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Connecticut Standards for Mathematics
(CCSS)
Standards for Mathematical Practice
Grade Three
Adopted from The Arizona Academic Content Standards
Grade Three Standards for Mathematical Practice
The K-12 Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop
in their students. This page gives examples of what the practice standards look like at the specified grade level.
Standards
Explanations and Examples
Students are expected to:
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.
1. Make sense of problems and
persevere in solving them.
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.
Students are expected to:.
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
2. Reason abstractly and
appropriate units involved and the meaning of quantities.
quantitatively.
Students are expected to:
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
3. Construct viable arguments
discussions involving questions like “How did you get that?” and “Why is that true?” They explain their
and critique the reasoning of
thinking to others and respond to others’ thinking.
others.
Students are expected to:
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,
4. Model with mathematics.
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.
Students are expected to:
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
5. Use appropriate tools
possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a
strategically.
table, and determine whether they have all the possible rectangles.
Students are expected to:
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
6. Attend to precision.
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.
Students are expected to:
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).
7. Look for and make use of
structure.
Students are expected to:
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
8. Look for and express
know to solve products that they don’t know. For example, if students are asked to find the product of 7 x
regularity in repeated
reasoning.
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?”
Adopted from The Arizona Academic Content Standards
Grade Three Pacing Guide
Torrington Math Resources Math Work Stations
(MWS), GWM, Van de Walle, MathLand (ML)
www.k-5mathteachingresources.com
Unit Title
1. Understanding
Multiplication and
Division
3 weeks plus 1 week for
enrichment, reteaching
3.OA.1
3.OA.2
3.MD.3
2. Connecting and Using
Multiplication and
Division
5 weeks plus 1 week for
enrichment, reteaching
3.OA.3
3.OA.4
3.OA.5
3.OA.6
.OA.7
3. Computing with Whole
Numbers
4 weeks plus 1 week for
enrichment, reteaching
3.OA.7
3.OA.8
3.OA.9
3.NBT.1
3.NBT.2
3.NBT.3
4. Exploring Measurement
and Data
3 weeks plus 1 week for
enrichment, reteaching
3.MD.1
3.MD.3
4 weeks plus 1 week for
enrichment, reteaching
3.MD.5
3.MD.6
3.MD.7
3.MD.8
3 weeks plus 1 week for
enrichment, reteaching
3.MD.8
3.G.1
3.G.2
7. Understanding
Fractions
3 weeks plus 1 week for
enrichment, reteaching
3.NF.1
3.NF.2
8. Reasoning about
Fraction Comparisons and
Equivalence
3 weeks plus 1 week for
enrichment, reteaching
3.NF.3
3.G.2
5. Understanding Area
and Perimeter
6. Reasoning about Twodimensional Shapes
Adopted from The Arizona Academic Content Standards
Standards
Pacing
3.MD.2
3.MD.4
CT Mathematics Unit Planning Organizers are designed to be a resource for developers
of curriculum. The documents feature standards organized in units with key concepts and
skills identified, and a suggested pacing guide for the unit. The standards for
Mathematical Practice are an integral component of CT Standards (CCSS) and are
evident highlighted accordingly in the units.
The information in the unit planning organizers can easily be placed into the
curriculum model in used at the local level during the revision process. It is expected
that local and/or regional curriculum development teams determine the “Big Ideas”
and accompanying “Essential Questions” as they complete the units with critical
vocabulary, suggested instructional strategies, activities and resources.
Note that all standards are important and are eligible for inclusion on the large scale
assessment to be administered during the 2014-15 school year. The Standards were
written to emphasize correlations and connections within mathematics. The priority and
supporting standard identification process emphasized that coherence. Standards were
identified as priority or supporting based on the critical areas of focus described in the
CT Standards, as well as the connections of the content within and across the K-12
domains and conceptual categories. In some instances, a standard identified as priority
actually functions as a supporting standard in a particular unit. No stratification or
omission of practice or content standards is suggested by the system of organization
utilized in the units.
Adopted from The Arizona Academic Content Standards
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 1- Understanding Multiplication and Division
Pacing: 3 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
.
Domain and Standards Overview
Operations and algebraic thinking
 Represent and solve problems involving multiplication and division.
 Understand properties of multiplication and the relationship between multiplication and division.
 Multiply and divide within 100.
 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
5
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 1- Understanding Multiplication and Division
Torrington Resources
GWM, Van de Walle, MathLand, Priority and Supporting CCSS
Basic Facts (Multiplication Division
GWM all of Topic 8
3.OA.1 Interpret products of
Need more story problems
whole numbers, e.g., interpret
k-5mathteachingresources.com
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.
GWM all of Topic 9
Need more story problems
3.OA.2
Interpret
wholenumber 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.
Explanations and Examples*
3.OA.1. 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.
3.OA.2. 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 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.
6
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 1- Understanding Multiplication and Division
Torrington Resources
GWM, Van de Walle, MathLand,
Basic Facts (Multiplication Division
Priority and Supporting CCSS
Explanations and Examples*
3.MD.3. Draw a scaled picture
graph and a scaled bar graph to
represent a data set with several
categories. Solve one- and twostep “how many more” and
“how many less” problems
using information presented in
scaled bar graphs. For example,
draw a bar graph in which each
square in the bar graph might
represent 5 pets.
3.MD.3. 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.
• 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.
How many more books did Juan read than Nancy?
•
7
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Single Bar Graphs: Students use both horizontal and vertical bar
graphs. Bar graphs include a title, scale, scale label, categories,
category label, and data.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 1- Understanding Multiplication and Division
Concepts
What Students Need to Know
Products
Quotient
Scaled Graph
 Picture
 Bar
Skills
What Students Need To Be Able To Do
INTERPRET (with whole numbers as a total
number of objects in groups)
Bloom’s Taxonomy Levels
INTERPRET (with whole numbers as the
number of objects in each share)
INTERPRET (with whole numbers as the
number of shares when objects are partitioned
into equal shares)
2
DRAW (to represent data set with several
categories)
SOLVE (one- and two-step problems using
information from graphs)
Essential Questions
2
2
3
3
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher
8
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
Pacing: 5 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
.
