Grade 3 Overview
Operations and Algebraic Thinking (OA)
 Represent and solve problems involving multiplication and
division. (1*, 2, 3, 4)
 Understand properties of multiplication and the relationship
between multiplication and division. (5, 6)
 Multiply and divide within 100. (7)
 Solve problems involving the four operations, and identify and
explain patterns in arithmetic. (8, 9)
Mathematical Practices (MP)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Number and Operations in Base Ten (NBT)
 Use place value understanding and properties of operations to
perform multi-digit arithmetic. (1, 2, 3)
Number and Operations—Fractions (NF)
 Develop understanding of fractions as numbers. (1, 2, 3)
Measurement and Data (MD)
 Solve problems involving measurement and estimation of
intervals of time, liquid volumes, and masses of objects. (1, 2)
 Represent and interpret data. (3, 4)
 Geometric measurement: understand concepts of area and
relate area to multiplication and to addition. (5, 6, 7)
 Geometric measurement: recognize perimeter as an attribute of
plane figures and distinguish between linear and area measures.
(8)
Geometry (G)
 Reason with shapes and their attributes. (1, 2)
*Numbers following cluster heading indicate standards included in the cluster.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
1
In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies
for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1);
(3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.
(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving
equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these
situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use
properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve
multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship
between multiplication and division.
(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions,
and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative
to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon
is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5
equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve
comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.
(3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size
units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area.
Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into
rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.
(4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles,
and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit
fraction of the whole.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
2
Standards for Mathematical Practice
Standards
Students are expected to:
Explanations and Examples
Mathematical Practices are
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
3.MP.1. Make sense of
problems and persevere in
solving them.
3.MP.2. Reason abstractly
and quantitatively.
3.MP.3. Construct viable
arguments and critique the
reasoning of others.
3.MP.4. Model with
mathematics.
3.MP.5. Use appropriate
tools strategically.
3.MP.6. Attend to
precision.
3.MP.7. Look for and make
use of structure.
In third grade, students know that doing mathematics involves solving problems and discussing
how they solved them. Students explain to themselves the meaning of a problem and look for
ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize
and solve problems. They may check their thinking by asking themselves, “Does this make
sense?” They listen to the strategies of others and will try different approaches. They often will
use another method to check their answers.
Third graders should recognize that a number represents a specific quantity. They connect the
quantity to written symbols and create a logical representation of the problem at hand,
considering both the appropriate units involved and the meaning of quantities.
In third grade, students may construct arguments using concrete referents, such as objects,
pictures, and drawings. They refine their mathematical communication skills as they participate
in mathematical discussions involving questions like “How did you get that?” and “Why is that
true?” They explain their thinking to others and respond to others’ thinking.
Students experiment with representing problem situations in multiple ways including numbers,
words (mathematical language), drawing pictures, using objects, acting out, making a chart, list,
or graph, creating equations, etc. Students need opportunities to connect the different
representations and explain the connections. They should be able to use all of these
representations as needed. Third graders should evaluate their results in the context of the
situation and reflect on whether the results make sense.
Third graders consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, they may use graph paper
to find all the possible rectangles that have a given perimeter. They compile the possibilities into
an organized list or a table, and determine whether they have all the possible rectangles.
As third graders develop their mathematical communication skills, they try to use clear and
precise language in their discussions with others and in their own reasoning. They are careful
about specifying units of measure and state the meaning of the symbols they choose. For
instance, when figuring out the area of a rectangle they record their answers in square units.
In third grade, students look closely to discover a pattern or structure. For instance, students use
properties of operations as strategies to multiply and divide (commutative and distributive
properties).
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
3
Standards for Mathematical Practice
Standards
Students are expected to:
Explanations and Examples
Mathematical Practices are
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
3.MP.8. Look for and
express regularity in
repeated reasoning.
Students in third grade should notice repetitive actions in computation and look for more shortcut
methods. For example, students may use the distributive property as a strategy for using
products they know to solve products that they don’t know. For example, if students are asked to
find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8
to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work by asking
themselves, “Does this make sense?”
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
4
Grade 3
Operations and Algebraic Thinking (OA) (4 Clusters)
3.OA Represent and solve problems involving multiplication and division (Cluster 1- Standards 1, 2, 3, and 4)
Essential Concepts
Essential Questions
 Just as addition and subtraction are related, so too, are multiplication
and division.
 Multiplication is finding an unknown product (the whole), and division is
finding an unknown factor (how many groups or how many in each
group--See Table 2).
 The position of the unknown can change, creating easier and more
difficult problem types.
 Common multiplication and division situations involve equal groups,
arrays/area and compare.
 Multiple models and strategies can be used to solve multiplication and
division problems.
3.OA.1
Standard 3.OA.1
Interpret products of whole
numbers, e.g., interpret 5 × 7
as the total number of objects
in 5 groups of 7 objects each.
For example, describe a
context in which a total
number of objects can be
expressed as 5 × 7.
Connections: 3.0A.3; 3.SL.1;
ET03-S1C4-01
3.OA.2
Standard 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
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.4. Model
with mathematics.
3.MP.7. Look for
and make use of
structure.
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.





Describe a context in which a total number of objects can be
expressed as 5 x 7.
Describe a context in which a number of groups or a number of shares
can be expressed as 56 ÷ 8.
What kinds of problems in your world might be modeled and solved
with multiplication/division?
How might multiplication help you solve a division problem?
How can estimation be useful when solving multiplication and division
problems?
Examples & Explanations
Students recognize multiplication as a means to determine the total number of objects when there
are a specific number of groups with the same number of objects in each group. Multiplication
requires students to think in terms of groups of things rather than individual things. Students learn
that the multiplication symbol ‘x’ means “groups of” and problems such as 5 x 7 refer to 5 groups of
7.
To further develop this understanding, students interpret a problem situation requiring multiplication
using pictures, objects, words, numbers, and equations. Then, given a multiplication expression
(e.g., 5 x 6) students interpret the expression using a multiplication context. (See Table 2) They
should begin to use the terms, factor and product, as they describe multiplication.
Students may use interactive whiteboards to create digital models.
Examples & Explanations
Students recognize the operation of division in two different types of situations. One situation
requires determining how many groups and the other situation requires sharing (determining how
many in each group). Students should be exposed to appropriate terminology (quotient, dividend,
divisor, and factor).
To develop this understanding, students interpret a problem situation requiring division using
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
5
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.MP.4. Model
with mathematics.
3.MP.7. Look for
and make use of
structure.
pictures, objects, words, numbers, and equations. Given a division expression (e.g., 24 ÷ 6)
students interpret the expression in contexts that require both interpretations of division. (See Table
2)
Students may use interactive whiteboards to create digital models.
Connections: 3.OA.3; 3.SL.1;
ET03-S1C4-01
3.OA.3
Standard 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. (See Table 2.)
Connections: 3.RI.7; ET03S1C1-01
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.4. Model
with mathematics.
3.MP.7. Look for
and make use of
structure.
Examples & Explanations
Students use a variety of representations for creating and solving one-step word problems, i.e.,
numbers, words, pictures, physical objects, or equations. They use multiplication and division of
whole numbers up to 10 x10. Students explain their thinking, show their work by using at least one
representation, and verify that their answer is reasonable.
Word problems may be represented in multiple ways:
 Equations: 3 x 4 = ?, 4 x 3 = ?, 12 ÷ 4 = ? and 12 ÷ 3 = ?
 Array:

Equal groups


Repeated addition: 4 + 4 + 4 or repeated subtraction
Three equal jumps forward from 0 on the number line to 12 or three equal jumps
backwards from 12 to 0
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
6
Examples of division problems:
 Determining the number of objects in each share (partitive division, where the size of the
groups is unknown):
o The bag has 92 hair clips, and Laura and her three friends want to share them
equally. How many hair clips will each person receive?

Determining the number of shares (measurement division, where the number of groups is
unknown)
Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4
bananas each day, how many days will the bananas last?
Starting
24
Day 1
24-4=
20
Day 2
20-4=
16
Day 3
16-4=
12
Day 4
12-4=
8
Day 5
8-4=
4
Day 6
4-4=
0
Solution: The bananas will last for 6 days.
Students may use interactive whiteboards to show work and justify their thinking
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
7
3.OA Represent and solve problems involving multiplication and division (Cluster 1- Standards 1, 2, 3, and 4)
3.OA.4
Standard 3.OA.4
Mathematical
Examples & Explanations
Determine the unknown
This standard is strongly connected to 3.OA.3 where students solve problems and determine
Practices
whole number in a
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 × ? = 48, 5 =  ÷
3, 6 × 6 = ?.
Connections: 3.AO.3; 3.RI.3;
3.SL.1;
ET03-S1C4-01
3.MP.1. Make
sense of problems
and persevere in
solving them.
unknowns in equations. Students should also experience creating story problems for given
equations. When crafting story problems, they should carefully consider the question(s) to be
asked and answered to write an appropriate equation. Students may approach the same story
problem differently and write either a multiplication equation or division equation.
3.MP.2. Reason
abstractly and
quantitatively.
Students apply their understanding of the meaning of the equal sign as ”the same as” to interpret
an equation with an unknown. When given 4 x ? = 40, they might think:
 4 groups of some number is the same as 40
 4 times some number is the same as 40
 I know that 4 groups of 10 is 40 so the unknown number is 10
 The missing factor is 10 because 4 times 10 equals 40.
3.MP.6. Attend to
precision.
3.MP.7. Look for
and make use of
structure.
Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown
in different positions.
Examples:
 Solve the equations below:
24 = ? x 6
72   = 9
 Rachel has 3 bags. There are 4 marbles in each bag. How many marbles does Rachel
have altogether? 3 x 4 = m
Students may use interactive whiteboards to create digital models to explain and justify their
thinking.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
8
3.OA Understand properties of multiplication and the relationship between multiplication and division (Cluster 2Standards 5 and 6)
Essential Concepts
Essential Questions
 There is an inverse relationship between multiplication and division.
Multiplication is finding an unknown product (the whole), and division is
finding an unknown factor (see Table 2).
 Understanding the properties of multiplication (commutative,
associative, distributive) helps us become efficient and flexible problem
solvers.





