the blue hat. How much does the red hat cost?
Larger Unknown
A red hat costs $C and that is A of the cost of a
blue hat. How much does a blue hat cost?
Multiplier Unknown
A red hat costs $C and a blue hat costs $B. What
fraction of the cost of the blue hat is the cost of
the red hat?
In equal groups, the roles of the factors differ. One factor is the number of objects in a group (like any quantity in addition and subtraction situations), and the
other is a multiplier that indicates the number of groups. So, for example, 4 groups of 3 objects are arranged differently than 3 groups of 4 objects. Thus there
are two kinds of division situations depending on which factor is the unknown (the number of objects in each group or the number of groups). In the array
situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the
situation. But rows and columns depend on the orientation of the array. If an array is rotated 90°, the rows become columns and the columns become rows.
This is useful for seeing the commutative property for multiplication 3.OA.5 in rectangular arrays and areas. This property can be seen to extend to equal group
situations when equal group situations are related to arrays by arranging each group in a row and putting the groups under each other to form an array. Array
situations can be seen as equal group situations if each row or column is considered as a group. Relating equal group situations to arrays, and indicating rows
or columns within arrays, can help students see that a corner object in an array (or a corner square in an area model) is not double counted: at a given time, it
is counted as part of a row or as a part of a column but not both. As noted in Table 3, row and column language can be difficult. The array problems given in
the table are of the simplest form in which a row is a group and equal groups language is used (“with 6 apples in each row”). Such problems are a good
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Third Grade
transition between the equal groups and array situations and can support the generalization of the commutative property discussed above. Problems in terms
of “rows” and “columns,” e.g., “The apples in the grocery window are in 3 rows and 6 columns,” are difficult because of the distinction between the number
of things in a row and the number of rows. There are 3 rows but the number of columns (6) tells how many are in each row. There are 6 columns but the
number of rows (3) tells how many are in each column. Students do need to be able to use and understand these words, but this understanding can grow over
3.MD.5–7
time while students also learn and use the language in the other multiplication and division situations. Grade 3 standards focus on area measurement.
Area problems where regions are partitioned by unit squares are foundational for Grade 3 standards because area is used as a model for single-digit
3.MD.
multiplication and division strategies.
Using a letter for the unknown quantity, the order of operations, and two-step word problems with all four
operations Students in Grade 3 begin the step to formal algebraic language by using a letter for the unknown quantity in expressions or equations for one and
two-step problems.3.OA.8 But the symbols of arithmetic, x for multiplication and ÷ for division, continue to be used in Grades 3, 4, and 5. Understanding and
using the associative and distributive properties (as discussed above) requires students to know two conventions for reading an expression that has more than
one operation:
1. Do the operation inside the parentheses before an operation outside the parentheses (the parentheses can be thought of as hands curved around
the symbols and grouping them).
2. If addition and subtraction is in the same equation as multiplication or division, students need to understand and utilize the order of operations
and complete the multiplication and division first. At Grades 3 through 5, parentheses can usually be used for such cases so that fluency with this rule can wait
until Grade 6. These conventions are often called the order of operations (Teacher tip: teachers have used PEMDAS (parenthesis, exponents, multiplication,
division, addition and then subtraction) or the mnemonic device “Please Excuse My Dear Aunt Sally” as a fun way for kids to remember these for years to
come) and can seem to be a central aspect of algebra. But actually they are just simple “rules of the road” that allow expressions involving more than one
operation to be interpreted unambiguously and thus are connected with the mathematical practice of communicating precisely (Math Practice 6). Use of
parentheses is important in displaying structure and thus is connected with the mathematical practice of making use of structure(Math Practice 7).
Parentheses are important in expressing the associative and especially the distributive properties. These properties are at the heart of Grades 3 to 5, the Grade
3 two-step word problems vary greatly in difficulty and ease of representation. More difficult problems may require two steps of representation and solution
rather than one. Use of two-step problems involving easy or middle difficulty adding and subtracting within 1,000 or one such adding or subtracting with one
step of multiplication or division can help to maintain fluency with addition and subtraction while giving the needed time to the major Grade 3 multiplication
and division standards.
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 The use of equal sized groups, arrays, and area models to multiplication and division situations.
 Problem-Solving strategies to multiply and divide to solve real-world problem situations.
 Knowledge of equal groups in multiplication and division to solve real-world problems.
 Problem-solving skills to find the unknown in both multiplication and division problems.
 Knowledge of arithmetic patterns with the patterns with the properties of operations to solve multiplication and division problems.
 The use of arrays to solve unknown factor problems. Example: Twenty stickers have been arranged on a sheet into 5 rows. How many columns
will there be? 5 x ? = 20
 The use of arrays to solve unknown product problems. Example: There are 7 bags with 3 apples in each bag for the field trip. How many apples
in all? 7 x 3 = ?
 Visual images and numerical patterns of multiplication and division will assist in solving problems.
 The properties of operations will help in performing computation as well as in problem-solving situations. (Distributive Property of Multiplication,
Commutative property of multiplication, identity, and zero.)
 How do modeling multiplication and division problems help in finding solutions?
 How can the strategy of breaking apart (decomposing) numbers make multiplication easier to understand?
 How can we use multiplication to solve division problems?
 How do multiples and factors relate to multiplication and division?
 How can modeling multiplication and divisions problems help in finding their solutions?
 What are the Properties of Operations?
 Model with equal groups by partitioning rectangles.
 Identify and work with factors and multiples.
 Solve equations for the unknown.
 Multiply and divide multi-digit whole numbers.
 Identify arithmetic patterns including, patterns in the addition and
 Multiply fractions by whole numbers.
multiplication table
 Solve multi-step problems.
 Solve basic problem-solving structures.
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 Multiplication and division facts.
 Problem-solving structures for arrays and for equal groups.
Priority Standards
 Interpret products of whole numbers as the total number of objects in
“so many” groups of “so many” objects each. (3.OA.1)
 Interpret whole-number quotients of whole numbers as the number of
objects in each share or as a number of equal shares. (3.OA.2)
 Determine the unknown whole number in a multiplication and division
equation relating three whole numbers. (3.OA.4)
 Apply properties of operations as strategies to multiply and divide.
