Grade 2 Mathematics, Quarter 2, Unit 2.1
Using Place Value Understanding and Properties
of Operations to Solve Addition Word Problems;
Building Foundations for Multiplication
Overview
Number of Instructional Days:
15
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Add within 100 to solve one- and two-step
word problems with unknowns in all positions.
Look for and make use of structure.
•
Solve word problems involving adding to and
putting together by using drawings and
equations.
•
Recognize if there is a pattern or a structure.
•
Recognize that some problems can be
complex.
•
Use a symbol for the unknown number
represented in the problem.
Look for and express regularity in repeated
reasoning.
•
Determine whether a group of objects has an
odd or even number of members.
•
Recognize if calculations are repeated.
•
•
Recognize general methods and shortcuts.
Write an equation to express an even number as
a sum of two equal addends (doubles facts).
•
•
Maintain oversight of the process while
attending to the details.
Skip count by 5s, 10s, and 100s
•
•
Evaluate the reasonableness of their results.
Use place value strategies and properties of
operations to add up to four two-digit numbers.
•
How can you use a number line to demonstrate
skip counting by 5? By 10? By 100?
•
What is the sum of
_____+_____+_____+_____= ? (two-digit
numbers)? How do you know you are accurate?
•
How does place value help you solve this
problem? What other strategies could you use
to solve the problem?
Essential Questions
•
How can you show your thinking to prove if a
group of objects has an even or an odd
number?
•
What strategies would you use to solve this
problem?
•
What information from this problem do you
need to find the answer?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 13 Grade 2 Mathematics, Quarter 2, Unit 2.1
Using Place Value Understanding and Properties of Operations to Solve
Addition Word Problems; Building Foundations for Multiplication (15 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Operations and Algebraic Thinking
2.OA
Represent and solve problems involving addition and subtraction.
2.OA.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving
situations of adding to, taking from, putting together, taking apart, and comparing, with
unknowns in all positions, e.g., by using drawings and equations with a symbol for the
unknown number to represent the problem.1
1
See Glossary, Table 1.
Work with equal groups of objects to gain foundations for multiplication.
2.OA.3
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g.,
by pairing objects or counting them by 2s; write an equation to express an even number as a
sum of two equal addends.
Number and Operations in Base Ten
2.NBT
Understand place value.
2.NBT.2
Count within 1000; skip-count by 5s, 10s, and 100s.
Use place value understanding and properties of operations to add and subtract.
2.NBT.6
Add up to four two-digit numbers using strategies based on place value and properties of
operations.
Common Core Standards for Mathematical Practice
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 14 Grade 2 Mathematics, Quarter 2, Unit 2.1
Using Place Value Understanding and Properties of Operations to Solve
Addition Word Problems; Building Foundations for Multiplication (15 days)
8
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them
to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically
proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In grade 1, students added and subtracted within 20 with fluency to 10 to solve one-step word problems
involving situations of adding to, taking from, putting together, taking apart, and comparing with
unknowns in all positions. Students also added three whole numbers whose sum was less than or equal to
20. Students used objects, drawings, and equations with a symbol for the unknown number. Also in grade
1, students used place value understanding and properties of operations to add within 100, a two-digit and
one-digit number or a two-digit number and a multiple of ten. Students understood that in adding twodigit numbers, one adds tens and tens, ones and ones and sometimes it’s necessary to compose a ten.
Current Learning
Early in the year, students add within 100 to solve one-step word problems involving adding to and
putting together with unknowns in all positions using drawings and equations with a symbol to represent
the unknown. Students also add two two-digit numbers based on place value.
In this unit, students determine whether a group of objects have an odd or even number of members.
Since this is the first time students are working with two-step word problems and odd/even
differentiation, these concepts should be taught at the developmental level of instruction. This is also the
first time students are exposed to the mathematical language and concept of skip counting. Students are
also adding up to four two-digit numbers using strategies based on place value and properties of
operations. In this unit, students learn to write equations to express an even number as the sum of two
equal addends.
Teachers may find it challenging to not teach the algorithm but rather lay the solid foundation for place
value. Students may find it challenging to recognize that the answer to the first step in the word problem
is necessary to complete the second step. To support students in maintaining an oversight of the process,
it may be beneficial to model the use of multiple strategies, such as using successive drawings, or some
combination of a diagram and an equation for the second step.
Later in the year, students subtract within 100 to solve one and two-step word problems involving
situations of taking from, taking apart, and comparing with unknowns in all positions. Students also skip
count within 1,000 by fives, tens, and hundreds.