3.OA.5
Standard 3.OA.5
Apply properties of operations
as strategies to multiply and
divide. (Students need not
use formal terms for these
properties.) Examples: If 6 ×
4 = 24 is known, then 4 × 6 =
24 is also known.
(Commutative property of
multiplication.) 3 × 5 × 2 can
be found by 3 × 5 = 15, then
15 × 2 = 30, or by 5 × 2 = 10,
then 3 × 10 = 30. (Associative
property of multiplication.)
Knowing that 8 × 5 = 40 and
8 × 2 = 16, one can find 8 × 7
as 8 × (5 + 2) = (8 × 5) + (8 ×
2) = 40 + 16 = 56.
(Distributive property.)
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.4. Model
with mathematics.
How are multiplication and division related?
How might you rewrite this multiplication problem as a division
problem? Mary has three friends. She gave each friend four stickers.
How many stickers did she give her friends in all?
Can the order of the factors be reversed in a multiplication problem? If
so, is this always true? Why or why not?
Can the order of the numbers be reversed in a division problem? If so,
is this always true? Why or why not?
How might you decompose this array to help you solve the
multiplication problem 6 x 7?
Examples & Explanations
Students represent expressions using various objects, pictures, words and symbols in order to
develop their understanding of properties. They multiply by 1 and 0 and divide by 1. They change
the order of numbers to determine that the order of numbers factors does not make a difference in
multiplication (but does make a difference in division). Given three factors, they investigate
changing the order of how they multiply the numbers to determine that changing the order of the
factors does not change the product. They also decompose numbers to build fluency with
multiplication.
Models help build understanding of the commutative property:
3.MP.7. Look for
and make use of
structure.
3.MP.8. Look for
and express
regularity in
repeated
reasoning.
Connections: 3.OA.1; 3.OA.3;
Example: 3 x 6 = 6 x 3
In the following diagram it may not be obvious that 3 groups of 6 is the same as 6 groups of 3. A
student may need to count to verify this.
is the same quantity as
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
9
3.RI 4; 3.RI.7; 3.W.2; ET03S1C4-01
Example: 4 x 3 = 3 x 4
An array explicitly demonstrates the concept of the commutative property.
4 rows of 3, or 4 x 3
3 rows of 4, or 3 x 4
Students are introduced to the distributive property of multiplication over addition as a strategy for
using products they know to solve products they don’t know. For example, if students are asked to
find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to
arrive at 40 + 16 or 56. Students should learn that they can decompose either of the factors. It is
important to note that the students may record their thinking in different ways.
5 x 8 = 40
2 x 8 = 16
56
7 x 4 = 28
7 x 4 = 28
56
To further develop understanding of properties related to multiplication and division, students use
different representations and their understanding of the relationship between multiplication and
division to determine if the following types of equations are true or false.
 0 x 7 = 7 x 0 = 0 (Zero Property of Multiplication)
 1 x 9 = 9 x 1 = 9 (Multiplicative Identity Property of 1)
 3x6=6x3
(Commutative Property)
 8÷2=2÷8
(Students are only to determine that these are not equal)
 2x3x5=6x5
 10 x 2 < 5 x 2 x 2
 2 x 3 x 5 = 10 x 3
 0x6>3x0x2
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
10
3.OA Understand properties of multiplication and the relationship between multiplication and division (Cluster 2Standards 5 and 6)
3.OA.6
Standard 3.OA.6
Mathematical
Examples & Explanations
Understand division as an
Multiplication and division are inverse operations and that understanding can be used to find the
Practices
unknown-factor problem. For
example, find 32 ÷ 8 by
finding the number that
makes 32 when multiplied by
8.
3.MP.1. Make
sense of problems
and persevere in
solving them.
Connections: 3.OA.4; 3.RI.3
3.MP.7. Look for
and make use of
structure.
unknown. Fact family triangles demonstrate the inverse operations of multiplication and division by
showing the two factors and how those factors relate to the product and/or quotient.
Examples:
 3 x 5 = 15
 15 ÷ 3 = 5
5 x 3 = 15
15 ÷ 5 = 3
15
X or ÷
3
5
Students use their understanding of the meaning of the equal sign as “the same as” to interpret an
equation with an unknown. When given 32 ÷
= 4, students may think:
 4 groups of some number is the same as 32
 4 times some number is the same as 32
 I know that 4 groups of 8 is 32 so the unknown number is 8
 The missing factor is 8 because 4 times 8 is 32.
Equations in the form of a ÷ b = c and c = a ÷ b need to be used interchangeably, with the unknown
in different positions.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
11
3.OA Multiply and divide within 100 (Cluster 3- Standard 7)
Essential Concepts
Essential Questions
 The foundation for fluency is based on the study of patterns and
relationships in multiplication and division facts.



3.OA.7
Standard 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.
Connections: 3.OA.3; 3.OA.5
Mathematical
Practices
3.MP.2. Reason
abstractly and
quantitatively.
3.MP.7. Look for
and make use of
structure.
3.MP.8. Look for
and express
regularity in
repeated
reasoning.
What strategies help you solve an unknown fact?
o How can you use known facts to help you find unknown facts?
If you don’t know 6x9, how can you use 6x10 to help?
What properties help you solve an unknown fact?
How can you explain the patterns observed in multiplication and
division combinations/facts?
Examples & Explanations
By studying patterns and relationships in multiplication facts and relating multiplication and division,
students build a foundation for fluency with multiplication and division facts. Students demonstrate
fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing
fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately,
and skill in performing them flexibly, accurately, and efficiently.
Strategies students may use to attain fluency include:
 Multiplication by zeros 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
General Note: Students should have exposure to multiplication and division problems presented in
both vertical and horizontal forms.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
12
3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic (Cluster 4Standards 8 and 9)
Essential Concepts
Essential Questions
 The four basic arithmetic operations are interrelated, and the properties
of each may be used to understand the others.
 Some mathematical problems may require multiple steps to solve.
 Unknowns in an equation can be represented by a letter or symbol (a or
b rather than ∆ or
).
 Estimation can be used to determine the reasonableness of an answer.
3.OA.8
Standard 3.OA.8
Solve two-step word
problems using the four
operations. Represent these
problems using equations
with a letter standing for the
unknown quantity. Assess the
reasonableness of answers
using mental computation
and estimation strategies
including rounding. (This
standard is limited to
problems posed with whole
numbers and having wholenumber answers; students
should know how to perform
operations in the conventional
order when there are no
parentheses to specify a
particular order (Order of
Operations).
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.




How are addition, subtraction, multiplication and division related?
What strategies help you determine the reasonableness of an
answer?
What numeric patterns do you see? Use the properties of
operation(s) to explain why those patterns exist.
How can estimation be useful when solving multiplication and division
problems?
Examples & Explanations
Students should be exposed to multiple problem-solving strategies (using any combination of
words, numbers, diagrams, physical objects or symbols) and be able to choose which ones to use.
Examples:
 Jerry earned 231 points at school last week. This week he earned 79 points. If he uses 60
points to earn free time on a computer, how many points will he have left?
3.MP.2. Reason
abstractly and
quantitatively.
3.MP.4. Model
with mathematics.
A student may use the number line above to describe his/her thinking, “231 + 9 = 240 so
now I need to add 70 more. 240, 250 (10 more), 260 (20 more), 270, 280, 290, 300, 310
(70 more). Now I need to count back 60. 310, 300 (back 10), 290 (back 20), 280, 270,
260, 250 (back 60).”
3.MP.5. Use
appropriate tools
strategically.
A student writes the equation, 231 + 79 – 60 = m and uses rounding (230 + 80 – 60) to
estimate.
A student writes the equation, 231 + 79 – 60 = m and calculates 79-60 = 19 and then
calculates 231 + 19 = m.
Connections: 3.OA.4; 3.OA.5;
3.OA.6; 3.OA.7; 3.RI.7

The soccer club is going on a trip to the water park. The cost of attending the trip is $63.
Included in that price is $13 for lunch and the cost of 2 wristbands, one for the morning and
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Grade 3
Board Approved 03/27/2012
13
one for the afternoon. Write an equation representing the cost of the field trip and determine the
price of one wristband.
w
w
13
63
The above diagram helps the student write the equation, w + w + 13 = 63. Using the
diagram, a student might think, “I know that the two wristbands cost $50 ($63-$13) so one
wristband costs $25.” To check for reasonableness, a student might use front end
estimation and say 60-10 = 50 and 50 ÷ 2 = 25.
When students solve word problems, they use various estimation skills which include identifying
when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate
method of estimation, and verifying solutions or determining the reasonableness of solutions.
Estimation strategies include, but are not limited to:
 using benchmark numbers that are easy to compute
 front-end estimation with adjusting (using the highest place value and estimating from the
front end making adjustments to the estimate by taking into account the remaining
amounts)
 rounding and adjusting (students round down or round up and then adjust their estimate
depending on how much the rounding changed the original values)
Tucson Unified School District
Mathematics Curriculum
Grade 3
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3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic (Cluster 4Standards 8 and 9)
3.OA.9
Standard 3.OA.9
Mathematical
Examples & Explanations
Identify arithmetic patterns
Students need ample opportunities to observe and identify important numerical patterns related to
Practices
(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.
Connections: 3.SL.1; ET03S1.C3.01
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.2. Reason
abstractly and
quantitatively.
3.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
3.MP.6. Attend to
precision.
3.MP.7. Look for
and make use of
structure.
operations. They should build on their previous experiences with properties related to addition and
subtraction. Students investigate addition and multiplication tables in search of patterns and explain
why these patterns make sense mathematically. For example:
 Any sum of two even numbers is even.
 Any sum of two odd numbers is even.
 Any sum of an even number and an odd number is odd.
 The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two
equal groups.
 The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles
(multiples of 2) in a multiplication table fall on horizontal and vertical lines.
 The multiples of any number fall on a horizontal and a vertical line due to the commutative
property.
 All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other
multiple of 5 is a multiple of 10.
Students also investigate a hundreds chart in search of addition and subtraction patterns. They
record and organize all the different possible sums of a number and explain why the pattern makes
sense.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
15
Additional Domain Information – Operations and Algebraic Thinking (OA)
Key Vocabulary




Array
Dividend
Division/divide
Divisor




Equal
Equal
groups/shares
Equation
Estimation





Factor
Multiples
Operation
Product
Multiplication/multiply

Properties:
commutative,
associative, identity,
distributive, zero


Quotient
Square number
Example Resources

Books
 Young Mathematicians at Work: Constructing Multiplication and Division by Catherine Twomey Fosnot and Maarten Dolk


Developing Number Concepts, Book 3 Place Value, Multiplication and Division by Kathy Richardson
Teaching Student-Centered Mathematics, Grades K-3 by John A Van de Walle and LouAnn H. Lovin