(3.OA.5)
 Understand division as an unknown-factor problem. (3.OA.6)
 Fluently multiply and divide within 100, using various strategies.
(3.OA.7)
 Identify arithmetic patterns (including patterns in the
addition table or multiplication table). (3.OA.9)
 Explain arithmetic patterns using properties of operations.
(3.OA.9)
 Use multiplication and division within 100 to solve word problems
in situations involving equal groups, arrays, and measurement
quantities. (3.OA.3)
 Use drawings and equations with a symbol for the unknown number to
represent the problem. (3.OA.3)
 Solve two-step word problems using the four operations. (3.OA.8)
 Represent these problems using equations with a letter standing for
the unknown quantity. (3.OA.8)
 Assess the reasonableness of answers using mental computation and
estimation strategies including rounding. (3.OA.8)
 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. (3.OA.7)
 Multiply one-digit whole numbers by multiples of 10 in the range
10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and
properties of operations. (3.NBT.3)
19
Third Grade
 Repeated verbalization of processes along with an anchor chart highlighting terms and steps.
 Anchor charts highlighting mathematical vocabulary specific to unit.
Desired Outcomes:
Standards:
Represent and solve problems involving multiplication and division.
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 x 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 each. 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 multiplication or division equation relating three whole numbers. For example, determine the unknown
number that makes the equation true in each of the equations 8 x ? = 48, 5 = □ ÷ 3, 6 x 6 = ?.
3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 is known, then 4 x 6 = 24 is also known. (Commutative property of
multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find
8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)
Understand properties of multiplication and the relationship between multiplication and division.
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.
Multiply and divide within 100.
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.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.8 Solve two-step word problems using the four operations (+, -, x, ÷.) 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.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties
of operations.
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Third Grade
1.
Make
sense
of
problems
and persevere in solving them. Students demonstrate their ability to persevere and identify appropriate strategies to solve
*
multiplication and division problems embedded within sophisticated problem-solving situations.
*2. Reason abstractly and quantitatively. Students demonstrate reasoning by justifying and explaining products of whole numbers as groups of objects (equal
groups/equal sharing). Students will make the connection between quantity and area models of multiplication and division.
*3. Construct viable arguments and critique the reasoning of others. Students will explain why specific multiplication and division strategies work. They will
also listen to each other and explain what their peers have said.
*4. Model with mathematics. In this unit, students are asked to use various modalities to show multiplication and division situations (drawings, arrays,
objects, etc.). Students are asked to communicate how their visuals represent these situations.
*5. Use appropriate tools strategically. Students will use concrete models to represent multiplication and division situations.
*6. Attend to precision. Students represent and describe the process of computations using inverse operations to justify their work.
*7. Look for and make use of structure. Students will recognize and identify patterns existing within and between multiplication and division. Students will
utilize parentheses to display the structure of these problems, i.e., 2(3 x 4) or 15 – (2 x 3). Students use this knowledge when applying strategies to evaluate
real-world situations of multiplication and division embedded within various problem-solving structures.
*8. Look for express regularity in repeated reasoning. Students will observe commonalities within and between multiplication and division.
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Third Grade
Critical Terms
Supplemental Terms
English:
Spanish:
English:
Spanish:
 Decompose
 Descomponer
 Inverse operation
 Operaciones inversas
 Multiple
 Múltiple
 Commutative property
 Propiedad conmutativa de la
multiplicación
 Equation
 Ecuación
 Property
 Propiedad
 Operations
 Operaciones
 Equal groups
 Grupos iguales
 Identity Property of
 Propiedad de identidad de la
Multiplication
multiplicación
 Equal shares
 Partes iguales
 Known Fact
 Operación conocida
 Fact family/related facts
 Operaciones relacionadas
 Zero Property of
 Propiedad del cero de la
 Fact family
 Familia de operaciones
Multiplication
multiplicación
 Reasonableness
 Razonabilidad
 Associative Property of
 Propiedad asociativa de la
 Mental computation
 Computación mental
Multiplication
multiplicación
 Array
 Arreglo
 Distributive Property
 Propiedad distributiva
 Product
 Producto
 Evaluate
 Evaluar
 Factor
 Factor
 Expression
 Expresión
 Divisor
 Divisor
 Variable
 Variable
 Dividend
 Dividendo
 Remainder
 Residuo
 Quotient
 Cociente
 Strategies
 Estrategia
 Multiplication
 Multiplicación
 Patterns
 Patrones
 Division
 División
 Unknown
 Incógnita
 Estimation
 Estimación
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Third Grade
Unit Four: Fractions
Approximate Time Frame: 4-6 Weeks February – Mid March
Grade 2 students have partitioned circles and rectangles into two, three, or four equal shares. They have used fractional language such as halves, thirds, half
of, a third of, etc., and described the whole as two halves, three thirds, four fourths. Students have begun to recognize that equal shares of identical wholes
need not have the same shape, the basis of equivalency.
Focus of the Unit:
In this unit, Grade 3 students begin their work on fractions in a more formal mathematical sense. This unit involves the sharing of a whole being partitioned,
however models in Grade 3 include only area models and linear models (number lines).* Beginning with unit fractions students model the whole as the sum
of fractional parts, and build an understanding that the size of the part is relative to the size of the whole. Students will also use fractions to represent
numbers equal to, less than, and greater than one. Comparison of fractions is done using visual models (circles, rectangles, squares) and strategies based on
observations of equal numerators or denominators. *NOTE: Set models are not addressed in Grade 3.
Understanding that fractional parts must be equal-sized is an important concept for
students to develop. (See diagram.) This development of congruence should be modeled
with a variety of area models, such as tiles or pattern blocks, for students to gain this
idea of same size, same shape. Students will need this understanding to solve word
problems requiring them to create and reason about fair share.
Linear model of fractional parts begins for the first time in Grade 3 as students position
fractions on number lines between whole numbers. Positioning fractions on the number line
can model location, measurement of distance, and also serves as a precursor to addition.
Furthermore, number lines will allow for modeling of equivalencies of fractions.
Finally, comparing fractions requires students in Grade 3 to reason about their size. Both area models and linear models can be used for this purpose.
Grade 4 students will compare fractions with different numerators and different denominators, compose and decompose fractions and mixed numbers, and
add and subtract fractions with like denominators. They will use the four operations to solve word problems involving distances, intervals of time, liquid
volumes, masses of objects, and money. Finally Grade 4 students will understand decimal notation for fractions and compare decimal fractions.