Domain and Standards Overview
Operations and algebraic thinking
 Represent and solve problems involving multiplication and division.
 Understand properties of multiplication and the relationship between multiplication and division.
 Multiply and divide within 100.
 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
9
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
Priority and Supporting
CCSS
Minilessons for Early
3.OA.5. Apply
Multiplication and
properties of
Division
operations as strategies
Basic Math Facts in to multiply and
Multiplication
and divide.2 Examples: If 6
Division
× 4 = 24 is known, then
GWM 14.1, 14.2
4 × 6 = 24 is also
GWM 20.3, 20.4
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.)
2 Students
need not use
formal terms for these
properties.
Explanations and Examples*
3.OA.5. 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 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 does not change the product. They also decompose numbers to build fluency with
multiplication.
Models help build understanding of the commutative property:
Example: 3 x 6 = 6 x 3
In the following diagram it may not be obvious that 3 groups of 6 is the
groups of 3. A student may need to count to verify this.
same as 6
is the same quantity as
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
Continued on next page
3.OA.5. Continued
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
10
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
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
GWM 8.1
GWM 14.5
GWM 20.5
(need more
story problems)
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.
7 x 4 = 28
7 x 4 = 28
56
3.OA.3. 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:
Continued on next page
3.OA.3. Continued
• Equal groups
11
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
• 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
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?
Continued on next page
12
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
3.OA.3. Continued
 Determining the number of shares (measurement division, where the number of
groups is unknown)
o 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
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
24
24-4=20 20-4=16 16-4=12 12-4=8
8-4=4
4-4=0
Solution: The bananas will last for 6 days.
GWM 14.4
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 × ? =
__÷ 3, 6 × 6 =
?
Students may use interactive whiteboards to show work and justify their thinking.
3.OA.4. This standard is strongly connected to 3.AO.3 when students solve problems and
determine 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.
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.
Continued on next page
13
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
3.OA.4 Continued
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=Δ
• Rachel has 3 bags. There are 4 marbles in each bag. How many marbles does Rachel
have altogether? 3 x 4 = m
GWM 14.4
Students may use interactive whiteboards to create digital models to explain and justify their
thinking.
3.OA.6. Multiplication and division facts are inverse operations and that understanding can be
used to find the 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.
3.OA.6.
Understand
division as an unknownfactor problem. For
example, find 32 ÷ 8 by
finding the number that
makes
32
when Examples:
multiplied by 8.
• 3 x 5 = 15
• 15 ÷ 3 = 5
5 x 3 = 15
15 ÷ 5 = 3
Continued on next page
3.OA.6. Continued
14
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
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.
3.OA.7. 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.
Basic Math Facts in 3.OA.7.
Fluently
Multiplication
and multiply and divide
Division
within
100,
using
strategies such as the
GWM Topic 14.3, 14.4, relationship between
GWM Topic 20.1, 20.2, multiplication
and
20.3, 20.4
division (e.g., knowing
that 8 × 5 = 40, one Strategies students may use to attain fluency include:
• Multiplication by zeros and ones
knows that 40 ÷ 5 = 8)
• Doubles (2s facts), Doubling twice (4s), Doubling three times (8s)
or
properties
of
• Tens facts (relating to place value, 5 x 10 is 5 tens or 50)
operations. By the end
• Five facts (half of tens)
of grade 3, know from
• Skip counting (counting groups of __ and knowing how many groups have been
memory all products of
counted)
two one-digit numbers.
• Square numbers (ex: 3 x 3)
: Strategies students may use to attain fluency include:
• 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 1 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
15
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
Bloom’s Taxonomy Levels
Concepts
What Students Need to Know
Properties of Operations
Skills
What Students Need To Be Able To Do
APPLY (as strategies to multiply and divide)
Word Problems
SOLVE (using multiplication and division
within 100 by using drawing and equations with
symbol for unknown)
3
Unknown Whole Number
DETERMINE (in multiplication and division
equations relating three whole numbers)
3
Division
UNDERSTAND (as an unknown-factor
problem)
2
MULTIPLY (fluently within 100 using
strategies)
DIVIDE (fluently within 100 using strategies)
3
KNOW (of two one-digit numbers)
1
Products
16
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
3
3
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 2- Connecting and Using Multiplication and Division
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
17
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Pacing: 4 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
Domain and Standards Overview
Operations and algebraic thinking
 Represent and solve problems involving multiplication and division.
 Understand properties of multiplication and the relationship between multiplication and division.
 Multiply and divide within 100.
 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Number and Operations in Base Ten
 Use place value understanding and properties of operations to perform multi-digit arithmetic.
18
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Priority and Supporting CCSS
GWM 2.3 (Topic 2 is a
second grade outcome)
GWM 3.3, 3.4, 3.5
(3.1 and 3.2 are grade 2
outcomes)
GWM 5 (all)
GWM 12 (all)
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.*
*
A range of algorithms may be used.
Explanations and Examples*
3.NBT.2. 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?
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, not 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)
19
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Priority and Supporting CCSS
GWM 12.2
More needed
Explanations and Examples*
3.NBT.1. Use place value 3.NBT.1. Students learn when and why to round numbers. They identify possible
understanding to round whole answers and halfway points. Then they narrow where the given number falls between
numbers to the nearest 10 or the possible answers and halfway points. They also understand that by convention if
100.
a number is exactly at the halfway point of the two possible answers, the number is
rounded up.
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.
Basic Math Facts in 3.OA.7. Fluently multiply and
Multiplication
and divide within 100, using
Division
strategies
such
as
the
relationship
between
GWM Topic 14.3, 14.4,
multiplication and division
GWM Topic 20.1, 20.2, (e.g., knowing that 8 × 5 = 40,
20.3, 20.4
one knows 40 ÷ 5 = 8) or
properties of operations. By
the end of Grade 3, know
from memory all products of
two one-digit numbers.
3.OA.7. 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.
Continued on next page
20
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Priority and Supporting CCSS
GWM 1.3, 1.4
Need more
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.