Technology
 http://multiplication.com/internet_resources.htm Multiplication online games, download games, activities, and lessons.
 http://nlvm.usu.edu/en/nav/grade_g_2.html Virtual manipulatives for grades 3 - 5.
 http://www.topmarks.co.uk/Interactive.aspx?cat=23 Interactive white board links/games/activities for multiplication and division
 http://www.topmarks.co.uk/Interactive.aspx?cat=29 Interactive white board links for enhancing problem solving skills.
 The National Library of Virtual Manipulatives: http://nlvm.usu.edu/en/nav/vlibrary.html
 http://investigations.terc.edu/library/Games_23.cfm#a_place Online games and activities
 http://gtdifferentiation.sites.fcps.org/Grade3 Extension menus for gifted students in multiplication
 https://docs.google.com/viewer?a=v&pid=sites&srcid=cGZsZXhvbmxpbmUubmV0fHd3d3xneDo0YmQ1ZmY2YTU3Y2FjNjIz Extension
menu for differentiation/gifted students in multiplication
 http://www.pflugervilleisd.net/curriculum/math/place_value_model.cfm#enrichment_extension Extension menus for differentiation/gifted
students in multiplication
 http://www.internet4classrooms.com/skill_builders/multiplication_math_third_3rd_grade.htm links to multiplication activities
 http://www.internet4classrooms.com/skill_builders/word_problems_math_third_3rd_grade.htm links to story problem/problem solving
activities
 http://www.prometheanplanet.com/en-us/Search/resources/grade/3-5/country/unitedstates/language/english/?Keywords=division&SortField=relevance Division anchor charts, interactive PowerPoint presentations

Exemplary Lessons
 http://illustrativemathematics.org/standards/k8 Illustrative Mathematics Project
 Promethean Planet
http://www.prometheanplanet.com/en-us/Resources/Item/92241/relating-multiplication-and-division
http://www.prometheanplanet.com/en-us/Resources/Item/40851/lesson-4-5-third-grade-everyday-mathematics (multiplication)
http://www.prometheanplanet.com/en-us/Resources/Item/68997/word-problems
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
16

NCTM:
http://illuminations.nctm.org/LessonsList.aspx?grade=2&standard=1

http://illuminations.nctm.org/LessonDetail.aspx?ID=U109 (4 lessons)
In this unit, students explore several meanings and representations of multiplication (number line, equal sets, arrays, and balanced
equations). They also learn about the order (commutative) property of multiplication, the results of multiplying by 1 and by 0, and the
inverse property of multiplication.
Assessments
All assessments used must align with our TUSD Curriculum and hold everyone involved accountable for the important mathematical concepts of this
domain. That is, all assessments for this domain must focus on assessing the degree to which third grade learners can, within the parameters specified
in the 9 standards included in this domain,
 represent and solve problems involving multiplication and division,

demonstrate understanding of properties of multiplication and the relationship between multiplication and division,

multiply and divide within 100, and

solve problems involving the four operations, and identify and explain patterns in arithmetic all.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
observation checklists, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative
assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted
resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
Arizona, as a state, is a governing member of the Partnership for Assessment of Readiness for College and Career (PARCC) Consortium
(http://www.parcconline.org/), one of two consortia funded by the US Dept. of Education to provide “next generation K-12 ELA and Mathematics
assessments” and tools for classroom use to assist teachers in assessing learners in formative ways. Teachers and administrators will be
informed of PARCC updates received by mathematics specialists via the Curriculum Connection and Elementary Edition.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students don’t understand story problems.
Maintain student focus on the meaning of the actions and number relationships, and encourage them to model the problem or draw as needed. Students
often depend on key words, a strategy that often is not effective. For example, they might assume that the word left always means that subtraction must
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
17
be used. Providing problems in which key words are used to represent different operations is essential. For example, the use of the word left in this
problem does not indicate subtraction: Suzy took the 28 stickers she no longer wanted and gave them to Anna. Now Suzy has 50 stickers left. How many
stickers did Suzy have to begin with? Students need to analyze word problems and avoid using key words to solve them.
Students don’t interpret multiplication by considering one factor as the number of groups and the other factor as the number in each group.
Have students model multiplication situations with manipulatives or pictorially. Have students write multiplication and division word problems.
Students solve multiplication word problems by adding or division problems by subtracting.
Students need to consider whether a word problem involves taking apart or putting together equal groups. Have students model word problems and
focus on the equal groups that they see.
Students believe that you can use the commutative property for division.
For example, students think that 3÷15 =5 is the same as 15÷3=5. Have students represent the problem using models to see the difference between
these two equations. Have them investigate division word problems and understand that division problems give the whole and an unknown, either the
number of groups or the number in each group.
Students don’t understand the relationship between addition/multiplication and subtraction/division.
Multiplication can be understood as repeated addition of equal groups; division is repeated subtraction of equal groups. Provide students with word
problems and invite students to solve them. When students solve multiplication problems with addition, note the relationship between the operations of
addition and multiplication and the efficiency that multiplication offers. Do the same with division problems and subtraction.
Students don’t understand the two types of division problems.
Division problems are of two different types--finding the number of groups (“quotative” or “measurement”) and finding the number in each group
(“partitive” or “sharing”). Make sure that students solve word problems of these two different types. Have them create illustrations or diagrams of each
type, and discuss how they are the same and different. Connect the diagrams to the equations.
Students use the addition, subtraction, multiplication or division algorithms incorrectly.
Remember that the traditional algorithms are only one strategy. Partial sums, partial products and partial quotients are examples of alternative strategies
that highlight place value and properties of operations. Have students solve problems using multiple models, including numbers, pictures, and words.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
18
Number and Operations in Base Ten (NBT) (1 Cluster)
3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic (Cluster 1Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 Four-digit numbers compose and decompose into units of thousands,
hundreds, tens and ones.
 Place value is essential to developing number sense and multiple
efficient strategies for computing with numbers.
 A digit in one place is ten times more than the same digit in a place to
the right. The base-ten number system is based on the idea that a unit
of higher value is created by grouping ten of the previous value units.
This process can be repeated to obtain larger and larger units of higher
value.
 There is a relationship between addition and subtraction (inverse
operations).
 There are patterns when multiplying by multiples of 10.
 Place value understanding is the foundation for being able to estimate
and round numbers.
 The most familiar form of estimation is rounding, which is a way of
changing the numbers in the problem to others that are easier to
compute mentally.
3.NBT.1
Standard 3.NBT.1
Use place value
understanding to round whole
numbers to the nearest 10 or
100.
Connections: 3.OA.5; 3.SL.1;
ET03-S1C4.01
Mathematical
Practices
3.MP.5. Use
appropriate tools
strategically.






What does place value mean? What is place? How does it affect its
value? For example, what is 10 more than 43?
How can the patterns of multiples of 10 help you solve problems?
How does place value help you estimate?
How do halfway points help you round?
How do the properties of addition and subtraction help you solve
problems?
How are estimating and rounding similar to and different from each
other?
Examples & Explanations
Students learn when and why to round numbers. They identify possible answers and halfway
points. Then they narrow where the given number falls between the possible answers and halfway
points. They also understand that by convention if a number is exactly at the halfway point of the
two possible answers, the number is rounded up.
3.MP.7. Look for
and make use of
structure.
3.MP.8. Look for
and express
regularity in
repeated
reasoning.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
19
Example: Round 178 to the nearest 10.
Step 1: The answer is either 170 or
180.
Step 2: The halfway point is 175.
Step 3: 178 is between 175 and 180.
Step 4: Therefore, the rounded
number is 180.
3.NBT.2
Standard 3.NBT.2
Fluently add and subtract
within 1000 using strategies
and algorithms based on
place value, properties of
operations, and/or the
relationship between addition
and subtraction.
Mathematical
Practices
3.MP.2. Reason
abstractly and
quantitatively.
3.MP.7. Look for
and make use of
structure.
Connections: ET03-S1C1-01
3.MP.8. Look for
and express
regularity in
repeated
reasoning.
Examples & Explanations
Problems should include both vertical and horizontal forms, including opportunities for students to
apply the commutative and associative properties. Adding and subtracting fluently refers to
knowledge of procedures, knowledge of when and how to use them appropriately, and skill in
performing them flexibly, accurately, and efficiently. Students explain their thinking and show their
work by using strategies and algorithms, and verify that their answer is reasonable. An interactive
whiteboard or document camera may be used to show and share student thinking.
Example:
 Mary read 573 pages during her summer reading challenge. She was only required to read
399 pages. How many extra pages did Mary read beyond the challenge requirements?
Continued on next page
Students may use several approaches to solve the problem including the traditional algorithm.
Examples of other methods students may use are listed below:
 399 + 1 = 400, 400 + 100 = 500, 500 + 73 = 573, therefore 1+ 100 + 73 = 174 pages
(Adding up strategy)
 400 + 100 is 500; 500 + 73 is 573; 100 + 73 is 173 plus 1 (for 399, to 400) is 174
(Compensating strategy)
 Take away 73 from 573 to get to 500, take away 100 to get to 400, and take away 1 to get
to 399. Then 73 +100 + 1 = 174 (Subtracting to count down strategy)
 399 + 1 is 400, 500 (that’s 100 more). 510, 520, 530, 540, 550, 560, 570, (that’s 70 more),
571, 572, 573 (that’s 3 more) so the total is
1 + 100 + 70 + 3 = 174 (Adding by tens or hundreds strategy)
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
20
3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic (Cluster 1Standards 1, 2, and 3)
3.NBT.3
Standard 3.NBT.3
Mathematical
Examples & Explanations
Multiply one-digit whole
Students use base ten blocks, diagrams, or hundreds charts to multiply one-digit numbers by
Practices
numbers by multiples of 10 in
the range 10–90 (e.g., 9 × 80,
5 × 60) using strategies
based on place value and
properties of operations.
Connections:; 3.NBT.1;
3NBT.5 (commutative
property); 3.SL.1; ET03S1C1-01
3.MP.2. Reason
abstractly and
quantitatively.
3.MP.7. Look for
and make use of
structure.
multiples of 10 from 10-90. They apply their understanding of multiplication and the meaning of the
multiples of 10. For example, 30 is 3 tens and 70 is 7 tens. They can interpret 2 x 40 as 2 groups of
4 tens or 8 groups of ten. They understand that 5 x 60 is 5 groups of 6 tens or 30 tens and know
that 30 tens is 300. After developing this understanding they begin to recognize the patterns in
multiplying by multiples of 10.
Students may use manipulatives, drawings, document camera, or interactive whiteboard to
demonstrate their understanding.
3.MP.8. Look for
and express
regularity in
repeated
reasoning.
Additional Domain Information – Number and Operations in Base Ten (NBT)
Key Vocabulary



Base Ten
Commutative and
Associative
properties
Equation



Expanded notation
Halfway point
Landmark numbers



Multiple
Multiples of 10
Multiplication



Place Value
Properties
Rounding

Standard form
Example Resources

Books
 Young Mathematicians at Work: Constructing Multiplication and Division by Catherine Twomey Fosnot and Maarten Dolk