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Third Grade
From the 3-5 Number and Operations – Fractions progression document, pp. 2, 3, and 4
Grade 3 students start with unit fractions (fractions with numerator 1), which are formed by partitioning a whole into equal parts and taking one part, e.g., if
a whole is partitioned into 4 equal parts then each part is 1/4 of the whole, and 4 copies of that part make the whole. Next, students build fractions from unit
3.NF.1
They read any fraction this way, and
fractions, seeing the numerator 3 of 3/4 as saying that 3/4 is the quantity you get by putting 3 of the 1/4’s together.
in particular there is no need to introduce “proper fractions" and “improper fractions" initially; 5/3 is the quantity you get by combining 5 parts together
when the whole is divided into 3 equal parts.
The number line and number line diagrams On the number line, the whole is the unit interval, that
is, the interval from 0 to 1, measured by length. Iterating this whole to the right marks off the whole
numbers, so that the intervals between consecutive whole numbers, from 0 to 1, 1 to 2, 2 to 3, etc.,
are all of the same length, as shown. Students might think of the number line as an infinite ruler.
To construct a unit fraction on a number line diagram, students partition the unit interval into
3 intervals of equal length and recognize that each has length 1/3. They locate the number 1/3 on
the number line by marking off this length from 0, and locate other fractions with denominator 3 by
marking off the number of lengths indicated by the numerator.
3.NF.2
Students sometimes have difficulty perceiving the unit on a number line diagram. When locating a
fraction on a number line diagram, they might use as the unit the entire portion of the number
line that is shown on the diagram, for example indicating the number 3 when asked to show 3/4
on a number line diagram marked from 0 to 4. Although number line diagrams are important
representations for students as they develop an understanding of a fraction as a number, in the
early stages of the Fraction Progression they use other representations such as area models, tape
diagrams, and strips of paper. These, like number line diagrams, can be subdivided, representing
an important aspect of fractions.
The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the
point on the number line reached by marking off 5 times the length of the unit interval from 0, so
5/3 is the point obtained in the same way using a different interval as the basic unit of length,
namely the interval from 0 to 1/3.
Comparing fractions Previously, in Grade 2, students compared lengths using a standard measurement unit.2.MD.3 In Grade 3 they build on this idea to
compare fractions with the same denominator. They see that for fractions that have the same denominator, the underlying unit fractions are the same
size, so the fraction with the greater numerator is greater because it is made of more unit fractions. For example, segment from 0 to 3/4 is shorter
Priority Standards
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Third Grade
3.NF.3d
than the segment from 0 to 5/4 because it measures 3 units of 1/4 as opposed to 5 units of 1/4. Therefore 3/4 < 5/4 .
Students also see that for
unit fractions, the one with the larger denominator is smaller, by reasoning, for example, that in order for more (identical) pieces to make the same
whole, the pieces must be smaller. From this they reason that for fractions that have the same numerator, the fraction with the smaller denominator
is greater. For example 2/5 > 2/7, because 1/7 < 1/5, so 2 lengths of 1/7 is less than 2 lengths of 1/5.
As with equivalence of fractions, it is important in comparing fractions to make sure that each fraction refers to the same whole. As students move towards
thinking of fractions as points on the number line, they develop an understanding of order in terms of position. Given two fractions—thus two points on the
number line—the one to the left is said to be smaller and the one to right is said to be larger. This understanding of order as position will become important in
Grade 6 when students start working with negative numbers.
 Problem-solving skills to understand fractions as they relate to real-world problem situations, such as in measurement, cooking, pizza, money, music, etc.
 The size of the fractional part is relative to the size of the whole.
 Fractions represent quantities where a whole is divided into equal-sized parts using models, manipulatives, words, and/or number lines.
 Fractions can be used as a tool to understand and model quantities and relationships.
 Fractions are composed of unit fractions.
 Fractions that represent equal-sized quantities are equivalent.
 What do fractions represent?
 What makes fractions equivalent?
 Identify and work with more fifths, tenths, twelfths and/or fractions
 Divide shapes (circles and rectangles) into no more than 4 equal
with unlike denominators.
sections and use vocabulary terminology to describe.
 How to visually represent and identify a fraction using a variety of
 Divide shapes into parts with equal areas. (3.G.2)
models.
 Represent the area of each part as a unit fraction. (3.G.2)
 How to recognize and generate simple equivalent fractions.
 Represent a whole using unit fractions. (3.NF.1)
 How to compare fractions with the same numerator or denominator
 Use the term numerator to indicate the number of parts and
by reasoning about their size.
denominator to represents the total number of parts a whole is
 How to divide shapes into equal sections (halves, thirds, fourths,
partitioned into. (3.NF.1)
sixths and eighths)
 Represent a fraction as the composition of unit fractions. (3.NF.1)
 Divide a number line diagram into equal segments and label the
appropriate fractional parts. (3.NF.2)
 Model equivalent fractions using manipulatives, pictures, or number
Priority Standards
25
Third Grade
line diagrams and explain in words why the fractions are equivalent.
(3.NF.3)
 Represent whole numbers as fractions using area models, number
line diagrams, and numbers. (3.NF.3)
 Compare two fractions with the same numerator or same denominator
using visual models, symbols and words. (3.NF.3)
 Recognize that comparisons are valid only when the two
fractions refer to identical wholes. (3.NF.3)
 Anchor charts and visuals highlighting mathematical vocabulary specific to fractions.
Desired Outcomes:
Standards:
Develop understanding of fractions as numbers
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 fractions as number on number line. Represent fractions on 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., ½ = 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 comparison with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
Reason with shapes and their attributes.
3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
Priority Standards
26
Third Grade
*1. Make sense of problems and persevere in solving them. Students demonstrate their ability to persevere and utilize reasoning to make sense of part- partwhole relationships.
2. Reason abstractly and quantitatively. Students will reason about the size of fractions. Students will make the connection between area models and linear
models of fractions.
*3. Construct viable arguments and critique the reasoning of others. Students may construct arguments using concrete models of fractions to reason about the
whole as they examine the fractional parts. They explain their thinking to others and respond to others’ thinking.