Explanations and Examples*
3.OA.7 Continued
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
3.OA.9. Students need ample opportunities to observe and identify important
numerical patterns related to 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.
Continued on next page
21
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Priority and Supporting CCSS
Explanations and Examples*
3.OA.9. Continued
• 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.
Possibly Grade 4 and 5
topic
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. *
*A
range of algorithms may be
used.
GWM 2.4
More needed
3.OA.8 Solve two-step word
problems using the four
operations. Represent these
problems using equations with
a letter standing for the
unknown quantity. Assess the
reasonableness of answers
using mental computation and
estimation strategies including
rounding.3
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.
3.NBT.3. Students use base ten blocks, diagrams, or hundreds charts to multiply onedigit numbers by 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.OA.8. 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.
Continued on next page
22
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Priority and Supporting CCSS
Explanations and Examples*
This standard is limited to 3.OA.8. Continued
problems posed with whole Examples:
numbers and having whole• Jerry earned 231 points at school last week. This week he earned 79 points.
number answers; students
If he uses 60 points to earn free time on a computer, how many points will he
should know how to perform
have left?
operations in the conventional
order when there are no
parentheses to specify a
particular order (Order of
Operations).
3
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).”
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.
Continued on next page
23
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Priority and Supporting CCSS
Explanations and Examples*
3.OA.8. Continued
• 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 one for the afternoon. Write an equation
representing the cost of the field trip and determine the price of one
wristband.
w
w
63
13
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)
24
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Concepts
What Students Need to Know
Whole Numbers
Word Problems
Reasonableness of answers
Skills
What Students Need To Be Able To Do
ADD (fluently within 1000 using strategies and algorithms based on
place value, properties and/or relationship between addition and
subtraction)
SUBTRACT (fluently within 1000 using strategies and algorithms
based on place value, properties and/or relationship between addition
and subtraction)
ROUND (to nearest 10 or 100 using place value understanding)
MULTIPLY AND DIVIDE fluently within 100
USE strategies such as the relationship between multiplication and
division
KNOW from memory all products of two one-digit numbers
MULTIPLY (fluently (one-digit by multiples of 10 in the range of
10-90 using place value strategies and properties of operations)
SOLVE (two-step using the four operations)
REPRESENT (using equations with letter standing for unknown)
Bloom’s Taxonomy Levels
3
3
3
1
3
3
3
ASSESS (using mental computation and estimations strategies
including rounding)
4
MULTIPLY (fluently within 100 using strategies)
DIVIDE (fluently within 100 using strategies)
3
3
Products
KNOW (of two one-digit numbers)
1
Arithmetic Patterns
IDENTIFY (including patterns in the addition or multiplication
tables)
EXPLAIN (using properties of operations)
2
25
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
4
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
26
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
NOTE: Multiplication and division facts are not included in the unit assessments but should be included in classroom assessment.
1. What is 28 rounded to the nearest ten? Show or explain how you found your answer using the number line.
Answer: 30. Student locates 28 on the number line and shows or explains that 28 is closer to 30 than 20, or is more than halfway between 20 and 30
and is therefore closer to 30.
Partial Credit: Correct answer 30, with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that 28 is closer to 30 than 20 on the number line.
No Credit: Incorrect answer with an incorrect or missing explanation.
2. What is 574 rounded to the nearest hundred? Show or explain your answer using the number line.
Answer: 600. Student locates the approximate location of 574 on the number line and shows or explains that 574 is closer to 600 than 500, or is
more than halfway between 500 and 600 and is therefore closer to 600.
Partial Credit: Correct answer 600, with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that 574 is closer to 600 than 500 on the number line.
No Credit: Incorrect answer with an incorrect or missing explanation.
27
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
3. What is 91 rounded to the nearest ten? Show or explain how you found your answer.
Answer: 90. Student uses a number line or another model to show or explain that 91 is closer to 90 than 100, or is less than halfway between 90 and
100 and is therefore closer to 90.
Partial Credit: Correct answer 90, with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that 91 is closer to 90 than 100 on the number line.
No Credit: Incorrect answer with an incorrect or missing explanation.
4. What is 773 rounded to the nearest ten? Show or explain your answer using the number line.
Answer: 770. Student locates 773 on the number line and shows or explains that 773 is closer to 770 than 780, or is less than halfway between 770
and 780 and is therefore closer to 770.
Partial Credit: Correct answer 770, with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that 773 is closer to 770 than 780 on the number line.
No Credit: Incorrect answer with an incorrect or missing explanation.
28
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
5. What is 45 rounded to the nearest ten? Show or explain how you found your answer.
Answer: 50. Student shows or explains that 45 is half way between 40 and 50 and understands that by convention, it is rounded up to the next ten.
Partial Credit: Correct answer 50 with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that 45 is halfway between 40 and 50 and by convention is rounded to 50.
No Credit: Incorrect answer with an incorrect or missing explanation.
6. What is 650 rounded to the nearest hundred? Show or explain how you found your answer.
Answer: 700. Student shows or explains that 650 is half way between 600 and 700 and understands that by convention, it is rounded up to the next
hundred.
Partial Credit: Correct answer 700 with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that 650 is halfway between 600 and 700 and by convention is rounded to 700.
No Credit: Incorrect answer with an incorrect or missing explanation.
29
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
7. Solve this problem. Show or explain how you found your answer.
527
- 284
Answer: 243. Students explain their thinking and show their work by using strategies and algorithms. For example:
 284 + 16 = 300, 300 + 200 = 500, 500 + 27 = 527, therefore 16 + 200 + 27 = 243 (Adding up strategy)
 300 + 200 is 500; 500 + 27 is 527; 200 + 27 is 227 plus 16 (for 284, not 300) is 243 (Compensating strategy)
 Take away 27 from 527 to get to 500, take away 200 to get to 300, and take away 16 to get to 284. Then 27 +200 + 16 = 243 (Subtracting to
count down strategy)
 284 + 16 is 300. Count by 100s 400, 500 (that’s 200 more). Count by tens, 510, 520, (that’s 20 more), 521, 522, 523, 524, 525, 526, 527
(that’s 7 more) so the total is 16 + 200 + 20 + 7 = 243 (Adding by tens or hundreds strategy)
 Rename 527 to 4 hundreds, 12 tens and 7 ones and subtract 2 hundreds, 8 tens and 4 ones (Standard algorithm)
Partial Credit: Correct answer 243 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 243.