Developing Number Concepts, Book 3 Place Value, Multiplication and Division by Kathy Richardson
Teaching Student-Centered Mathematics, Grades K-3 by John A Van de Walle and LouAnn H. Lovin
Technology
 http://illuminations.nctm.org/WebResourceList.aspx?Ref=2&Std=0&Grd=0 (Specific 3rd Grade Resources include:
 http://edweb.sdsu.edu/courses/edtec670/Cardboard/Card/N/NumberClub.html (Place Value Game)
 http://nlvm.usu.edu/en/nav/grade_g_2.html Virtual manipulatives for grades 3 – 5.
 http://www.topmarks.co.uk/Interactive.aspx?cat=21 Interactive white board activities/lessons/games on place value.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
21



http://investigations.terc.edu/library/Games_23.cfm#a_place online games and activities
http://www.internet4classrooms.com/skill_builders/place_value_math_third_3rd_grade.htm Links to place value activities
http://www.prometheanplanet.com/en-us/Search/resources/grade/3-5/country/unitedstates/language/english/?Keywords=place+value&SortField=relevance Place Value anchor charts, white board activities, and
PowerPoint presentations
Assessments
All assessments used must align with our TUSD Curriculum and hold everyone involved accountable for the important mathematical concepts of this
domain. That is, all assessments for this domain must focus on assessing the degree to which third grade learners can, within the parameters specified
in the 3 standards included in this domain
 use place value understanding and properties of operations to perform multi-digit arithmetic.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
observation checklists, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative
assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted
resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
Arizona, as a state, is a governing member of the Partnership for Assessment of Readiness for College and Career (PARCC) Consortium
(http://www.parcconline.org/), one of two consortia funded by the US Dept. of Education to provide “next generation K-12 ELA and Mathematics
assessments” and tools for classroom use to assist teachers in assessing learners in formative ways. Teachers and administrators will be
informed of PARCC updates received by mathematics specialists via the Curriculum Connection and Elementary Edition.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students misinterpret the value of digits in multi-digit numbers.
Students need to understand that when you have ten of one unit, you also have one unit of the next higher value. Frequently refer to a place value chart
and connect the digits to conceptual models, i.e., place value blocks and pictorial representations. Have students create multiple ways to represent
numbers, such as 132 can be made of 1 hundred, 3 tens and 2 ones, or 1 hundred, 1 ten and 22 ones, or 12 tens and 12 ones. When explaining
strategies used, students must identify the unit value; e.g., when adding 492 and 265, they state that they are adding “two hundred” to “four hundred”,
i.e., the 2 in 265 is named “two hundred”, rather than “two”.
Students believe that subtraction is commutative.
After students have discovered and applied the commutative property for addition, ask them to investigate whether this property works for subtraction.
Have students share and discuss their reasoning and guide them to conclude that the commutative property does not apply to subtraction.
Students misunderstand the meaning of the equal sign.
The equal sign means “is the same quantity as” but many primary students believe the equal sign tells you that the “answer is coming up” to the right of
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
22
the equal sign. Students need to see equations written multiple ways. It is important to model equations in various ways 28 = 20 + 8, or 19 + 8 = 20 + 7.
Number and Operations--Fractions (NF) (1 Cluster)
3.NF Develop understanding of fractions as numbers (Grade 3 expectations in this domain are limited to fractions with denominators 2,
3, 4, 6, and 8.) (Cluster 1- Standards 1, 2, and 3)
Essential Concepts
Essential Questions











A fraction is a number.
A fraction is a quantity when a whole is partitioned into equal parts.
The whole that the fraction refers to must be specified.
Given congruent shapes, “equal parts” can refer to non-congruent
parts that measure the same.
Unit fractions are the basic building blocks of fractions in the same way
that 1 is the basic building block of whole numbers.
As the number of equal parts in the whole increases, the size of the
fractional pieces decreases.
The denominator represents the number of equal parts in the whole.
The numerator is the count of the number of equal parts.
Equivalent fractions represent the same size or the same point on a
number line.
When comparing fractions, each fraction must refer to the same whole.
Fractions with common numerators or common denominators can be
compared by reasoning about the number of parts or the size of the
parts.
3.NF.1
Standard 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.
Connections: ET03-S1C2-02;
ET03-S1C4-02
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.









How can fractions be represented?
How does the denominator affect the size of the pieces?
What do the denominator and numerator represent in a fraction?
How can you compare unit fractions with same denominators? (i.e. 1/8
and 3/8)
How can you compare fractions with the same numerator? (i.e. 3/6 and
3/ )
4
How can you use visual models to compare simple equivalent
fractions?
What makes some fractions equivalent?
How can fractions be represented on a number line?
Which is greater: 2/8 or 2/3? What is your reasoning?
Examples & Explanations
Some important concepts related to developing understanding of fractions include:
 Understand fractional parts must be equal-sized
Example
Non-example
3.MP.4. Model
with mathematics
3.MP.7. Look for
and make use of
structure.
These are thirds
Tucson Unified School District
Mathematics Curriculum
These are NOT thirds
Grade 3
Board Approved 03/27/2012
23
Continued on next page
 The number of equal parts tell how many make a whole
 As the number of equal pieces in the whole increases, the size of the fractional pieces
decreases
 The size of the fractional part is relative to the whole
o The number of children in one-half of a classroom is different than the number of
children in one-half of a school. (the whole in each set is different therefore the half
in each set will be different)
 When a whole is cut into equal parts, the denominator represents the number of equal
parts
 The numerator of a fraction is the count of the number of equal parts
o ¾ means that there are 3 one-fourths
o Students can count one fourth, two fourths, three fourths
Students express fractions as fair sharing, parts of a whole, and parts of a set. They use various
contexts (candy bars, fruit, and cakes) and a variety of models (circles, squares, rectangles,
fraction bars, and number lines) to develop understanding of fractions and represent fractions.
Students need many opportunities to solve word problems that require fair sharing.
To develop understanding of fair shares, students first participate in situations where the number of
objects is greater than the number of children and then progress into situations where the number
of objects is less than the number of children.
Examples:
 Four children share six brownies so that each child receives a fair share. How many
brownies will each child receive?
 Six children share four brownies so that each child receives a fair share. What portion of
each brownie will each child receive?
 What fraction of the rectangle is shaded? How might you draw the rectangle in another
way but with the same fraction shaded?
Solution:
Tucson Unified School District
Mathematics Curriculum
2
1
or
4
2
Grade 3
Board Approved 03/27/2012
24
Continued on next page
What fraction of the set is black?
3.NF.2
Standard 3.NF.2
Understand a fraction as a
number on the number line;
represent fractions on a
number line diagram.
a. Represent a fraction 1/b
on a number line
diagram by defining the
interval from 0 to 1 as
the whole and
partitioning it into b equal
parts. Recognize that
each part has size 1/b
and that the endpoint of
the part based at 0
locates the number 1/b
on the number line.
b. Represent a fraction a/b
on a number line
diagram by marking off a
lengths 1/b from 0.
Recognize that the
resulting interval has
size a/b and that its
endpoint locates the
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
Solution:
2
6
Solution:
1
3
Examples & Explanations
Students transfer their understanding of parts of a whole to partition a number line into equal parts.
There are two new concepts addressed in this standard which students should have time to
develop.
1. On a number line from 0 to 1, students can partition (divide) it into equal parts and
recognize that each segmented part represents the same length.
3.MP.4. Model
with mathematics
3.MP.7. Look for
and make use of
structure.
2. Students label each fractional part based on how far it is from zero to the endpoint.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
25
number a/b on the
number line.
Continued on next page
An interactive whiteboard may be used to help students develop these concepts.
Connections: 3.RI.7; 3.SL.1;
ET03-S1C4-01
3.NF Develop understanding of fractions as numbers (Cluster 1- Standards 1, 2, and 3) (Grade 3 expectations in this domain
are limited to fractions with denominators 2, 3, 4, 6, and 8.)
3.NF.3
Standard 3.NF.3
Explain equivalence of
fractions in special cases,
and compare fractions by
reasoning about their size.
a. Understand two fractions
as equivalent (equal) if
they are the same size, or
the same point on a
number line.
b. Recognize and generate
simple equivalent
fractions, e.g., 1/2 = 2/4,
4/6 = 2/3). Explain why
the fractions are
equivalent, e.g., by using
a visual fraction model.
c. Express whole numbers
as fractions, and
recognize fractions that
are equivalent to whole
numbers. Examples:
Express 3 in the form 3 =
3/1; recognize that 6/1 =
6; locate 4/4 and 1 at the
same point of a number
line diagram.
d. Compare two fractions
with the same numerator
or the same denominator
by reasoning about their
size. Recognize that
Mathematical
Practices
Examples & Explanations
An important concept when comparing fractions is to look at the size of the parts and the number of
1
1
3.MP.1. Make
sense of problems
and persevere in
solving them.
the parts. For example, 8 is smaller than 2 because when 1 whole is cut into 8 pieces, the pieces
are much smaller than when 1 whole is cut into 2 pieces.
3.MP.2. Reason
abstractly and
quantitatively.
Students recognize when examining fractions with common denominators, the wholes have been
divided into the same number of equal parts. So the fraction with the larger numerator has the
larger number of equal parts.
5
2
< 6
6
3.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
3.MP.4. Model
with mathematics.
To compare fractions that have the same numerator but different denominators, students
understand that each fraction has the same number of equal parts but the size of the parts are
different. They can infer that the same number of smaller pieces is less than the same number of
bigger pieces.
3
8
<
3
4
3.MP.6. Attend to
precision.
3.MP.7. Look for
and make use of
structure.
3.MP.8. Look for
and express
regularity in
repeated
reasoning.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
26
comparisons are valid
only when the two
fractions refer to the
same whole. Record the
results of comparisons
with the symbols >, =, or
<, and justify the
conclusions, e.g., by
using a visual fraction
model.
Connections: 3.NF.1; 3NF.2;
3.RI.3; 3.SL.1; 3.SL.3; ET03S1C4-01
Additional Domain Information – Number and Operations-Fractions (NF)
Key Vocabulary



Denominator
Equal parts
Equivalent



Fraction
Numerator
Unit Fraction

Whole
Example Resources

Books





Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents by Catherine Twomey Fosnot and Maarten Dolk
Teaching Student-Centered Mathematics, Grades K-3 by John A Van de Walle and LouAnn H. Lovin
Teaching Student-Centered Mathematics, Grades 3-5 by John A Van de Walle and LouAnn H. Lovin
Beyond Pizzas and Pies, by Julie McNamara and Meghan M. Shaughnessy
Extending Children’s Mathematics: Fractions & Decimals, Susan B Empson, and Linda Levi

Technology
 Promethean Planet
http://www.prometheanplanet.com/enus/Search/resources/country/unitedstates/language/english/?Keywords=fractions&SortField=relevance (Multiple flip charts about
fractions.)