Teachers should refer to Table 1 of CCSS (p. 88) for a variety of rich word problem examples.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 15 Grade 2 Mathematics, Quarter 2, Unit 2.1
Using Place Value Understanding and Properties of Operations to Solve
Addition Word Problems; Building Foundations for Multiplication (15 days)
Future Learning
The following year, students will use place value understanding and properties of operations to perform
multidigit arithmetic. They will fluently add and subtract within 1,000 using strategies and algorithms.
Students will also use their place value understanding to round whole numbers to the nearest 10 or 100.
In grade 3, students will build on their understanding of addition and subtraction to solve two-step word
problems using all four operations. In addition, students will apply their understanding of skip counting to
identify and explain patterns, and relate skip counting to multiplication strategies.
Additional Findings
According to Principles and Standards for School Mathematics, when students solve word problems,
“explaining their pictorial and written solutions helped them articulate their thinking and make it precise”
(p. 119).
The book also states, “In developing the meaning of addition and subtraction with whole numbers,
students should also encounter the properties of operations, such as the commutativity and the
associativity of addition” (p. 83).
“Recognizing the inverse relationship between addition and subtraction can allow students to be flexible
in using strategies to solve problems” (p. 83).
“Students need many instructional experiences to develop their understanding of the system, including
how numbers are written. They should understand, for example, that multiples of ten provide bridges
when counting (e.g., 38, 39, 40, 41) and that “ten” is a special unit within a base-ten system” (p. 81).
“Most work with two-step word problems should involve single-digit addends as grade 2 students are still
developing proficiency.” (K–5 Operations and Algebraic Thinking, Learning Progressions, p. 18)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 16 Grade 2 Mathematics, Quarter 2, Unit 2.2
Develop Understanding of Place Value and
Properties of Operations Through Comparison
and Written Representation of Numbers Within
1,000
Overview
Number of Instructional Days:
15
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Skip-count by 5s, 10s, and 100s within 1,000.
Look for and make use of structure.
•
Compare two three-digit numbers by using the
symbols <, =, > to record the comparisons.
•
Recognize if there is a pattern or a structure.
•
Add within 1,000 using concrete models or
drawings and strategies based on place value,
properties of operations and relate the strategy
to a written method.
•
Use the structure of the place value system to
make sense of skip counting, comparing, and
adding whole numbers.
•
•
Look for and express regularity in repeated
reasoning.
Compose and decompose 10s and 100s to add
three-digit numbers.
Explain why addition strategies work, using
place value and the properties of operations.
•
Recognize if calculations are repeated.
•
Recognize general methods and shortcuts.
•
Maintain oversight of the process while
attending to the details.
•
Evaluate the reasonableness of their results.
Essential Questions
•
How can you use a number line to show skip
counting by 5s? 10s? 100s?
•
How does place value help you compare
numbers?
•
What symbol would you use to compare these
two numbers?
•
What is your strategy for adding these
numbers?
•
How do you know that these two numbers are
equal? Not equal?
•
How is adding two digit-numbers like adding
three-digit numbers? How is it different?
•
Which number is greater? How do you know?
•
How do your strategies for adding two-digit
numbers work for three-digit numbers?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 17 Grade 2 Mathematics, Quarter 2, Unit 2.2
Develop Understanding of Place Value and Properties of Operations
Through Comparison and Written Representation of Numbers
Within 1,000 (15 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Number and Operations in Base Ten
2.NBT
Understand place value.
2.NBT.2
Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.4
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits,
using >, =, and < symbols to record the results of comparisons.
Use place value understanding and properties of operations to add and subtract.
2.NBT.7
Add and subtract within 1000, using concrete models or drawings and strategies based on
place value, properties of operations, and/or the relationship between addition and subtraction;
relate the strategy to a written method. Understand that in adding or subtracting three-digit
numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and
sometimes it is necessary to compose or decompose tens or hundreds.
2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties of
operations.3
3
Explanations may be supported by drawings or objects.
Common Core Standards for Mathematical Practice
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 18 Grade 2 Mathematics, Quarter 2, Unit 2.2
8
Develop Understanding of Place Value and Properties of Operations
Through Comparison and Written Representation of Numbers
Within 1,000 (15 days)
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them
to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically
proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In grade 1, students compared two-digit numbers and recorded their results using <, =, >. They added
within 100 using concrete models or drawings and strategies based on place value and the properties of
operations. They related the strategy to a written method and explained the strategy used. They
understood that when adding two-digit numbers it’s sometimes necessary to compose a ten. Also in grade
1, students extended the counting sequence to 120 by starting at any given number less than 120.
Current Learning
Early in the year, students learn the concept of skip counting by 5s, 10s, and 100s. They add within 100
using two two-digit numbers and strategies of place value. They explain why addition strategies work
using place value. Students also compose and decompose tens and hundreds when they add three-digit
numbers.