No Credit: Incorrect answer with an incorrect or missing explanation.
NOTE: To raise the ceiling on this item, or to ensure that students are able to use multiple methods, ask students to show how to find the answer two
different ways.
30
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
8. Solve this problem. Show or explain how you found your answer.
326
+ 416
Answer 742. Students explain their thinking and show their work by using strategies and algorithms. For example:
 300 + 400 = 700, 20 + 10 = 30, 6 + 6 = 12, therefore 700 + 30 + 12 = 742 (Adding by place values strategy)
 326 + 420 = 746, 746 – 4 = 742 (Compensating strategy)
Partial Credit: Correct answer 243 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 243.
No Credit: Incorrect answer with an incorrect or missing explanation.
NOTE: To raise the ceiling on this item, or to ensure that students are able to use multiple methods, ask students to show how to find the answer two
different ways.
9. Which expression means the same as 60 + 180 + 40?
A. 100 + 180*
B. 18 + 10
C. 60 + 1840
D. 780 + 40
31
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
10. What number is missing in the number sentence?
139 +
 = 436
Answer: 297
11. What number is missing in the number sentence?
 - 459 = 286
Answer: 745
12. Which expression has the same value as 485 + 215?
A. 400 + 200 + 100*
B. 400 + 200 + 90
C. 400 + 200 + 15
D. 400 + 200 + 10
32
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
13. On Monday, 507 students bought lunch from the cafeteria. Another 249 children brought their own lunches from home. How many more
children bought lunch from the cafeteria than brought their lunches from home? Show or explain how you found your answer.
Answer: 258 students. Student showed or explained how they found the answer by subtracting 249 from 507.
Partial Credit: Correct answer 258 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 258.
No Credit: Incorrect answer with an incorrect or missing explanation.
NOTE: To raise the ceiling on this item, or to ensure that students are able to use multiple methods, ask students to show how to find the answer two
different ways.
14. Tim had 573 baseball cards in his collection. For his birthday, his friends gave him some more. Now Tim has 802 baseball cards. How
many cards did his friends give him? Show or explain how you found your answer.
Answer: 229 baseball cards. Student shows or explains how they found the answer by subtracting 573 from 802.
Partial Credit: Correct answer 229 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 229.
No Credit: Incorrect answer with an incorrect or missing explanation.
NOTE: To raise the ceiling on this item, or to ensure that students are able to use multiple methods, ask students to show how to find the answer two
different ways.
33
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
15. Sharon read 835 pages during her summer reading challenge. She was only required to read 492 pages. How many extra pages did Sharon
read beyond the challenge requirements? Show or explain how you found your answer.
Answer: 343 pages. Student shows or explains how they found the answer by subtracting 492 from 835.
Partial Credit: Correct answer 343 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 343.
No Credit: Incorrect answer with an incorrect or missing explanation.
NOTE: To raise the ceiling on this item, or to ensure that students are able to use multiple methods, ask students to show how to find the answer two
different ways.
16. On Monday, Jan picked 137 apples to be sold at the orchard. On Tuesday, she picked 284 apples. How many apples did Jan pick during both
days? Show or explain how you found your answer.
Answer: 421 apples. Student shows or explains how they found the answer by adding 137 and 284..
Partial Credit: Correct answer 421 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 421.
No Credit: Incorrect answer with an incorrect or missing explanation.
NOTE: To raise the ceiling on this item, or to ensure that students are able to use multiple methods, ask students to show how to find the answer two
different ways.
34
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
17. What number is missing in the number sentence? Show or explain how you found your answer.
385 -
 = 292
Answer 93. Student shows or explains how to find the answer using strategies and algorithms.
Partial Credit: Correct answer 93 with an incorrect or missing explanation, OR an incorrect answer with an explanation that would result in the
correct answer 93.
No Credit: Incorrect answer with an incorrect or missing explanation.
18. Will the sum of two even addends always be even or odd? Show or explain how you found your answer.
Answer: Even. Student shows or explains that the sum of two even addends will always be even.
Partial Credit: Correct answer even, with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that the sum of two even addends is always even.
No Credit: Incorrect answer with an incorrect or missing explanation.
35
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
19. Jason is counting by fives. He started at the number 17. Will he say the number 70? Show or explain how you found your answer.

Answer: No. The explanation could include:
o
If Jason started counting by fives with a multiple of 5 (a number that ended with 0 or 5), then he would say the number 70 because
multiples of 5 always end in 0 or 5, but starting with 17 he wouldn’t say the number 70, OR
o If Jason counted by fives and started with 17, which is 2 more than 15, he would say the number 72 which is 2 more than 70 .
o If Jason counted by fives and started at 17, he would say the numbers 67 and 72 and not 70.
Partial Credit: Correct answer no with an incorrect or missing explanation, OR an incorrect answer with an explanation that demonstrates
understanding that Jason would not say 70 when counting by multiples of five starting with 17.
No Credit: Incorrect answer with an incorrect or missing explanation.
20. There are 2 chairs in the first row, 3 chairs in the second row, 5 chairs in the third row, 8 chairs in the fourth row, and 12 chairs in the fifth
row. How many chairs will there be in the eighth row? Show or explain how you found your answer.
Answer: 30. Student explains that the pattern is “add 1, and then add 2, 3, 4, 5, 6, and 7. Start with 2. Add 2 + 1 = 3, 3 + 2 = 5, 5 + 3 = 8, 8 + 4 = 12,
12 + 5 = 17, 17 + 6 = 23, 23 + 7 = 30.
Partial Credit: Correct answer with an incorrect or missing explanation, OR an incorrect answer with a correct explanation.
No Credit: Incorrect answer with and incorrect or missing explanation.
36
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
Students will need a hundreds chart for this question.