Exemplary Lessons
 http://illustrativemathematics.org/standards/k8 Illustrative Mathematics Project

NCTM
 http://illuminations.nctm.org/LessonsList.aspx?grade=2&standard=1
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
27

http://illuminations.nctm.org/LessonDetail.aspx?id=U112 (6 lessons)
In this unit, students explore relationships among fractions through work with the set model. This early work with fraction
relationships helps students make sense of basic fraction concepts and facilitates work with comparing and ordering fractions and
working with equivalency
Assessments
All assessments used must align with our TUSD Curriculum and hold everyone involved accountable for the important mathematical concepts of this
domain. That is, all assessments for this domain must focus on assessing the degree to which third grade learners are, within the parameters specified in
the 3 standards included in this domain,
 developing an understanding of fractions as numbers.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
observation checklists, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative
assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted
resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
Arizona, as a state, is a governing member of the Partnership for Assessment of Readiness for College and Career (PARCC) Consortium
(http://www.parcconline.org/), one of two consortia funded by the US Dept. of Education to provide “next generation K-12 ELA and Mathematics
assessments” and tools for classroom use to assist teachers in assessing learners in formative ways. Teachers and administrators will be
informed of PARCC updates received by mathematics specialists via the Curriculum Connection and Elementary Edition.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students misunderstand the meaning of the numerator and denominator.
Read fractions with meaning. Example: ¾ read, “3 out of 4 equal parts.” Have students count by fractions and highlight the different roles that the
numerator and denominator have. Continually connect the vocabulary to models.
Students believe that fractions are not numbers.
Use number lines to demonstrate placement of fractions and whole numbers.
Students believe that the larger the denominator, the larger the piece.
This can result from students incorrectly memorizing “the larger the denominator the smaller the piece.” Rather than simple memorization, have students
make sense of this relationship themselves. For example, have students investigate whether they would prefer to eat one-hundredth of a pizza or onefourth of a pizza. Have them defend their answer in terms of what you’ve heard other students say, that 100 is more than 4, so one-hundredth must be
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
28
greater.
Students believe that the numerator alone determines the size of the fraction.
Fractions are a part-to-whole relationship. Have students create models of fractions, and associate the written fraction to the relationship between that
part to its whole. Have students confront this relationship using a wide variety of fraction models. Continually connect the vocabulary for fraction names to
models.
Students create models that do not represent equal groups.
Create models that demonstrate equal parts.
Students have difficulty perceiving the unit on a number line diagram.
In the early stages of instruction, use area models and paper strips to highlight the importance of identifying the whole. Subdividing these models can
transfer to subdividing a linear unit.
Measurement and Data (MD) (4 Clusters)
MD Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects
(Cluster 1- Standards 1 and 2)
Essential Concepts
Essential Questions
 Standardized measurement is used to describe and quantify the world.
 Elapsed time is a measure of the duration of an event. It can be
determined using addition and/or subtraction on a number line.
 Liquid volume measures the amount of liquid (or other pourable
substance) a container can hold. It can be measured with metric units
such as liters and milliliters.
 Mass is the amount of matter in an object and, on Earth, is measured in
the same way as weight.
 Mass can be measured using metric units such as grams and
kilograms.
 Benchmark measurements help to develop an understanding of
incremental units and increases familiarity with units, e.g., a large plastic
bottle of soda is 2 liters.
3.MD.1
Standard 3.MD.1
Tell and write time to the
nearest minute and measure
time intervals in minutes.
Solve word problems
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in





Why does “what” we measure influence “how” we measure?
What units and tools are used to measure
o How long it takes to get to school in the morning?
o How much water you drink in one day?
o Which is heavier, e.g., a pencil or a calculator?
Estimate and then measure the liquid volume of this container. Why
did you choose that tool? How close was your estimate?
What is the relationship between grams and kilograms? Give an
example that illustrates the convenience of this relationship.
What is the relationship between liters and milliliters? How does
knowing this relationship help to measure?
Examples & Explanations
Students in second grade learned to tell time to the nearest five minutes. In third grade, they
extend telling time and measure elapsed time both in and out of context using clocks and number
lines.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
29
involving addition and
subtraction of time intervals in
minutes, e.g., by representing
the problem on a number line
diagram.
Connections: 3.RI.3; 3.RI.7;
ET03-S1C4-01
3.MD.2
Standard 3.MD.2
Measure and estimate liquid
volumes and masses of
objects using standard units
of grams (g), kilograms (kg),
and liters (l). (Excludes
compound units such as cm 3
and finding the geometric
volume of a container.) 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.
Excludes multiplicative
comparison problems
(problems involving notions of
“times as much”; see Table
2).
solving them.
3.MP.4. Model
with mathematics.
Continued on next page
Students may use an interactive whiteboard to demonstrate understanding and justify their
thinking.
3.MP.6. Attend to
precision.
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.2. Reason
abstractly and
quantitatively,
Examples & Explanations
Students need multiple opportunities weighing classroom objects and filling containers to help them
develop a basic understanding of the size and weight of a liter, a gram, and a kilogram. Milliliters
may also be used to show amounts that are less than a liter.
Example:
Students identify 5 things that weigh about one gram. They record their findings with words and
pictures. (Students can repeat this for 5 grams and 10 grams.) This activity helps develop gram
benchmarks. One large paperclip weighs about one gram. A box of large paperclips (100 clips)
weighs about 100 grams so 10 boxes would weigh one kilogram.
3.MP.4. Model
with mathematics.
3.MP.5. Use
appropriate tools
strategically.
3.MP.6. Attend to
precision.
Connections: SC03-S1C2-04;
3.RI.3; 3.RI.4; 3.SL.3;
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
30
3.MD Represent and interpret data (Cluster 2- Standards 3 and 4)
Essential Concepts
Essential Questions
 Data can be collected and represented in many ways, including graphs
or line plots.
 The foundation of a line plot is a number line; an ‘X’ is made above the
corresponding value using whole and mixed number (halves and
fourths) units on the line for every corresponding piece of data.
 Labeling graphs or line plots helps to interpret the representation.
 Graphs can be read to compare and contrast information.
 Scaled intervals are important for accurate graph representations.
3.MD.3
Standard 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.
Connections: 3.OA.1; 3.SL.2;
ET03-S1C3-01
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.


How can you represent your data in a way that makes sense?
How are the parts of a graph helpful to a reader? What might be the
result of not having those parts?
Continued on next page
 What comparison problems can you create from your data?
 Why do intervals need to be in equal increments?
Examples & Explanations
Students should have opportunities reading and solving problems using scaled graphs before
being asked to draw one. The following graphs all use five as the scale interval, but students
should experience different intervals to further develop their understanding of scale graphs and
number facts.

3.MP.4. Model
with mathematics.
Pictographs: Scaled pictographs include symbols that represent multiple units. Below is an
example of a pictograph with symbols that represent multiple units. Graphs should include
a title, categories, category label, key, and data.
3.MP.6. Attend to
precision.
3.MP.7. Look for
and make use of
pattern.
How many more books did Juan read than Nancy?

Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs
include a title, scale, scale label, categories, category label, and data.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
31
3.MD.4
Standard 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.
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.4. Model
with mathematics.
3.MP.6. Attend to
precision.
Connections: 3.NF.2; 3.SL.2;
ET03-S1C4-01
Examples & Explanations
Students in second grade measured length in whole units using both metric and U.S. customary
systems. It’s important to review with students how to read and use a standard ruler including
details about halves and quarter marks on the ruler. Students should connect their understanding
of fractions to measuring to one-half and one-quarter inch. Third graders need many opportunities
measuring the length of various objects in their environment.
Continued on next page
Some important ideas related to measuring with a ruler are:
 The starting point of where one places a ruler to begin measuring
 Measuring is approximate. Items that students measure will not always measure exactly ¼,
½ or one whole inch. Students will need to decide on an appropriate estimate length.
 Making paper rulers and folding to find the half and quarter marks will help students
develop a stronger understanding of measuring length
Students generate data by measuring and create a line plot to display their findings. An example of
a line plot is shown below:
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
32
3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition (Cluster
3- standards 5, 6, and 7)
Essential Concepts
Essential Questions
 Area is the two dimensional space inside a region.
 Area is an attribute of plane figures and is measured in square units.
 There is a relationship between area and the operations of multiplication
and addition.
3.MD.5
Standard 3.MD.5
Recognize area 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.
Mathematical
Practices
3.MP.2. Reason
abstractly and
quantitatively.




What is the area of this 4 by 6-inch figure? Prove your answer by
using both addition and multiplication.
Why does multiplying side lengths determine the area of a rectangle?
Will it always work?
How can you decompose this figure to identify its area?
How can you use the side lengths of this figure that are given to
determine the side lengths that are not given?
Examples & Explanations
Students develop understanding of using square units to measure area by:
 Using different sized square units
 Filling in an area with the same sized square units and counting the number of square units
 An interactive whiteboard would allow students to see that square units can be used to
cover a plane figure.
3.MP.4. Model
with mathematics.
3.MP.5. Use
appropriate tools
strategically.
3.MP.6. Attend to
precision.
Connections: 3.RI.4; 3.RI.7;
ET03-S1C1-01
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
33
3.MD.6
Standard 3.MD.6
Measure areas by counting
unit squares (square cm,
square m, square in, square
ft, and improvised units).
Connections: ET03-S1C1-01
3.MD.7
Standard 3.MD.7
Relate area to the operations
of multiplication and addition.
a. Find the area of a
rectangle with wholenumber 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 wholenumber 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 × b and a ×
c. Use area models to
represent the distributive
property in mathematical
reasoning.
Mathematical
Practices
3.MP.5. Use
appropriate tools
strategically.
3.MP.6. Attend to
precision.
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.2. Reason
abstractly and
quantitatively.
Examples & Explanations
Using different sized graph paper, students can explore the areas measured in square centimeters
and square inches. An interactive whiteboard may also be used to display and count the unit
squares (area) of a figure.
Examples & Explanations
Students tile areas of rectangles, determine the area, record the length and width of the rectangle,
investigate the patterns in the numbers, and discover that the area is the length times the width.
Example:
Joe and John made a poster that was 4’ by 3’. Mary and Amir made a poster that was 4’ by 2’.
They placed their posters on the wall side-by-side so that that there was no space between them.
How much area will the two posters cover?
Students use pictures, words, and numbers to explain their understanding of the distributive
property in this context.
3.MP.4. Model
with mathematics.
3.MP.5. Use
appropriate tools
strategically.
3.MP.6. Attend to
precision.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
34
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.
Example:
Students can decompose a rectilinear figure into different rectangles. They find the area of the
figure by adding the areas of each of the rectangles together.
Connections: 3.OA.5; 3.OA.7;
3.RI.3; 3.RI.4; 3.RI.7; 3.SL.1;
ET03-S1C4-01
3.MD Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and
area measures (Cluster 4- Standard 8)
Essential Concepts
Essential Questions
 Perimeter is an attribute of plane figures that can be measured.
 There is a relationship between area and perimeter; area is the space
within the perimeter, perimeter is the border of an area.
 Two or more shapes with the same area do not necessarily have the
same perimeter. Two or more shapes with the same perimeter do not
necessarily have the same area.
3.MD.8
Standard 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.
Mathematical
Practices
3.MP.1. Make
sense of problems
and persevere in
solving them.
3.MP.4. Model
with mathematics.
3.MP.7. Look for
and make use of
structure.
Connections: 3.RI.3; 3.RI.4;