4. Model with mathematics. In this unit, students representing fractions and wholes in multiple ways including numbers, words (mathematical
language), drawing pictures, objects, etc. Both area models of fractions and linear models will be used.
* 5. Use appropriate tools strategically. Students will use concrete models to represent part-part-whole relationships.
*6. Attend to precision. Students represent and use clear and precise mathematical language in their descriptions of fractions as specifying the whole.
7. Look for and make use of structure. Students will recognize and utilize the structure of the part-part-whole relationships between various fractional pieces.
*8. Look for express regularity in repeated reasoning. Students will observe commonalities within the various models for fractional pieces and what they
represent.
English:
 Fraction
 Equivalent
 Denominator
 Numerator
 Unit fraction
 Partition
 Comparison
 Half
 Third
 Fourth
 Sixth
 Eighth
Critical Terms:
Spanish:
 Fracciona
 Equivalente
 Denominador
 Numerito
 Fracción unitaria
 Partición
 Comparación
 Mitad
 Tercio
 Cuartos
 Sextos
 Octavos
Supplemental Terms:
English:
Spanish:
 Part/part/whole
 Parte/parte/total
 Linear measurement (using a
 Medidas lineales (using a
fraction to show distance)
fraction to show distance)
 Equal parts
 Partes iguales
 Equal distance (intervals)
 Distancia igual (intervals)
 Equivalence
 Equivalencia
 Reasonable
 Razonable
 Justify
 Justifica
Priority Standards
27
Third Grade
Priority Standards
28
Third Grade
Unit Five: Measurement
Approximate Time Frame: 5-7 Weeks March – April
Second Grade students extended their work with telling time to the hour and half-hour in First Grade in order to tell (orally and in writing) the time indicated on
both analog and digital clocks to the nearest five minutes. Students made connections between skip counting by 5s and telling time to the nearest five minutes
on an analog clock. Students also indicated if the time is in the morning (a.m.) or in the afternoon/evening (p.m.) as they recorded the time. In addition, they
partitioned circles and rectangles into 2, 3 or 4 equal shares (regions). Concept exploration with paper strips and pictures provided a base for time
measurement vocabulary terms “halves”, “fourths “ and “quarters.” They have used vocabulary and visuals for quarter hours.
Students in Grade 2 have extended addition and subtraction skills to within 100 and become fluent. They will have had experience representing and solving
one- and two-step situational problems involving addition and subtraction within 100, including relationships to length
Prior to the study of area, third grade students developed an understanding of the concept of perimeter through various experiences, such as walking around the
perimeter of a room, using a geoboard, or using technology. Initially, students counted unit lengths with sides marked. Later, the lengths of the sides were
labeled with numerals. They found the perimeter of objects; used addition to find perimeters; and recognized the patterns that exist when finding the sum of the
lengths and widths of rectangles. Students also used tools, such as geoboards, tiles, graph paper or technology to find all the possible rectangles that have a
given perimeter. They recorded all the possibilities using dot or graph paper, compiled the possibilities into an organized list or a table, and determined whether
they had all the possible rectangles. This experience allowed students to reason about connections between their representations, side lengths, and the
perimeter of the rectangles. In addition, given a perimeter and a length or width, students used objects or pictures to find missing length or width.
During students’ introduction to multiplication readiness in Grade 2, they explored areas of rectangular shapes, leading to the understanding that
rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students
will connect area to repeated addition, and justify using addition to determine the area of a rectangle. This skill can be applied to the study of area in third
grade.
As an extension from first grade, second grade students posed a question, determined up to four categories of possible responses, collected data,
represented data on a picture graph or bar graph, and interpreted the results. They used selected graphs to identify specific aspects of the data collected,
including the sum of responses, which category had the most/least responses, and differences/similarities between the four categories. They also solved
simple one-step problems using information from graphs.
Focus of the Unit:
In Grade 3, students use their knowledge and experience with the number line, from 2nd grade, to extend multi-digit addition, subtraction, multiplication
and division skills to solve problems involving elapsed time, area and perimeter.
Priority Standards
29
Third Grade
Students will solve elapsed time interval problems. Clock models or pre-determined (intervals every 5 or 15 minutes)/open number lines (intervals
determined by students) could be used to determine solutions. Students should be given opportunities to determine time interval units and size of jumps on
their number line. Elapsed time problems should include start, change and ending time situations.
Grade 3 students should have a variety of opportunities to develop an understanding that perimeter is the boundary of a plane/2-dimensional figure, such use
rubber bands on geoboards to create shapes with different perimeters, tracing a shape on paper or a whiteboard, making polygons with straws. The length of
the perimeter of a polygon is the sum of the lengths of the sides. Students should count unit length marks of the sides of a polygon during early learning
experiences. They need to count the length-units and not the end-points of a polygon. Eventually, numerals can be used to label side lengths. Next, students
use a ruler to mark unit lengths and measure lengths of polygons. Gradually students move on to measure everyday items such as area rugs, desk tops, etc.;
use addition to find perimeters; and recognize the patterns that exist when finding the sum of the lengths and widths of rectangles. Students should have
opportunities to use tools strategically, such as geoboards, graph paper or technology to find all the possible rectangles that have a given perimeter (e.g., find
the rectangles with a perimeter of 16 cm). They draw all the possible figures using dot or graph paper, record the possibilities into an organized chart, and
verify whether they have all the possible rectangles. Building on this experience, they will make logical connections about the relationship of their
representations, side lengths/widths, and the perimeter of the rectangles. Also, given a perimeter and a length or width, students will find the unknown length
or width.
Third grade students will solve one- and two-step problems involving picture graphs, bar graphs and line graphs. Students should have a variety of learning
experiences involving reading and problem solving using horizontal and vertical scaled graphs with different intervals to further develop their understanding of
scale graphs and number facts prior to creating one. Working with scaled graphs relates to students’ understanding of counting patterns and repeated addition.
While exploring data concepts, students should pose a question, collect data, analyze data, and interpret data. It is important that they graph data that is related
to personal experiences, e.g., reading genre choices, extracurricular activities. Scaled pictographs include a title, symbols that represent multiple units, categories,
category label and data. At the right is an example of a pictograph with symbols that represent multiple units. Bar graphs include a title, scale, scale label,
categories, category label, and data.