Source: http://thinkmath.edc.org/index.php/Hundreds_chart
21. Circle all of the numbers on the hundreds chart that have the same number in the tens and ones places, 11, 22, 33, 44, 55… Explain why these
numbers fall on a diagonal line on the chart.
Answer: Student circles all of the multiples of 11 from 11 to 99 and explains that the numbers fall on a diagonal line because they increase by 11.
Increasing by 10 results in numbers in a vertical line. Increasing by 11 makes the next number shift one more place to the right, resulting in a
diagonal line.
Partial Credit: Student circles all multiples of 11 with an incorrect or missing explanation, OR does not circle all the repeating-digit number with an
explanation that demonstrates understanding that the numbers increase by 11.
No Credit: Student does not circle all doubles numbers with an incorrect or missing explanation.
37
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
22. Jack had some money. He spent $15 on a book and $20 on a shirt. He had $38 left. How much money did he start with? Show or explain
how you found your answer.
Answer: $73. Student explains the answer using any combination of words, numbers, diagrams, physical objects or symbols. For example:

Jack spent $15 + $20 = $35.
 - $35 = $38. I added $35 + $38 to find the amount of money he started with, OR
Partial Credit: Correct answer with an incorrect or missing explanation, OR an incorrect answer with a correct explanation.
No Credit: Incorrect answer with and incorrect or missing explanation.
23. Dan had 253 cards in his baseball card collection. His mom bought him 102 more cards. Then he gave 45 cards to his friend. About how
many baseball cards does Dan have now? Show or explain how you found your answer.
Answer: About 300 cards. Student explains the answer using an estimation strategy, for example:
 253 is about 250, 102 is about 100. 45 is about 50. 250 + 100 = 350. 350 – 50 = 300. (Used benchmark numbers)

253
350
+ 102
- 45
300 + 50 = 350
About 300 (Front-end estimation with adjusting)
Partial Credit: Correct answer with an incorrect or missing explanation, OR an incorrect answer with a correct explanation.
No Credit: Incorrect answer with and incorrect or missing explanation.
38
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
24. Eva wants to ride the Ferris wheel, the roller coaster, and the log ride. The Ferris wheel costs 60 tickets, the roller coaster costs 30 tickets and
the log ride costs 40 tickets. Eva has 50 tickets. How many more tickets should Eva buy?
Answer: 80 tickets
25. Anna had 158 coins in her piggy bank. Her brother gave her 234 coins. Then Anna gave 126 of the coins to her friend Pete. How many coins
does Anna have left?
Answer: 266 coins
26. Gavin walked 123 feet from his house to the corner. Then he walked 248 feet from the corner to the store. Later, he came home the same
way. How many feet did Gavin walk in all? Show or explain how you found your answer.
Answer: 742 feet. Student explains the answer using any combination of words, numbers, diagrams, physical objects or symbols.
Partial Credit: Correct answer with an incorrect or missing explanation, OR an incorrect answer with a correct explanation.
No Credit: Incorrect answer with and incorrect or missing explanation.
27. Jacob spent 360 minutes raking, mowing his lawn and planting flowers. He spent 110 minutes raking and 70 minutes mowing the lawn.
How many minutes did he spend planting flowers? Show or explain how you found your answer.
Answer: 180 minutes. Student explains the answer using any combination of words, numbers, diagrams, physical objects or symbols.
Partial Credit: Correct answer with an incorrect or missing explanation, OR an incorrect answer with a correct explanation.
No Credit: Incorrect answer with and incorrect or missing explanation.
39
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 3 - Computing with Whole Numbers
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
28. Lyla made 24 cupcakes, and Jack made 12 cupcakes for a party. They want to put an equal number of cupcakes on two tables. How many
cupcakes will they put on each table? Show or explain how you found your answer.
Answer: 18 cupcakes. Student explains the answer using any combination of words, numbers, diagrams, physical objects or symbols.
Partial Credit: Correct answer with an incorrect or missing explanation, OR an incorrect answer with a correct explanation.
No Credit: Incorrect answer with and incorrect or missing explanation.
29. Solve this problem
3 × 70 =
Show or explain how you found your answer.
Answer: 210. Student explains how to find the answer using strategies based on place value and properties of operations, for example:
 70 equals 7 tens. 3 groups of 7 tens equal 21 tens. 21 tens = 210
30. Solve this problem
5 × 40 =
Show or explain how you found your answer.
Answer: 200. Student explains how to find the answer using strategies based on place value and properties of operations, for example:
 5 groups of 4 tens equals 20 tens. 20 tens = 200
31. Solve this problem
9 × 80 =
Show or explain how you found your answer.
Answer: 720. Student explains how to find the answer using strategies based on place value and properties of operations, for example:
 80 equals 8 tens. 9 groups of 8 tens equal 72 tens. 72 tens = 720
40
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 4 - Exploring Measurement and Data
Pacing: 3 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
Domain and Standards Overview
Measurement and data
 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
 Represent and interpret data.
41
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 4 - Exploring Measurement and Data
GWM 18 (all)
GWM 19.2, 19.4, 19.5
Need more
Need metric
Possibly MathLand pp.
238-245
Priority and Supporting CCSS
Explanations and Examples*
3.MD.1. Tell and write time to
the nearest minute and
measure time intervals in
minutes. Solve word problems
involving addition and
subtraction of time intervals in
minutes, e.g., by representing
the problem on a number line
diagram.
3.MD.2. Measure and estimate
liquid volumes and masses of
objects using standard units of
grams (g), kilograms (kg), and
liters (l).6 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. 7
3.MD.1. 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.
Students may use an interactive whiteboard to demonstrate understanding and justify
their thinking.
3.MD.2. 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.
6
Excludes compound units such
as cm3 and finding the geometric
volume of a container.
7 Excludes multiplicative
comparison problems (problems
involving notions of “temes as
much”; see Glossary, Table 2).
42
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 4 - Exploring Measurement and Data
GWM 10.1, 10.2, 10.3,
10.4
Keep 16.4 for CMT
MathLand possibly Unit
3.MD.3. Draw a scaled picture
graph and a scaled bar graph to
represent a data set with several
categories. Solve one- and twostep “how many more” and “how
many less” problems using
information presented in scaled
bar graphs. For example, draw a
bar graph in which each square
in the bar graph might represent
5 pets.