How are arrays used to determine area and perimeter?
Can two shapes with the same perimeter have the same area? If so,
will this always be the case? Explain your reasoning.
Can two shapes with the same area have the same perimeter? If so,
will this always be the case? Explain your reasoning.
Examples & Explanations
Students develop an understanding of the concept of perimeter by walking around the perimeter of
a room, using rubber bands to represent the perimeter of a plane figure on a geoboard, or tracing
around a shape on an interactive whiteboard. They find the perimeter of objects; use addition to
find perimeters; and recognize the patterns that exist when finding the sum of the lengths and
widths of rectangles.
Students use geoboards, tiles, and graph paper to find all the possible rectangles that have a given
perimeter (e.g., find the rectangles with a perimeter of 14 cm.) They 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.
Given a perimeter and a length or width, students use objects or pictures to find the missing length
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
35
3.RI.7;
ET03-S1C3-01; ET03-S1C201; ET03-S1C2-02
or width. They justify and communicate their solutions using words, diagrams, pictures, numbers,
and an interactive whiteboard.
Students 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 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 then investigate the perimeter
of the rectangles with an area of 12.
Area
12 sq. in.
12 sq. in.
12 sq. in
12 sq. in
12 sq. in
12 sq. in
Length
1 in.
2 in.
3 in.
4 in.
6 in.
12 in.
Width
12 in.
6 in.
4 in.
3 in.
2 in.
1 in.
Perimeter
26 in.
16 in.
14 in.
14 in.
16 in.
26 in.
The patterns in the chart allow the students to identify the factors of 12, connect the results to the
commutative property, and discuss the differences in perimeter within the same area. This chart
can also be used to investigate rectangles with the same perimeter. It is important to include
squares in the investigation.
Additional Domain Information – Measurement and Data (MD)
Key Vocabulary





Analog/digital
Area/perimeter
Bar graph
Centimeter
Customary Units





Data
Grams
Graph
Key
Kilograms,





Line plot
Liquid
Liters
Mass
Meter





Metric Units
Milliliters
Minute/hour/second
Picture graph
Plane figures




Scale
Survey
Unit square/square
unit
Volume
Granite School Vocabulary List
http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/Documents/Vocabulary%20Documents/3rd%20Grade%20CCSS%20V
ocabulary%20Word%20List.pdf
Example Resources

Books
 Teaching Students-Centered Mathematics- Grades K-3, Van De Walle, 2006
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
36

Technology
 http://nlvm.usu.edu/en/nav/grade_g_2.html Virtual manipulatives for grades 3 - 5.
 http://www.echalk.co.uk/maths/dfes_numeracy/Assets/area_flash.swf Interactive teaching program on area.
 http://nlvm.usu.edu/en/nav/frames_asid_281_g_2_t_4.html A useful teaching tool for demonstrating shapes, perimeters and areas.
 http://www.topmarks.co.uk/Interactive.aspx?cat=28 Interactive white board activities/lessons and games for representing and interpreting
data.
 http://investigations.terc.edu/library/Games_23.cfm#a_place Online games and activities for place value
 http://gtdifferentiation.sites.fcps.org/Grade3 Extension menus for differentiation/gifted students in area and perimeter
 https://docs.google.com/viewer?a=v&pid=sites&srcid=cGZsZXhvbmxpbmUubmV0fHd3d3xneDo3OGQ5MDY2YzgxY2RiOGM4
Extension menu is measurement
 http://www.pflugervilleisd.net/curriculum/math/place_value_model.cfm#enrichment_extension Extension menu for measurement
 http://www.internet4classrooms.com/skill_builders/measurement_math_third_3rd_grade.htm links to measurement activities
 http://www.internet4classrooms.com/skill_builders/data_analysis_math_third_3rd_grade.htm links to data analysis
 http://www.prometheanplanet.com/en-us/Search/resources/grade/3-5/country/unitedstates/language/english/?Keywords=data+&SortField=relevance Data analysis white board activities, anchor charts and PowerPoint
presentations
 http://www.prometheanplanet.com/en-us/Search/resources/grade/3-5/country/unitedstates/language/english/?Keywords=perimeter&SortField=relevance Perimeter white board activities, anchor charts and PowerPoint
presentations
 http://www.prometheanplanet.com/en-us/Search/resources/grade/3-5/country/unitedstates/language/english/?Keywords=area&SortField=relevance Area white board activities, anchor charts and PowerPoint presentations

Exemplary Lessons
 http://illustrativemathematics.org/standards/k8 Illustrative Mathematics Project
 Promethean Planet
http://www.prometheanplanet.com/en-us/Search/resources/country/united
states/language/english/?Keywords=measurement&SortField=relevance (This page has links to multiple flipcharts for measurement.)

NCTM
 http://illuminations.nctm.org/LessonsList.aspx?grade=2&standard=4 (measurement)
 http://illuminations.nctm.org/LessonsList.aspx?grade=2&standard=5 (data analysis)
 http://illuminations.nctm.org/LessonDetail.aspx?id=L635 (1 lesson) Students become familiar with the language/vocabulary of
measurement, they gain an understanding of measuring length by estimating, and they make comparisons with tools
 http://illuminations.nctm.org/LessonDetail.aspx?id=U149 (4 lessons) Students conduct surveys and represent data in a variety of ways.
They also find and compare measures of center. This unit includes four lessons centered around a food court, where students create and
use menus in a meaningful way.
Interactive Links for Student Practice
Amphi School District: http://www.amphi.com/departments--programs/teaching-and-learning/math.aspx
Vesey Elementary: http://edweb.tusd.k12.az.us/vesey/ComputerLab/3rd_grade.htm
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
37
Assessments
All assessments used must align with our TUSD Curriculum and hold everyone involved accountable for the important mathematical concepts of this
domain. That is, all assessments for this domain must focus on assessing the degree to which third grade learners can, within the parameters specified
in the 8 standards included in this domain,
 solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects,

represent and interpret data,

demonstrate an understanding of concepts of area and relate area to multiplication and to addition, and

recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
observation checklists, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative
assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted
resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
Arizona, as a state, is a governing member of the Partnership for Assessment of Readiness for College and Career (PARCC) Consortium
(http://www.parcconline.org/), one of two consortia funded by the US Dept. of Education to provide “next generation K-12 ELA and Mathematics
assessments” and tools for classroom use to assist teachers in assessing learners in formative ways. Teachers and administrators will be
informed of PARCC updates received by mathematics specialists via the Curriculum Connection and Elementary Edition.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions

Students confuse area and perimeter: Introduce the ideas separately. Create real world connections for these ideas. For example the
perimeter of a white board is illustrated by the metal frame; the area of the floor is illustrated by the floor tiles. Use the vocabulary of area and
perimeter in the context of the school day. For example, have students sit on the “perimeter” of the rug.

Students may have difficulty using known side lengths to determine unknown side lengths: Offer these students identical problems on
grid paper and without the gridlines. Have them compare the listed lengths to the gridlines that the lines represent. Transition students to
problems without gridlines, but have grid paper available for students to use to confirm their answers.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
38
Geometry (G) (1 Cluster)
3.G Geometry (Cluster 1- Standards 1 and 2)
Essential Concepts
Essential Questions
 Shapes can be compared and classified by their sides, angles and the
relationship between opposite sides.
 A single shape can belong in several categories.
 Shapes in different categories may have shared attributes that define a
larger category. (A rhombus and a rectangle are both quadrilaterals.)
 Shapes can be partitioned into parts with equal areas in several ways.
(Note: This section supports the standards in Grade 3 NF.)
3.G.1
Standard 3.G.1
Understand that shapes in
different categories (e.g.,
rhombuses, rectangles, and
others) may share attributes
(e.g., having four sides), and
that the shared attributes can
define a larger category (e.g.,
quadrilaterals). Recognize
rhombuses, rectangles, and
squares as examples of
quadrilaterals, and draw
examples of quadrilaterals
that do not belong to any of
these subcategories.
Mathematical
Practices
3.MP.5. Use
appropriate tools
strategically.
3.MP.6. Attend to
precision.




To which different categories does this shape belong? Support your
answer.
Which shape or shapes do not belong to this group? How do the
shape’s angles, sides and/or vertices support your choice?
Which attributes distinguish different shapes? (eg: How is a rhombus
different from a rectangle?)
In what different ways can you divide this square into four equal parts?
Examples & Explanations
In second grade, students identify and draw triangles, quadrilaterals, pentagons, and hexagons.
Third graders build on this experience 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
(see examples below) and identify squares, rectangles, and rhombuses as quadrilaterals.
3.MP.7. Look for
and make use of
structure.
Connections: 3.RI.3; 3.RI.4;
ET03-S2C2-01
3.G.2
Standard3.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
Mathematical
Practices
3.MP.2. Reason
abstractly and
quantitatively.
Examples & Explanations
Given a shape, students partition it into equal parts, recognizing that these parts all have the same
area. They identify the fractional name of each part as “one of four” and “one-fourth,” and are able
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
39
into 4 parts with equal area,
and describe the area of each
part as 1/4 of the area of the
shape.
Connections: 3.MD.7; 3.NF.1;
3.RI.7;
ET03-S1C1-01
3.MP. 4. Model
with mathematics.
to partition a shape into parts with equal areas in several different ways.
3.MP.5. Use
appropriate tools
strategically.
Additional Domain Information – Geometry (G)
Key Vocabulary



Angles
Attribute
Closed figure



Equal
One-fourth
One-half



One-third
Partition
Polygon



Quadrilateral
Rhombus
Shape



Sides
Trapezoid
Vertex/vertices
Example Resources

Books
 Teaching Students-Centered Mathematics- Grades K-3, Van De Walle, 2006

Technology
 http://gtdifferentiation.sites.fcps.org/Grade3 Extension menus for differentiation/gifted students in geometry
 http://investigations.terc.edu/library/Games_23.cfm#a_geometry Online geometry concentration games and tangram puzzles
 http://www.internet4classrooms.com/skill_builders/geometry_math_third_3rd_grade.htm links to geometry games and activities
 http://www.prometheanplanet.com/en-us/Search/resources/grade/3-5/country/unitedstates/language/english/?Keywords=geometry&SortField=relevance Anchor charts, white board activities, and PowerPoint presentations
 http://nlvm.usu.edu/en/nav/category_g_2_t_3.html Virtual geometry manipulatives
 http://classroom.jc-schools.net/basic/mathgeom.html geometry online games

Exemplary Lessons
 http://illustrativemathematics.org/standards/k8 Illustrative Mathematics Project
 http://www.prometheanplanet.com/en-us/Search/resources/country/unitedstates/language/english/?Keywords=geometry&SortField=relevance (This page has links to multiple flipcharts for geometry.)