Each
Priority Standards
stands for 5 pets
30
Third Grade
Building upon addition concepts relative to perimeter, third grade students will recognize area as an attribute of two-dimensional regions. They will 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, i.e. square cm, square in. Covering a region with “unit squares,” could include square tiles or shading on
grid or graph paper. Based on students’ development, they should have ample experiences filling a region with square tiles before transitioning to pictorial
representations on graph paper. Counting the square units to find the area could be done in metric, customary, or non-standard square units. Using different
sized graph paper, students will explore the areas measured in square centimeters and square inches. Students can use geoboards, tiles, graph paper, or
technology to find all the possible rectangles with a given area, e.g., find the rectangles that have an area of 12 square units. They will record all the
possibilities using dot or graph paper, compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles.
Students can then investigate the perimeter of the rectangles with a given area, such as 12. Further investigation will involve rectangles and squares with the
same perimeter.
Examples of Rectangles with an Area of 12:
*Adapted from North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards (Updated 2012).
Based on work in third grade, fourth graders will explore and discuss the pros and cons of various formulas for finding the perimeter length of a rectangle that
is measured in length (l) units by width (w) units. When solving perimeter problems that only label one length and one width of a rectangle, remembering the
basic formula for perimeter of a rectangle is P = 2 (l x w) will help students avoid the error of adding one length to one width. Students will progress from
marking all length units along the sides of a rectangle to showing just parts of units as a reminder of which kind of unit is being used. In Grade 4, students will
use multi-digit addition and subtraction to solve real-world problems and represent problems as equations with a variable in place of the unknown quantity.
They will apply the formulas for perimeter when solving real world and mathematical problems.
Based on work in third grade, fourth grade students learn to consider applications using rectangle area. Fourth graders use multiplication; spatially structuring
arrays, and area, they determine the formula for the area of a rectangle A= l x w. Students will generalize that, given a unit of length, a rectangle whose sides
have the length of 4 w (width) units and 3 l (length) units, can be portioned into 4w (width) rows of unit squares with 3l (length) squares in each row. The product
l x w gives the number of unit squares in the partition, thus the area measurement is l x w square units. These square units are derived from the length unit.
Repeated reasoning about how to calculate rectangle area will help students see area formulas as summaries of addition calculations. In Grade 4 and later
grades, rectangle area problems become more complex. Students learn to apply these understandings and formulas to the solution of real-world and
mathematical problems.
Priority Standards
31
Third Grade
Taken from K-5 Geometric Measurement Progression Document pages 16 and 18.
Students in Grade 3 learn to solve a variety of problems involving measurement and such attributes as length and area, liquid volume, mass and time.
Such work involves units of mass such as the kilogram.
3.MD.1, 3.MD.2
A few words on volume are relevant. Compared to the work in area, volume introduces more complexity, not only in adding a third dimension and thus
presenting a significant challenge to students’ spatial structuring, but also in the materials whose volumes are measured. These materials may be solid or fluid,
so their volumes are generally measured with one of two methods, e.g., “packing” a right rectangular prism with cubic units or “filling” a shape such as a right
circular cylinder. Liquid measurement, for many third graders, may be limited to a one-dimensional unit structure (i.e., simple iterative counting of height that is
not processed as three-dimensional). Thus, third graders can learn to measure with liquid volume and to solve problems requiring the use of the four arithmetic
operations, when liquid volumes are given in the same units throughout each problem. Because liquid measurement can be represented with one-dimensional
scales, problems may be presented with drawings or diagrams, such as measurements on a beaker with a measurement scale in milliliters.
Perimeter
Third graders focus on solving real-world and mathematical problems involving perimeters of polygons. A perimeter is the boundary of a two-dimensional
shape. For a polygon, the length of the perimeter is the sum of the lengths of the sides. Initially, it is useful to have sides marked with unit length marks,
allowing students to count the unit lengths. Later, the lengths of the sides can be labeled with numerals. As with all length tasks, students need to count the
length-units and not the end-points. Next, students learn to mark off unit lengths with a ruler and label the length of each side of the polygon. For rectangles,
parallelograms, and regular polygons, students can discuss and justify faster ways to find the perimeter length than just adding all of the lengths (MP3).
Rectangles and parallelograms have opposite sides of equal length, so students can double the lengths of adjacent sides and add those numbers or add lengths
of two adjacent sides and double that number. A regular polygon has all sides of equal length, so its perimeter length is the product of one side length and the
number of sides.
Perimeter problems for rectangles and parallelograms often give only the lengths of two adjacent sides or only show numbers for these sides in a drawing of
the shape. The common error is to add just those two numbers. Having students first label the lengths of the other two sides as a reminder is helpful. Students
then find unknown side lengths in more difficult “missing measurements” problems and other types of perimeter problems.
Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures
With strong and distinct concepts of both perimeter and area established, students can work on problems to differentiate their measures. For example, they can
find and sketch rectangles with the same perimeter and different areas or with the same area and different perimeters and justify their claims (MP3).
Problem solving involving measurement and estimation of intervals of time, liquid volumes, and masses of objects
Students in Grade 3 learn to solve a variety of problems involving measurement and such attributes as length and area, liquid volume, mass, and time.
Priority Standards
32
Third Grade
From the K-5 Geometric Measurement progression document, pg. 16 and 18.
Third graders focus on learning area. Students learn formulas to compute area, with those formulas based on, and summarizing, a firm conceptual foundation
about what area is. Students need to learn to conceptualize area as the amount of two-dimensional space in a bounded region and to measure it by choosing a
unit of area, often a square. A two-dimensional geometric figure that is covered by a certain number of squares without gaps or overlaps can be said to have
3.MD.5
an area of that number of square units.
Activities such as those in the Geometry Progression teach students to compose and decompose geometric
regions. To begin an explicit focus on area, teachers might then ask students which of three rectangles covers the most area. Students may first solve the
problem with decomposition (cutting and/or folding) and re-composition, and eventually analyses with area-units, by covering each with unit squares
(tiles).3.MD.5, 3.MD.6 Discussions should clearly distinguish the attribute of area from other attributes, notably length.