3.MD.3. 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.
• 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.
Continued on next page
3.MD.3. Continued
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.
43
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 4 - Exploring Measurement and Data
NEED MORE
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.
3.MD.4. 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
3.MD.4. Continued
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:
44
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 4 - Exploring Measurement and Data
Bloom’s Taxonomy Levels
Concepts
What Students Need to Know
Time
Skills
What Students Need To Be Able To Do
TELL (to the nearest minute)
WRITE (to the nearest minute)
Time intervals
MEASURE (in minutes)
SOLVE (word problems involving addition and subtraction of
intervals in minutes)
3
3
Liquid Volumes
MEASURE (using standard unit of liters)
ESTIMATE (using standard unit of liters)
3
3
Masses
MEASURE (using standard units of grams and kilograms)
ESTIMATE (using standard units of grams and kilograms)
3
3
Word Problems
SOLVE (using all four operations involving masses or volumes given
in the same units)
3
Scaled Graph
 Picture
 Bar
DRAW (to represent data set with several categories)
SOLVE (one- and two-step problems using information from graphs)
3
3
Measurement Data
GENERATE (by measuring lengths using rulers marked with halves
and fourths of an inch)
DISPLAY (in line plot in which horizontal scale is marked in whole
numbers, halves and quarters)
3
45
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
3
3
3
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 4 - Exploring Measurement and Data
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
46
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Pacing: 4 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
Domain and Standards Overview
Operations and algebraic thinking
 Represent and solve problems involving multiplication and division.
 Understand properties of multiplication and the relationship between multiplication and division.
 Multiply and divide within 100.
 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Measurement and data
 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
 Represent and interpret data.
 Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
 Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
47
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Priority and Supporting CCSS
GWM 21.1
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.
GWM 21.2
GWM 21.3
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).
3.MD.7. Relate area to the
operations of multiplication and
addition.
a. Find the area of a rectangle
with whole-number side
lengths by tiling it, and show
that the area is the same as
would be found by multiplying
the side lengths.
Explanations and Examples*
3.MD.5. 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.MD.6. 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.
3.MD.7. 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?
48
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Priority and Supporting CCSS
b. Multiply side lengths to find
areas of rectangles with wholenumber side lengths in the
context of solving real world
and mathematical problems,
and represent whole-number
products as rectangular areas
in mathematical reasoning.
Explanations and Examples*
Students use pictures, words, and numbers to explain their understanding of the
distributive property in this context.
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
Example: Students can decompose a rectilinear figure into different rectangles.
area models to represent the
They find the area of the figure by adding the areas of each of the rectangles
distributive property in
together.
mathematical reasoning.
d. Recognize area as additive.
Find areas of rectilinear
figures by decomposing them
into non-overlapping
rectangles and adding the
areas of the non-overlapping
parts, applying this technique
to solve real world problems.
49
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Priority and Supporting CCSS
GWM 21.3
NEED MORE
MathLand Geoboard
Spaces, Grade 4
MathLand Zoo
Enclosures, Grade 3
GWM 15.3, 15.4, 15.5
Topic 4.1, 4.3, MathLand
grade 4, p. 174, Metric
Olympics, and 15.1 and
15.2 for CMT
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.
Explanations and Examples*
3.MD.8. 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.
Continued on next page
3.MD.8. Continued
Given a perimeter and a length or width, students use objects or pictures to find
the missing length or width. They justify and communicate their solutions using
words, diagrams, pictures, numbers, and an interactive whiteboard.
50
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Priority and Supporting CCSS
Explanations and Examples*
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
with the same area. The chart can be used to investigate rectangles with the
same perimeter. It is important to include squares in the investigation.
51
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Concepts
What Students Need to Know
Area of Rectangle
Rectilinear Figures
Skills
What Students Need To Be Able To Do
RECOGNIZE (as an attribute of plane figures)
UNDERSTAND (measurement of)
MEASURE (by counting unit squares)
FIND (with whole number side lengths by tiling)
MULTIPLY (whole number side lengths in context of real world and
mathematical problems)
SHOW(measurement is same whether tiling or multiplying side
lengths)
SHOW (a and b+c is the sum of a x b and a x c with whole numbers,
using tiling)
RECOGNIZE (as additive)
Bloom’s Taxonomy Levels
2
2
3
3
3
3
3
2
FIND (area by decomposing into non-overlapping rectangles and
adding areas)
SOLVE (real world area problems involving rectilinear figures)
3
Distributive Property
REPRESENT (in mathematical reasoning using area models)
2
Products
REPRESENT (as rectangular areas in mathematical reasoning)
2
Perimeter of Polygons
SOLVE (real world and mathematical problems including given side
lengths and finding an unknown side length)
SHOW (rectangles with same perimeter and different areas)
SHOW (rectangles with same area and different perimeters)
3
Essential Questions
52
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
3
3
3
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 5 - Understanding Area and Perimeter
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
53
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 6 - Reasoning about 2-Dimensional Shapes
Pacing: 3 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
Domain and Standards Overview
Measurement and data
 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
 Represent and interpret data.
 Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
 Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Geometry
 Reason with shapes and their attributes.
Operations and algebraic thinking
 Represent and solve problems involving multiplication and division.
 Understand properties of multiplication and the relationship between multiplication and division.
 Multiply and divide within 100.
 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
54
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 6 - Reasoning about 2-Dimensional Shapes
Priority and Supporting CCSS
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.
Explanations and Examples*
3.MD.8. 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 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.
55
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
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.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 6 - Reasoning about 2-Dimensional Shapes
Priority and Supporting CCSS
Explanations and Examples*
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 with the
same area. The chart can be used to investigate rectangles with the same perimeter. It
is important to include squares in the investigation.
GWM 6.2, 6.3
Van de Walle
NEED MORE
3.G.1. Understand that shapes
in different categories (e.g.,
rhombuses, rectangles, and
others) may share attributes
(e.g., having four sides), and
that the shared attributes can
define a larger category (e.g.,
quadrilaterals). Recognize
rhombuses, rectangles, and
squares as examples of
quadrilaterals, and draw
examples of quadrilaterals
that do not belong to any of
these subcategories.