NCTM
 http://illuminations.nctm.org/LessonsList.aspx?grade=2&standard=1&standard=3
 http://illuminations.nctm.org/LessonDetail.aspx?id=L813
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
40
Assessments
All assessments used must align with our TUSD Curriculum and hold everyone involved accountable for the important mathematical concepts of this
domain. That is, all assessments for this domain must focus on assessing the degree to which third grade learners can, within the parameters specified
in the 2 standards included in this domain
 reason with shapes and their attributes.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
observation checklists, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative
assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted
resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
Arizona, as a state, is a governing member of the Partnership for Assessment of Readiness for College and Career (PARCC) Consortium
(http://www.parcconline.org/), one of two consortia funded by the US Dept. of Education to provide “next generation K-12 ELA and Mathematics
assessments” and tools for classroom use to assist teachers in assessing learners in formative ways. Teachers and administrators will be
informed of PARCC updates received by mathematics specialists via the Curriculum Connection and Elementary Edition.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students do not understand the relationship between squares and rectangles.
A square is a rectangle, but a rectangle is not a square. Make a compare/contrast graphic organizer to list the attributes of rectangles and squares. Also
have students use the definitions to differentiate between a square and a rectangle.
Students believe that the orientation of a shape changes the shape.
Students may not recognize these as the same shape. Be sure to model, investigate and discuss shapes in a variety of orientations and
in contexts.
Students believe that all quadrilaterals have parallel sides and that only regular polygons can be a shape.
Use definitions and models to show that a variety of shapes fit the definition of a quadrilateral or any polygon.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
41
Grade 2. Common multiplication and division situations.7
Unknown Product
Table 2
3x6=?
There are 3 bags with 6 plums in
each bag. How many plums are there
in all?
Equal
Groups
Measurement example.
You need 3 lengths of string, each 6
inches long. How much string will you
need altogether?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
Arrays,4
Area5
Area example.
What is the area of a 3 cm by 6 cm
rectangle?
A blue hat costs $6. A red hat costs 3
times as much as the blue hat. How
much does the red hat cost?
Compare
General
Measurement example.
A rubber band is 6 cm long. How long
will the rubber band be when it is
stretched to be 3 times as long?
General a x
b=?
Group Size Unknown
(“How many in each group?” Division)
3 x ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into 3
bags, then how many plums will be in
each bag?
Number of Groups Unknown
(“How many groups?” Division)
? x 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example.
You have 18 inches of string, which you
will cut into 3 equal pieces. How long
will each piece of string be?
Measurement example.
You have 18 inches of string, which you
will cut into pieces that are 6 inches
long. How many pieces of string will you
have?
If 18 apples are arranged into 3 equal
rows, how many apples will be in each
row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example.
A rectangle has area 18 square
centimeters. If one side is 3 cm long,
how long is a side next to it?
A red hat costs $18 and that is 3 times
as much as a blue hat costs. How much
does a blue hat cost?
Area example.
A rectangle has area 18 square
centimeters. If one side is 6 cm long,
how long is a side next to it?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
Measurement example.
A rubber band is stretched to be 18 cm
long and that is 3 times as long as it
was at first. How long was the rubber
band at first?
a x ? = p, and p ÷ a = ?
Measurement example.
A rubber band was 6 cm long at first.
Now it is stretched to be 18 cm long.
How many times as long is the rubber
band now as it was at first?
? x b = p, and p ÷ b = ?
7The
first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement
examples.
4The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in
the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.
5Area
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
42
COMPARATIVE ANALYSIS: GRADE 3 MATHEMATICS
PLANNING FOR CONTENT SHIFTS
2010 STANDARD
3.OA.1 Interpret products of whole
numbers, e.g., interpret 5 × 7 as the total
number of objects in 5 groups of 7 objects
each. For example, describe a context in
which a total number of objects can be
expressed as
5 × 7.
2008 PO
M03-S1C2-03 Demonstrate the concept of
multiplication and division using multiple
models.
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. (See Glossary, Table 2.)
M03-S1C2-03 Demonstrate the concept of
multiplication and division using multiple
models.
3.OA.4 Determine the unknown whole
number in a 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 × ? = 48, 5 = ? ÷ 3, 6 × 6 = ?.
PLAN
M03-S1C2-02 Create and solve word
problems based on addition, subtraction,
multiplication, and division.
M03-S1C2-03 Demonstrate the concept of
multiplication and division using multiple
models.
M03-S3C3-02 Use a symbol to represent an
unknown quantity in a given context.
M03-S3C3-03 Create and solve simple onestep equations that can be solved using
addition and multiplication facts.
M03-S1C2-04 Demonstrate fluency of
multiplication and division facts through 10.
M03-S1C2-05 Apply and interpret the concept
of multiplication and division as inverse
operations to solve problems.
M03-S3C3-02 Use a symbol to represent an
unknown quantity in a given context.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
43
2010 STANDARD
2008 PO
M03-S3C3-03 Create and solve simple onestep equations that can be solved using
addition and multiplication facts.
3.OA.5 Apply properties of operations as
strategies to multiply and divide. (Students
need not use formal terms for these
properties.) Examples: If 6 × 4 = 24 is
known, then 4 × 6 = 24 is also known.
(Commutative property of multiplication.) 3
× 5 × 2 can be found by 3 × 5 = 15, then 15
× 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16,
one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) +
(8 × 2) = 40 + 16 = 56. (Distributive
property.)
3.OA.6 Understand division as an
unknown-factor problem. For example, find
32 ÷ 8 by finding the number that makes 32
when multiplied by 8.
M03-S1C2-04 Demonstrate fluency of
multiplication and division facts through 10.
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.
M03-S1C2-04 Demonstrate fluency of
multiplication and division facts through 10.
3.OA.8 Solve two-step word problems using
the four operations. Represent these
problems using equations with a letter
standing for the unknown quantity. Assess
M03-S1C3-01 Make estimates appropriate to
a given situation or computation with whole
numbers.
PLAN
M03-S1C2-07 Apply commutative, identity,
and zero properties to multiplication and apply
the identity property to division.
M04-S1C2-05 Apply associative and
distributive properties to solve multiplication
and division problems.
(includes distributive property)
M03-S1C2-04 Demonstrate fluency of
multiplication and division facts through 10.
M03-S1C2-05 Apply and interpret the concept
of multiplication and division as inverse
operations to solve problems.
M03-S1C2-05 Apply and interpret the
concept of multiplication and division as
inverse operations to solve problems.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
44
2010 STANDARD
the reasonableness of answers using
mental computation and estimation
strategies including rounding. (This
standard is limited to problems posed with
whole numbers and having whole number
answers; students should know how to
perform operations in the conventional
order when there are no parentheses to
specify a particular order (Order of
Operations).
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
3.NBT.1 Use place value understanding to
round whole numbers to the nearest 10 or
100.
3.NBT.2 Fluently add and subtract within
1000 using strategies and algorithms based
on place value, properties of operations,
and/or the relationship between addition
and subtraction.
2008 PO
M03-S3C3-02 Use a symbol to represent an
unknown quantity in a given context.
PLAN
M04-S1C2-06 Apply order of operations with
whole numbers.
(order of operations)
M05-S3C3-01 Create and solve two-step
equations that can be solved using inverse
operations with whole numbers.
M03-S1C2-07 Apply commutative, identity,
and zero properties to multiplication and apply
the identity property to division.
M03-S3C1-01 Recognize, describe, extend,
create, and find missing terms in a numerical
sequence.
M03-S3C1-02 Explain the rule for a given
numerical sequence and verify that the rule
works.
M04-S1C2-05 Apply associative and
distributive properties to solve multiplication
and division problems.
M03-S1C3-01 Make estimates appropriate to
a given situation or computation with whole
numbers.
M02-S1C2-04 Apply and interpret the concept
of addition and subtraction as inverse
operations to solve problems.
M03-S1C1-01 Express whole numbers
through six digits using and connecting
multiple representations.
M03-S1C2-01 Add and subtract whole
numbers to four digits.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
45
2010 STANDARD
3.NBT.3 Multiply one-digit whole numbers
by multiples of 10 in the range 10–90 (e.g.,
9 × 80, 5 × 60) using strategies based on
place value and properties of operations.
3.NF.1 Understand a fraction 1/b as the
quantity formed by 1 part when a whole is
partitioned into b equal parts; understand a
fraction a/b as the quantity formed by a
parts of size 1/b.
3.NF.2 Understand a fraction as a number
on the number line; represent fractions on a
number line diagram.
a. Represent a fraction 1/b on a number
line diagram by defining the interval
from 0 to 1 as the whole and
partitioning it into b equal parts.
Recognize that each part has size 1/b
and that the endpoint of the part based
at 0 locates the number 1/b on the
number line.
b. Represent a fraction a/b on a number
line diagram by marking off a lengths
1/b from 0. Recognize that the
resulting interval has size a/b and that
its endpoint locates the number a/b on
the number line.
2008 PO
M03-S1C2-03 Demonstrate the concept of
multiplication and division using multiple
models.
PLAN
1.
M03-S1C2-04 Demonstrate fluency of
multiplication and division facts through 10.
M03-S1C2-07 Apply commutative, identity,
and zero properties to multiplication and apply
the identity property to division
M04-S1C2-05 Apply associative and
distributive properties to solve multiplication
and division problems.
M03-S1C1-05 Express benchmark fractions 2.
as fair sharing, parts of a whole, or parts of a
set.
3.
M03-S1C1-06 Compare and order benchmark
4.
fractions.
M04-S1C1-03 Express fractions as fair
sharing, parts of a whole, parts of a set, and
locations on a real number line
M03-S1C1-06 Compare and order benchmark
5.
fractions.
M04-S1C1-03 Express fractions as fair
sharing, parts of a whole, parts of a set, and
locations on a real number line
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
46
2010 STANDARD
3.NF.3 Explain equivalence of fractions in
special cases, and compare fractions by
reasoning about their size.
a. Understand two fractions as equivalent
(equal) if they are the same size, or the
same point on a number line.
b. Recognize and generate simple
equivalent fractions, e.g., 1/2 = 2/4, 4/6
= 2/3). Explain why the fractions are
equivalent, e.g., by using a visual
fraction model.
c. Express whole numbers as fractions,
and recognize fractions that are
equivalent to whole numbers.
Examples: Express 3 in the form 3 =