Third grade students will recognize perimeter as an attribute of plane figures and distinguish between linear and area measures with strong and distinct
concepts of both perimeter and area. Established students can work on problems to differentiate their measures. For example, they can find and sketch
rectangles with the same perimeter and different areas or with the same area and different perimeters and justify their claims (MP3).3.MD.8 Differentiating
perimeter from area is facilitated by having students draw congruent rectangles and measure, mark off, and label the unit lengths all around the perimeter
on one rectangle, then do the same on the other rectangle but also draw the square units. This enables students to see the units involved in length and area
and find patterns in finding the lengths and areas of non-square and square rectangles (MP7). Students can continue to describe and show the units
involved in perimeter and area after they no longer need these.
 Addition and subtraction problem solving skills to create and interpret picture and
bar graphs.
 Knowledge of perimeter to real-world problem solving situations.
 Area measurement to real-world problem solving situations.
 Add To, Result Unknown Example: If the area of a garden is measured in square feet, one side of the garden is 8 feet and another side is 2 feet. What
is the area of the garden? Solution: 8 + 8 =? or 2 + 2 + 2 + 2 = 16
 Information can be represented in bar graph, picture graph and line plot form. These graphs can be used to help us solve one and two- step math
problems.
 Area is additive.
 There is a relationship between area and multiplication.
 Metric measurement units are related to place value concepts/multiples of 10.
 Elapsed time is the interval of time, given a specific unit, from a starting time to an ending time.
 Perimeter and addition are related.
 A linear unit is used to measure perimeter.
Priority Standards
33
Third Grade
 Everyday objects have a variety of attributes, each of which can be measured in many ways.
 Area and addition are related.
 Perimeter and area are related.
 How does metric measurement connect to multiples of 10?
 How can understanding the relationship between addition and subtraction aid us in problem solving?
 How do we use data represented in bar graphs, picture graphs, and line plot to make sense of world around us?
 How does elapsed time help us to plan and organize real life responsibilities?
 How can understanding the relationship between addition and area aid in problem solving?
 Measure length and represent that data in a line plot.
 Fluently add & subtract multi-digit whole numbers using the
standard algorithm.
 Relate metric measurement to concepts and multiples of 10.
 Solve multi-step problems for perimeter involving more complex
 Know from memory with fluency and automaticity sums of all onepolygons and rectangles.
digit numbers.
 Solve multi-step problems for area involving more complex polygons
 Fluently add and subtract within 100.
and rectangles.
 Use addition and subtraction within 100 to solve one- and two- step
word problems involving situations of adding to, taking from, putting
together, taking apart, and comparing, with unknowns in all positions.
 Count and skip count within 1000 by 5s, 10s, 100s.
 Add up to four two-digit numbers with and without
regrouping.
 Subtract two digit numbers with and without regrouping.
 Use place value and properties of operations to explain why
addition and subtraction strategies work.
 Read and write numbers to 1000 using word, standard, and
expanded form.
 Mentally add and subtract 10 and 100 from a number 100-900.
 Compare 3-digit numbers using < = > symbols.
 Add two digit numbers with and without regrouping.
 Use a ruler to measure side length.
 Solve real world and mathematical problems involving
perimeters of polygons.
 Find the perimeter given the side lengths.
 Find an unknown side length.
 Exhibiting rectangles with the same perimeter.
Priority Standards
34
Third Grade
 Problem-solving structures for area/arrays and for equal groups.
 Metric measurements units for liquid volume and weight.
 Addition and subtraction computation and problem solving
strategies.
 A.M. represents time from midnight to noon.
 P.M. represents time from noon to midnight.
 60 min = 1 hour.
 Addition problem solving strategies.
 A square unit is used to measure area.
Priority Standards
 Generate measurement data by measuring lengths to the ¼ and ½ inch.
(3.MD.4)
 Show data in a line plot given a scale in ½, ¼, or whole numbers.
(3.MD.4)
 Tell and write time to the nearest minute. (3.MD.1)
 Solve word problems involving elapsed time. (3.MD.1)
 Use a number line or clocks to model elapsed time and record
calculations. (3.MD.1)
 Draw and label a picture graph, bar graph and line plot to represent a
data set (including the scale, title, categories, etc.). (3.MD.3)
 Solve one- and two-step “how many more” and “how many less”
problems using information presented in bar graphs. (3.MD.3)
 Solve real world and mathematical problems involving perimeters of
polygons, including finding the perimeter given the side lengths.
(3.MD.8)
polygons, including finding an unknown side length. (3.MD.8)
 Relate area to the operations of multiplication and addition. (3.MD.7)
 Recognize areas as an attribute of plane figures and understand
concepts of area measurement. (3.MD.5)
 Measure areas by counting unit squares (square cm, square m,
square in., square ft., and improvised units). (3.MD.6)
 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. (3.MD.7)
 Use area models to represent the distributive property in mathematical
reasoning. (3.MD.7)
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. (3.MD.8)
 Measure and estimate liquid volumes and masses of objects using
standard units of grams (g), kilograms (kg), and liters (l). (3.MD.2)
 Add, subtract, multiply, or divide to solve one-step word problems
35
Third Grade
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.2)
 Anchor Charts highlighting mathematical vocabulary specific to unit.
 Explicit instruction for time and perimeter measurement vocabulary, and picture graph and bar graph vocabulary.
 The use of visual tools such as analog clocks and digital clocks, tiles, geoboards, etc.
 Time interval number labels, including fractional partitions.
 Vertical and horizontal scaled graph models, including picture graphs and bar graphs
 An awareness of area measurement vocabulary.
 Labels for square measurement units.
 The use of visual tools such as tiles, geoboards, etc.
Desired Outcomes:
Standards:
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
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.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.
Represent and interpret data.
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.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.5 Recognize areas as an attribute of plane figures and understand concepts of area measurement.
a. 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.
b. 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).
Priority Standards
36
Third Grade
3.MD.7 Relate area to the operations of multiplication and addition.
a. 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.
b. 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.
c. 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 x b and a x c. Use area models to
represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
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.
*1. Make sense of problems and persevere in solving them. Students can tell if word problems involve addition or subtraction and choose a modality to
represent the problem. Problems should encompass real-world situations involving distance, time and graphs, money (either dollars or cents, but not
mixed).