3.G.1. 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.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.G.2. 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 and are able to
partition a shape into parts with equal areas in several different ways.
56
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 6 - Reasoning about 2-Dimensional Shapes
Priority and Supporting CCSS
Explanations and Examples*
57
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 6 - Reasoning about 2-Dimensional Shapes
Skills
What Students Need To Be Able To Do
SOLVE (real world and mathematical problems including given side
lengths and finding an unknown side length)
SHOW (rectangles with same perimeter and different areas)
SHOW (rectangles with same area and different perimeters)
Bloom’s Taxonomy
Levels
3
Shapes
UNDERSTAND (those in different categories may share attributes)
UNDERSTAND (that shared attributes can define a larger category)
PARTITION (into parts with equal areas)
EXPRESS (area of each part as a unit fraction of the whole shape)
2
2
3
3
Quadrilaterals
RECOGNIZE (rhombus as example of)
RECOGNIZE (rectangle as example of)
RECOGNIZE (square as example of)
DRAW (examples of that do not belong to subcategories of rhombus,
rectangle or square)
Essential Questions
2
2
2
3
Concepts
What Students Need to Know
Perimeter of Polygons
3
3
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
58
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Pacing: 3 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
.
Domain and Standards Overview
Number and operations—fractions
 Develop understanding of fractions as numbers.
59
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
MORE WITH SETS
Priority and Supporting
CCSS
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. 5
Explanations and Examples*
3.NF.1. Some important concepts related to developing understanding of fractions include:
• Understand fractional parts must be equal-sized
Example
Non-example
5
Grade 3 expectations in
this domain are limited to
fractions with
denominators 2, 3, 4, 6,
and 8.
These are thirds
These are NOT thirds
• 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
 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
 ¾ means that there are 3 one-fourths
 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.
60
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Priority and Supporting
CCSS
Explanations and Examples*
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:
2
1
or
4
2
• What fraction of the set is black?
Solution:
2
6
1
3
3.NF.2. 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.
Solution:
NEED MORE
3.NF.2. Understand a
fraction as a number on the
number line; represent
fractions on a number line
diagram. 5
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.
a. Represent a fraction
1/b on a number line
61
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Priority and Supporting
CCSS
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.
Explanations and Examples*
2. Students label each fractional part based on how far it is from zero to the endpoint.
An interactive whiteboard may be used to help students develop these concepts.
5
Grade 3 expectations in
this domain are limited to
fractions with
denominators 2, 3, 4, 6,
and 8.
62
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Concepts
What Students Need to Know
Fraction
Skills
What Students Need To Be Able To Do
UNDERSTAND (1/b as the quantity formed by 1 part when a whole is
partitioned into b equal parts)
UNDERSTAND (a/b as the quantity formed by a parts of size 1/b)
UNDERSTAND (as a number on the number line)
REPRESENT (on a number line diagram)
Bloom’s Taxonomy Levels
2
2
2
3
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
63
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
1. Four children want to share a candy bar equally. What fraction of the candy bar will each child get?
Answer:
1
4
2. A pizza is cut into 8 equal slices. Joe ate 3 slices. What fractional part of the pizza did Joe eat?
Answer:
3
8
3. What fractional part of this set is shaded?
Answer:
3
5
64
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
4. What fractional part of the shape is shaded?
Answer:
3
8
5. What fractional part of the shape is shaded?
Answer:
1
3
65
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
6. What fraction best represents the shaded part of this shape?
Answer:
4
2
1
or
or
4
2
1
66
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
7. Look at the figure below.
a) What fraction of the figure is shaded?
b) Explain what the numerator and denominator of your fraction tells you about the figure.
Answer: a)
3
4
b) The numerator tells number of parts shaded and the denominator tells total number of equal parts.
Partial Credit: Student finds the correct answer with an incorrect or missing explanation, or finds an incorrect answer with a correct explanation.
No Credit: Student finds an incorrect answer with an incorrect or missing explanation.
67
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
8. Which figures below are divided into eighths? Explain how you know that the figures you chose are divided into eighths.
D
C
Answer: A and D because these figures are divided into 8 equal parts. B and C are divided into 8 unequal parts which are not eighths.
68
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
3
9. Which letter best represents on the number line below?
4
A
B
C
D
0
2
Answer: C
10. What fraction is shown by the X on the number line below?
0
Answer:
1
1
3
69
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
11. Divide this number line into sixths. Mark where
5
is located on the line with an X.
6
0
1
Answer: Student divides the number line into 6 equal sections (allow for sections that are not exact but appear to be an attempt to make all sections
5
equal) and marks the correct location of with an X.
6
5
Partial Credit: Student divides the number line into 6 equal sections but does not mark the location of OR does not divide the number line into 6
6
5
equal sections but marks the correct location of on the number line.
6
5
No Credit: Student does not divide the number line into 6 equal sections and does not mark the correct location of on the number line.
6
5
12. Which letter best represents on the number line below?
8
A
Answer: C
B
C
0
1
D
1
70
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 7 - Understanding Fractions
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
1
13. Which letter best represents on the number line below?
2
A
B
C
D
0
1
Answer: B
71
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Pacing: 3 weeks (plus 1 week for reteaching/enrichment)
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching
and learning.
Practices in bold are to be emphasized in the unit.
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.
Domain and Standards Overview
Number and operations—fractions
 Develop understanding of fractions as numbers.
Geometry
 Reason with shapes and their attributes.
72
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Need new unit
Priority and Supporting
CCSS
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
Explanations and Examples*
3.NF.3. An important concept when comparing fractions is to look at the size of the
parts and the number of the parts. For example, 1/8 is smaller than ½ because when 1
whole is cut into 8 pieces, the pieces are much smaller than when 1 whole of the same
size is cut into 2 pieces.