In this unit, students skip count by 5s, 10s, and 100s within 1,000. They also compare two three-digit
numbers and record their results using <, =, >. They add within 1,000 using concrete models or drawings
and strategies of properties of operations. Also in this unit, students explain why addition strategies work,
using the properties of operations.
Later in the year, students subtract with 1,000 using strategies based on the relationship of addition and
subtraction. They subtract three-digit numbers and explain why the subtraction strategies work using
place value and properties of operations.
Future Learning
In grade 3, students will fluently add and subtract within 1,000 using strategies and algorithms. Students
in grade 3 round whole numbers to the nearest 10 or 100. They will also use this knowledge to identify
patterns and explain them using properties of operations.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 19 Grade 2 Mathematics, Quarter 2, Unit 2.2
Develop Understanding of Place Value and Properties of Operations
Through Comparison and Written Representation of Numbers
Within 1,000 (15 days)
Additional Findings
Principles and Standards for School Mathematics states, ‘Students need many instructional experiences
to develop their understanding of the system, including how numbers are written. They should
understand, for example, that multiples of ten provide bridges when counting (e.g., 38, 39, 40, 41) and
that “ten” is a special unit within a base-ten system” (p. 81).
According to Benchmarks for Science Literacy, “Although children at this level are not yet comfortable
enough with numbers to succeed much in comparing magnitudes, they have knowledge of place value
using hundreds, tens, and ones to conceptualize an understanding of which number would be greater than,
less than, or equal to the other” (p. 277).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 20 Grade 2 Mathematics, Quarter 2, Unit 2.3
Recognize, Draw, and Partition Shapes Having
Specified Attributes
Overview
Number of Instructional Days:
10
(1 day = 45–60 minutes)
Content to be Learned
Mathematical Practices to Be Integrated
•
Recognize that shapes have specified attributes.
Look for and make use of structure.
•
Draw shapes with specified attributes.
•
•
Identify triangles, quadrilaterals, pentagons,
hexagons, and cubes.
Recognize the significant lines in geometric
figures.
•
Recognize complicated things, such as single
objects as being composed of several objects.
•
How do your drawings show shapes with
___(4) sides? What are their names?
•
How is it possible to partition this rectangle
into rows and columns of equal-sized squares?
What is the total number of squares?
•
Partition a rectangle into rows and columns of
same-sized squares.
•
Count to find the total of same-sized squares in
the partitioned shape.
Essential Questions
•
What are the defining attributes of a cube?
•
Which shape has only ___(3) angles?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 21 Grade 2 Mathematics, Quarter 2, Unit 2.3
Recognize, Draw, and Partition
Shapes Having Specified Attributes (10 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Geometry
2.G
Reason with shapes and their attributes.
2.G.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a
given number of equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and
cubes.
5
2.G.2
Sizes are compared directly or visually, not compared by measuring.
Partition a rectangle into rows and columns of same-size squares and count to find the total
number of them.
Common Core Standards for Mathematical Practice
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
In kindergarten and first grade, students have had multiple experiences with 2-D and 3-D geometry. In
first grade, students distinguished between defining attributes and non-defining attributes. They built and
drew shapes to possess defining attributes. They partitioned circles and rectangles into two and four equal
shares. They understood that decomposing shapes into more equal shares created smaller shares. They
have had experience with two-dimensional and three-dimensional shapes.
Current Learning
In grade 2, students recognize and draw shapes having specified attributes at the reinforcement level of
instruction. They identify triangles, quadrilaterals, pentagons, hexagons, and cubes. They compare sizes
directly or visually. They do not compare by measuring. At a developmental level, students in grade 2
partition a rectangle into rows and columns of same-sized squares, and they count to find the total amount
in each row and column. The standards in this unit are addressed in their entirety. It is applied in future
units, but not directly taught.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 22 Grade 2 Mathematics, Quarter 2, Unit 2.3
Recognize, Draw, and Partition
Shapes Having Specified Attributes (10 days)
Future Learning
In third grade, students will understand that shapes in different categories will share attributes. Students
will use their knowledge of partitioning rectangles as a foundation to developing conceptual
understanding of area, fractions, and the use of arrays in multiplication.
Additional Findings
According to Principles and Standards for School Mathematics, students “should learn to represent 2and 3-dimensional shapes with drawings, block constructions, dramatizations, and words …. Teachers
must help students gradually incorporate conventional terminology into their descriptions of 2- and 3dimensional shapes. However, terminology itself should not be the focus of the PreK–2 geometry
program” (p. 97).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 23 Grade 2 Mathematics, Quarter 2, Unit 2.3
Recognize, Draw, and Partition
Shapes Having Specified Attributes (10 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 24