*2. Reason abstractly and quantitatively. Students will demonstrate their abstract and quantitative reasoning by adding and subtracting 3-digit numbers to
see if solutions are reasonable.
*3. Construct viable arguments and critique the reasoning of others. Students will create visual models of word problems (manipulative or picture) and
explain their models. Given a perimeter and a length or width, students use objects or pictures to find the missing length or width. They justify and
communicate their solutions using words, diagrams, pictures, numbers, and an interactive whiteboard. They will also listen to each other, explain what their
peers have said, and give reasons why they agree or disagree.
*4. Model with mathematics. In this unit, students demonstrate modeling by transferring raw data to graphical representations and to real-world
situations. Students can transfer between modalities to solve real-world problems.
*5. Use appropriate tools strategically. Strategic use of tools is demonstrated when students draw hands on an analog clock, given a digital clock. It can be
demonstrated when students use number lines to represent time or other visual models to represent real-world problems and estimations. It can also be
demonstrated when students use manipulatives to explain their 3-digit calculations.
*6. Attend to precision. Students demonstrate precision by fluently adding and subtracting 3-digit numbers and telling time accurately to the minute. They
use precise language to describe their strategies.
*7. Look for and make use of structure. Students demonstrate this standard by identifying the type of story problem (adding to, taking from, putting
together, taking apart, comparing). They also demonstrate understanding of structure by building and describing 3-digit numbers and explaining their value,
by analyzing the structure of a graph, and showing the location of an unknown in relation to an equation or visual model.
*8. Look for express regularity in repeated reasoning. When students can use the clock structure and relate it to a number line, and when they set up a
scaled graph, they also use repeated reasoning. When students can use the structures of word problems to solve problems, they also demonstrate their ability
to use repeated reasoning.
Priority Standards
37
Third Grade
English:
 Survey
 Data
 Key
 Scale
 Picture Graph
 Pictograph
 Bar Graph
 Analyze
 Quarter Inch (1/4)
 Perimeter
 Area
 Square Unit
 Analog Clock
 Capacity
 Digital clock
 Gram (g)
 Kilogram (kg)
 Liquid Volume
 Liter (L)
 Milliliter (mL)
 Mass
 Metric Unit
 Time Interval
 Unit
 Frequency Table
 Half Inch (1/2)
 Interpret
 Line plot
 Tally Mark(s)
 Tally Chart
 Composite figure
 Formula
Critical Terms
Supplemental Terms
Spanish:
English:
Spanish:
 Encuesta
 Model
 Modelo
 Datos
 Fact Family
 Familia de operaciones
 Clave
 Addition
 Adición
 Escala
 Addend
 Sumando
 Gráfica con imágenes
 Subtraction
 Restas
 Pictograma
 A.M./P.M.
 A.M./P.M.
 Gráfica de barras
 Symbol
 Símbolo
 Analiza
 Title Labels
 Etiqueta del titulo
 Cuarto de pulgada (1/4)
 Compare
 Compara
 Perímetro
 How many more/less
 Cuantos más/menos
 Área
 Chart
 Tabla
 Unidad cuadrada
 Quarter to/’till
 Un cuarto a
 Reloj analógico
 Quarter of
 Un cuarto de
 Capacidad
 Quarter past
 Un cuarto después de
 Reloj digital
 Midnight
 Media noche
 Gramo (g)
 Noon
 Medio día
 Kilogramo (kg)
 Digit
 Digito
 Volumen líquido
 Greater Than
 Mayor que
 Litro (L)
 Lesson Than
 Menor que
 Mililitro (mL)
 Hour
 Hora
 Masa
 Minute
 Minuto
 Unidad métrica
 Nonstandard units
 Unidad no estándar
 Intervalo de tiempo
 Gap
 Espacio
 Unidad
 Overlap
 Solapo
 Tabla de frecuencias
 Tiling
 Embaldosar
 Media pulgada (1/2)
 Linear
 Lineal
 Interpreta
 Formula
 Formula
 Diagrama lineal
 Graph
 Gráfica
 Marcas de conteo
 Array
 Arreglo
 Tabla de conteo
 Figura compuesta
 Formula
Priority Standards
38
Third Grade









Unit Square
Elapsed Time
Category
Equal
Square cm, m, in, and ft
Length
Width
Side Length
U.S. Customary










Metric System

Unidad cuadrada
Tiempo transcurrido
Categoría
Igual
Cuadrado (cm, m, in, and ft)
Longitud
Ancho
Longitud del lado
Sistema tradicional de EE.
UU.
Sistema métrico
Priority Standards
39
Third Grade
Priority Standards
40
Third Grade
Unit Six: Geometry
Approximate Time Frame: 1-2 Weeks May
In second grade, students identify and draw triangles, quadrilaterals, pentagons, and hexagons based on attributes.
Grade 2 students have partitioned circles and rectangles into two, three, or four equal shares. They have used fractional language such as halves, thirds, half
of, a third of, etc., and described the whole as two halves, three thirds, four fourths. Students have begun to recognize that equal shares of identical wholes
need not have the same shape, the basis of equivalency.
Focus of the Unit:
Third graders build on experience in second grade and further investigate quadrilaterals (technology may be used during this exploration). Students recognize
shapes that are and are not quadrilaterals by examining the properties of the geometric figures. They conceptualize that a quadrilateral must be a closed figure
with four straight sides and begin to notice characteristics of the angles and the relationship between opposite sides. Students should be encouraged to provide
details and use proper vocabulary when describing the properties of quadrilaterals. They sort geometric figures and identify squares, rectangles, and rhombuses
as quadrilaterals.
Understanding that fractional parts must be equal-sized is an important concept for students to develop.
(See diagram.) This development of congruence should be modeled with a variety of area models, such
as tiles or pattern blocks, for students to gain this idea of same size, same shape. Students will need this
understanding to solve word problems requiring them to create and reason about fair share.
Fourth grade students will build a firm foundation of several shape categories; these categories can be the foundation for thinking about the relationships
between classes. Students will classify 2-dimensional shapes by attributes and drawing shapes that fit specific categories, including classification by their
angles and angle sizes as they relate to geometric properties of perpendicularity and parallelism.
Priority Standards
41
Third Grade
From the K-6 Geometry progression document, p. 13
Students analyze, compare, and classify two-dimensional shapes by their properties (see the
3.G.1
footnote on p. 3).