Examples:
Use fraction bars
1 2 4 2
 , 
2 4 6 3
Use number lines
1/6
2/6
3/6
4/6
5/6
0
Continued on next page
6/6
1
¼
73
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
1
0
2/4
¾
4/4
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Priority and Supporting
CCSS
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.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
Explanations and Examples*
3.NF.3. Continued
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.
2 5

6 6
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 3

8 4
3.G.2. 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 and are able to
partition a shape into parts with equal areas in several different ways.
74
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Bloom’s Taxonomy Levels
Concepts
What Students Need to Know
Equivalence of Fractions
Skills
What Students Need To Be Able To Do
UNDERSTAND (as same size or same point on number line)
RECOGNIZE (simple equivalent fractions)
GENERATE (simple equivalent fractions)
EXPLAIN (why fractions are equivalent using visual models)
Fractions
COMPARE (two with same numerator or denominator by reasoning about
size)
RECOGNIZE (comparisons are only valid when referring to same whole)
RECORD (results of comparisons using >, =, <)
JUSTIFY (results of comparisons)
EXPRESS (whole numbers as fractions)
RECOGNIZE (when equivalent to whole numbers)
4
PARTITION (into parts with equal areas)
EXPRESS (area of each part as a unit fraction of the whole shape)
3
3
Shapes
2
2
2
2
2
1
5
2
2
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment Consortium
(SBAC) and has input into the development of the assessment.
Essential Questions
Corresponding Big Ideas
75
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
1
1. Which fraction below is equivalent to ?
3
2
a)
3
b)
2
*
6
c)
1
6
d)
3
6
2. Which point on this number line shows the location of the fraction,
A
B
C
D
• •
•
•
3
?
3
Answer: C
76
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
1
1
3. Is
more than or less than
? Show or explain how you found your answer.
4
2
1
1
1
1
less than
OR
is more than . Student explains the answer (answers may vary), for example, student draws a picture of two
4
4
2
2
1
1
1
1
congruent shapes, one
shaded and the other shaded and explains that covers less area than or vice versa. OR, Student explains using the
2
4
4
2
size of each part when a shape is cut into two equal parts compared to a congruent shape cut into 4 equal parts fourths are smaller than halves).
Answer:
Partial Credit: Student has the correct answer with an incorrect or missing explanation OR has an incorrect answer with a correct explanation.
No Credit: Student has an incorrect answer with an incorrect or missing explanation.
4. Which inequality is true?
1.
2
6
2.
2
6
>
3.
2
6
>
4
6
4.
6
6
<
1
6
<
5
6
*
5
6
77
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
5. Kim ate 3/8 of a pizza. Jon ate 3/4 of a pizza that was the same size as Kim’s. Who ate more pizza? Show or explain how you found your
answer.
Answer: John ate more pizza than Kim, OR Kim ate less pizza than Jon. Student explains the answer (answers may vary), for example, student draws
3
3
3
3
a picture of two congruent circles, one shaded and the other shaded and explains that covers less area than or vice versa. OR, Student
8
4
8
4
explains using the size of each part when a shape is cut into eight equal parts compared to a congruent shape cut into 4 equal parts (eighths are
smaller than fourths).
Partial Credit: Student has the correct answer with an incorrect or missing explanation OR has an incorrect answer with a correct explanation.
No Credit: Student has an incorrect answer with an incorrect or missing explanation.
78
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
1
6. Is of the triangle shaded? Explain why or why not.
3
Answer: Student correctly answers no and explanation shows understanding that fractional parts must be equal.
Partial Credit: Student correctly answers no with an incorrect or missing explanation OR answers yes with an explanation that shows understanding
that fractional parts must be equal.
No Credit: Student answers no with an incorrect or missing explanation.
79
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
7. The pictures below show how two students divided a shape into fourths.
A
B
Sam claims that one of the shapes is not divided into fourths. Is Sam correct? Explain why you agree or disagree with Sam.
Answer: Sam is correct. Shape A is not divided into fourths because the parts are not equal in size.
Partial Credit: Student correctly answers that Sam is correct with an incorrect or missing explanation OR incorrectly answers that Sam is not correct
with an explanation that indicates that the student understands that the four parts must be equal.
No Credit: Student incorrectly answers that Sam is not correct with an incorrect or missing explanation.
80
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
8. Show two different ways to divide the shape into eighths.
Answer: Student shows two different ways to divide the shape into 8 equal parts. Possible answers include:
Partial Credit: Student shows one way to divide the shape into eight equal parts
No Credit: Student shows two ways to divided the shape into 8 unequal parts or into parts other than eighths.
81
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
1
9. Is of the hexagon shaded? Explain why or why not.
6
Answer: No because the hexagon is not divided into 6 equal parts.
Partial Credit: Student correctly answers no with an incorrect or missing explanation OR answers yes with an explanation that shows understanding
that fractional parts must be equal.
No Credit: Student answers yes with an incorrect or missing explanation.
82
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
10. Mike’s mom ordered enough small pizzas that were all the same size for his birthday party.



Some pizzas were cut into four slices and some were cut into eight slices.
2
4
John got
of a pizza and Sarah got
of a pizza.
4
8
John said he received less pizza than Sarah.
Is John correct? Show or explain why you agree or disagree with John.
Answer: No, John is not correct. Student shows or explains using pictures that
2
4
1
is equivalent to , or that both fractions equal .
4
8
2
Partial Credit: Student correctly answers no, with an incorrect or missing explanation, OR answers yes with an explanation that indicates
2
4
1
understanding that and are equivalent or that both fractions equal .
4
8
2
No Credit: Student answers yes with an incorrect or missing explanation.
83
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Grade 3 Mathematics
Unit 8 - Reasoning about Fraction Comparisons and Equivalence
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
11. Look at the shaded part of the shape below.
Is
1
of the rectangle shaded? Show or explain how you found your answer.
4
Answer: Yes. Student shows or explains that
2
1
is equivalent to .
8
4
Partial Credit: Student correctly answers yes with an incorrect or missing explanation, OR answers no with an explanation that demonstrates
2
1
understand that is equivalent to .
8
4
No Credit: Student answers no with an incorrect or missing explanation.
84
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.