3/1; recognize that 6/1 = 6; locate 4/4
and 1 at the same point of a number
line diagram.
d. Compare two fractions with the same
numerator or the same denominator by
reasoning about their size. Recognize
that comparisons are valid only when
the two fractions refer to the same
whole. Record the results of
comparisons with the symbols >, =, or
<, and justify the conclusions, e.g., by
using a visual fraction model.
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.
2008 PO
PLAN
6.
M03-S1C1-06 Compare and order benchmark
7.
fractions.
M03-S1C1-06 Compare and order benchmark
8.
fractions
M04-S1C1-01 Express whole numbers,
fractions, decimals, and percents using and
connecting multiple representations.
M03-S1C1-05 Express benchmark fractions 9.
as fair sharing, parts of a whole, or parts of a
set.
M04-S1C1-01 Express whole numbers,
fractions, decimals, and percents using and
connecting multiple representations.
M03-S1C1-06 Compare and order benchmark
10.
fractions.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
47
2010 STANDARD
3.MD.2 Measure and estimate liquid
volumes and masses of objects using
standard units of grams (g), kilograms (kg),
and liters (l). (Excludes compound units
such as cm3 and finding the geometric
volume of a container.) 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. (Excludes multiplicative
comparison problems (problems involving
notions of “times as much”; see Glossary,
Table 2).
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.
2008 PO
M03-S1C3-01 Make estimates appropriate to
a given situation or computation with whole
numbers.
PLAN
M04-S3C3-02 Create and solve one-step
equations that can be solved using addition,
subtraction, multiplication, and division of
whole numbers.
M04-S4C4-02 Apply measurement skills to
measure length, mass, and capacity using
metric units.
M04-S4C4-03 problems involving conversions
within the same measurement system.
M02-S2C1-01 Collect, record, organize, and
display data using pictographs, frequency
tables, or single bar graphs. (extends beyond
pictographs)
M03-S1C2-02 Create and solve word
problems based on addition, subtraction,
multiplication, and division. (extends to word
problems based on all operations)
M03-S2C1-01 Collect, record, organize, and
display data using frequency tables, single
bar graphs, or single line graphs. (extends
beyond scaled bar graph)
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.
M03-S4C4-02 APPLY MEASUREMENT
SKILLS TO MEASURE LENGTH, WEIGHT,
AND CAPACITY USING US CUSTOMARY
UNITS. (DOES NOT INCLUDE MAKING A
LINE PLOT AND EXTENDS BEYOND
LENGTH)
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
48
2010 STANDARD
3.MD.5 Recognize area as an attribute of
plane figures and understand concepts of
area measurement.
3.MD.5
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).
2008 PO
PLAN
M04-S4C4-04 Solve problems involving
perimeter of 2-dimensional figures and area of
rectangles. (includes area of rectangles only
and extends to include perimeter)
M05-S4C4-05 Solve problems involving area
and perimeter of regular and irregular
polygons using reallotment of square units.
M04-S4C4-04 Solve problems involving
perimeter of 2-dimensional figures and area of
rectangles (includes area of rectangles only
and extends to include perimeter).
M05-S4C4-05 Solve problems involving area
and perimeter of regular and irregular
polygons using reallotment of square units.
M03-S4C4-04 Determine the area of a
rectangular figure using an array model.
M05-S4C4-05 Solve problems involving area
and perimeter of regular and irregular
polygons using reallotment of square units.
3.MD.7 Relate area to the operations of
multiplication and addition.
a. Find the area of a rectangle with wholenumber side lengths by tiling it, and
show that the area is the same as
would be found by multiplying the side
lengths.
M03-S4C4-04 Determine the area of a
rectangular figure using an array model.
M04-S4C4-04 Solve problems involving
perimeter of 2-dimensional figures and area of
rectangles (includes area of rectangles only
and extends to perimeter)
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
49
2010 STANDARD
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.
2008 PO
M03-S4C4-04 Determine the area of a
rectangular figure using an array model.
3.MD.7
c. Use tiling to show in a concrete case
that the area of a rectangle with wholenumber side lengths a and b + c is the
sum of a × b and a × c. Use area
models to represent the distributive
property in mathematical reasoning.
M03-S4C4-04 Determine the area of a
rectangular figure using an array model.
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.
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.
PLAN
M04-S1C2-05 Apply associative and
distributive properties to solve multiplication
and division problems.
(includes distributive property)
M03-S4C4-04 Determine the area of a
rectangular figure using an array model. (does
not explicitly include decomposing shapes)
M03-S4C4-05 Measure and calculate
perimeter of 2-dimensional figures.
(addresses perimeter only)
M04-S4C4-04 Solve problems involving
perimeter of 2-dimensional figures and area of
rectangles. (addresses both perimeter and
area)
M04-S4C4-05 Describe the change in
perimeter or area when one attribute (length
or width) of a rectangle changes. (addresses
relationship between area and perimeter)
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
50
2010 STANDARD
3.G.1 Understand that shapes in different
categories (e.g., rhombuses,\ rectangles,
and others) may share attributes (e.g.,
having four sides), and that the shared
attributes can define a larger category (e.g.,
quadrilaterals). Recognize rhombuses,
rectangles, and squares as examples of
quadrilaterals, and draw examples of
quadrilaterals that do not belong to any of
these subcategories.
2008 PO
M02-S4C1-01 Describe and compare the
attributes of polygons up to six sides using the
terms side, vertex, point, and length. (does
not include drawing examples)
3.G.2 Partition shapes into parts with equal
areas. Express the area of each part as a
unit fraction of the whole. For example,
partition a shape into 4 parts with equal
area, and describe the area of each part as
1/4 of the area of the shape.
3.MP.1 Make sense of problems and
persevere in solving them.
M03-S1C1-05 Express benchmark fractions
as fair sharing, parts of a whole, or parts of a
set. (includes parts of a whole; area is not
addressed)
PLAN
M02-S4C1-02 Justify which objects in a
collection match a given geometric
description.
M03-S5C2-01 Analyze a problem situation to
determine the question(s) to be answered.
M03-S5C2-02 Identify relevant, missing, and
extraneous information related to the solution
to a problem.
M03-S5C2-03 Select and use one or more
strategies to efficiently solve the problem and
justify the selection.
M03-S5C2-04 Determine whether a problem
to be solved is similar to previously solved
problems, and identify possible strategies for
solving the problem.
M03-S5C2-05 Represent a problem situation
using any combination of words, numbers,
pictures, physical objects, or symbols.
M03-S5C2-06 Summarize mathematical
information, explain reasoning, and draw
conclusions.
M03-S5C2-07 Analyze and evaluate whether
a solution is reasonable, is mathematically
correct, and answers the question.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
51
2010 STANDARD
3.MP.2 Reason abstractly and
quantitatively.
3.MP.3 Construct viable arguments and
critique the reasoning of others.
3.MP.4 Model with mathematics.
3.MP.5 Use appropriate tools strategically.
3.MP.6 Attend to precision.
3.MP.7 Look for and make use of
structure.
2008 PO
M03-S5C2-08 Make and test conjectures
based on data (or information) collected from
explorations and experiments.
M03-S5C2-06 Summarize mathematical
information, explain reasoning, and draw
conclusions.
M03-S5C2-06 Summarize mathematical
information, explain reasoning, and draw
conclusions.
M03-S5C2-08 Make and test conjectures
based on data (or information) collected from
explorations and experiments.
M03-S5C2-03 Select and use one or more
strategies to efficiently solve the problem and
justify the selection.
M03-S5C2-04 Determine whether a problem
to be solved is similar to previously solved
problems, and identify possible strategies for
solving the problem.
M03-S5C2-05 Represent a problem situation
using any combination of words, numbers,
pictures, physical objects, or symbols.
M03-S5C2-03 Select and use one or more
strategies to efficiently solve the problem and
justify the selection.
M03-S5C2-07 Analyze and evaluate whether
a solution is reasonable, is mathematically
correct, and answers the question.
M03-S5C2-08 Make and test conjectures
based on data (or information) collected from
explorations and experiments.
M03-S5C2-06 Summarize mathematical
information, explain reasoning, and draw
conclusions.
M03-S5C2-06 Summarize mathematical
information, explain reasoning, and draw
conclusions.
Tucson Unified School District
Mathematics Curriculum
PLAN
Grade 3
Board Approved 03/27/2012
52
2010 STANDARD
3.MP.8 Look for and express regularity in
repeated reasoning.
MOVEMENT
MOVED TO GRADE 4
MOVED TO GRADE 2
2008 PO
M03-S5C2-08 Make and test conjectures
based on data (or information) collected from
explorations and experiments.
M03-S5C2-07 Analyze and evaluate whether
a solution is reasonable, is mathematically
correct, and answers the question.
2008 PO
M03-S1C1-04 Sort whole numbers into sets
and justify the sort.
REMOVED
M03-S1C2-06 Describe the effect of
operations (multiplication and division) on the
size of whole numbers.
M03-S2C1-02 Formulate and answer
questions by interpreting and analyzing
displays of data, including frequency tables,
single bar graphs, or single line graphs.
M03-S2C3-01 Represent all possibilities for a
variety of counting problems using arrays,
charts, and systematic lists; draw conclusions
from these representations.
MOVED TO GRADE 4
MOVED TO GRADE 4
REMOVED
REMOVED
PLAN
M03-S1C1-02 Compare and order whole
numbers through six digits by applying the
concept of place value.
M03-S1C1-03 Count and represent money
using coins and bills to $100.00.
REMOVED
REMOVED
PLAN
M03-S2C3-02 Solve a variety of problems
based on the multiplication principle of
counting.
M03-S2C4-01 Color complex maps using the
least number of colors and justify the coloring.
M03-S2C4-02 Investigate properties of
vertex-edge graphs
 circuits in a graph,
 weights on edges, and
 shortest path between two vertices.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
53
REMOVED
MOVED TO GRADE 5
REMOVED
MOVED TO GRADE 4
REMOVED
REMOVED
MOVED TO KINDERGARTEN
MOVED TO KINDERGARTEN
REMOVED
MOVED TO GRADE 4
MOVED TO GRADE 4
M03-S2C4-03 Solve problems using vertexedge graphs.
M03-S3C2-01 Recognize and describe a
relationship between two quantities, given by
a chart, table, or graph, in which quantities
change proportionally, using words, pictures,
or expressions.
M03-S3C2-02 Translate between the different
representations of whole number
relationships, including symbolic, numerical,
verbal, or pictorial.
M03-S3C3-01 Record equivalent forms of
whole numbers to six digits by constructing
models and using numbers.
M03-S4C1-01 Describe sequences of 2dimensional figures created by increasing the
number of sides, changing size, or changing
orientation.
M03-S4C1-02 Recognize similar figures.
M03-S4C1-03 Identify and describe 3dimensional figures including their relationship
to real world objects: sphere, cube, cone,
cylinder, pyramids, and rectangular prisms.
M03-S4C1-04 Describe and compare
attributes of two- and three-dimensional
figures.
M03-S4C2-01 Identify a translation, reflection,
or rotation and model its effect on a 2dimensional figure.
M03-S4C2-02 Identify, with justification, all
lines of symmetry in a 2-dimensional figure.
M03-S4C4-03 Convert units of length, weight,
and capacity
 inches or feet to yards,
 ounces to pounds, and
 cups to pints, pints to quarts, quarts
to gallons.
Tucson Unified School District
Mathematics Curriculum
Grade 3
Board Approved 03/27/2012
54