They explicitly relate and combine these classifications. Because they
have built a firm foundation of several shape categories, these categories can be the raw
material for thinking about the relationships between classes. For example, students can form
larger, super ordinate, categories, such as the class of all shapes with four sides, or
quadrilaterals, and recognize that it includes other categories, such as squares, rectangles,
rhombuses, parallelograms, and trapezoids. They also recognize that there are quadrilaterals
that are not in any of those subcategories.
Students learn to draw shapes with pre-specified attributes, without making prior assumptions
regarding their classification.MP1 For example, they could solve the problem of making a shape
with two long sides of the same length and two short sides of the same length that is not a
rectangle.
 Knowledge of shapes by drawing representations in different categories that share attributes and recognize those shapes (quadrilaterals) in real world
settings.
Priority Standards
42
Third Grade
 Objects can be described and compared using their geometric attributes.
 Figures are categorized according to their attributes.
 The size of the fractional part is relative to the size of the whole.
 Fractions represent quantities where a whole is divided into equal-sized parts
 How can 2-dimensional shapes be described?
 How are geometric figures constructed?
 Grade 2 students work with shapes as they recognize, identify and
 Connect lines to lines of symmetry in two-dimensional figures.
draw various shapes based upon attributes.
 Identify angles.
 Divide shapes (circles and rectangles) into no more than 4 equal
 Classify shapes by the properties of their angles.
sections and use vocabulary terminology to describe.
 Geometric shapes that represent quadrilaterals.
 Divide shapes into parts with equal areas. (3.G.2)
 Shapes are categorized.
 Represent the area of each part as a unit fraction. (3.G.2)
 Quadrilaterals are two-dimensional.
 Analyze, compare, and classify 2-dimensional shapes by their
properties. (3.G.1)
 Draw shapes with pre-specified attributes. (3.G.1)
 Investigate, describe, and reason about decomposing and
composing quadrilaterals to make other quadrilaterals.
(3.G.1)
 Understand that shapes in different categories may share
attributes and belong to a larger category. (3.G.1)
 Recognize and draw examples of more complex quadrilaterals.
(3.G.1)



Labeling shapes on anchor charts and providing manipulates in those shapes for identification/matching purposes.
Identifying real world objects that represent various quadrilaterals.
Relating the terms ‘quad’ and ‘lateral’ to students’ native languages.
Priority Standards
43
Third Grade
Desired Outcomes:
Standards:
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 ¼ of the area of the shape.
* 1. Make sense of problems and persevere in solving them. Students demonstrate their ability to persevere by drawing shapes with pre-specified attributes.
*2. Reason abstractly and quantitatively. Students demonstrate reasoning by justifying and explaining attributes of quadrilaterals in words and drawings.
*3. Construct viable arguments and critique the reasoning of others. Students will be able to explain why specific shapes are called
quadrilaterals. They will also listen to each other and explain what their peers have said.
*4. Model with mathematics. In this unit, students are asked to use various modalities and model shapes with manipulatives or drawings. They are asked to
communicate how their visuals represent these shapes.
*5. Use appropriate tools strategically. Students will use concrete models to represent shapes. Students will use concrete models to represent shapes.
*6. Attend to precision. Students precisely solve problems such as finding all the possible different compositions of quadrilaterals that make other
quadrilaterals.
*7. Look for and make use of structure. Students will observe, identify, and categorize quadrilaterals based upon attributes.
*8. Look for and express regularity in repeated reasoning. Students will notice commonalities in attributes.
Priority Standards
44
Third Grade
English:
 Quadrilateral
 Rhombus
 Rectangle
 Square
 Attribute
 Endpoint
 Pentagon
 Hexagon
 Trapezoid
 Parallelogram
 Parallel
 Angle
 Polygon
 2-Dimensional
 Octagon
 Ray
 Right Angle
 Right Triangle
 Triangle
 Vertex
 Plane Figure
 Segment
Critical Terms
Spanish:
English:
 Cuadrilátero
 Geometric
 Rombo
 Kite
 Rectángulo
 Degree
 Cuadrado
 Compare
 Atributo
 Flat
 Extremo
 Solid
 Pentágono
 3-Dimensional
 Hexágono
 Adjacent
 Trapecio
 Equal parts
 Paralelogramo
 Fraction
 Paralelo
 Partition
 Angulo
 Denominator
 Polígono
 Numerator
 Bidimensional
 Unit fraction
 Octágono
 Half
 Semirrecta
 Third
 Angulo recto
 Fourth
 Triangulo rectángulo
 Intersect
 Triangulo
 Vértice
 Figura plana
 Segmento
Supplemental Terms
Spanish:
 Geométrico
 Cometa
 Grado
 Compara
 Plano
 Solido
 Tridimensional
 Adyacente
 Partes iguales
 Fracción
 Partición
 Denominador
 Numerador
 Fracción unitaria
 Mitad
 Tercio
 Cuarto
 Secante
Priority Standards
45
Third Grade
Priority Standards
46
References:
Common Core State Standards for Mathematics. (2010). : Common Core State Standards Initiative.
Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action: Implementing the Common Core
Standards for Mathematical Practice, K-8. : Heinemann.
Illinois Early Learning Developmental Standards (2013). Springfield: Illinois State Board of Education.
Illinois Mathematics Curriculum Model Units. (2013). Springfied: Illinois State Board of Education .
K-8 Publishers' Criteria for the Common Core State Standards for Mathematics. (2013, April 9). . Retrieved
November 1, 2014, from http://www.corestandards.org/assets/Math_Publishers_Criteria_K8_Summer%202012_FINAL.pdf
Marzano, R. J., & Simms, J. A. (2013). Vocabulary for the Common Core. : Marzano Research Laboratory.
My Math. (2013). Columbus: McGraw-Hill Education.
PARCC Model Content Frameworks Mathematics . (2012). Washington: Partnership for Assessment of
Readiness for College and Careers .
Tomlinson, C. A., & McTighe, J. (2006). Integrating Differentiated Instruction & Understanding by Design
Connecting Content and Kids. Alexandria, Va.: Association for Supervision and Curriculum Development.

community unit district 308: mathematics curriculum guide

Transcription

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