Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2
Standards for Mathematical Practice
Critical Area: Building Fluency with addition and subtraction
FOCUS for Grade 2
Supporting Work
20% of Time
2.OA.C.3-4
2.MD.C.7-8
2.MD.D.9-10
Major Work
Additional Work
70% of time
10% of Time
2.OA.A.1
2.G.A.1-2-3
2.OA.B.2
2.NBT.A.1-2-3-4
2.NBT.B.5-6-7-8-9
2.MD.A.1-2-3-4
2.MD.B.5-6
Required fluency: 2.OA.B.2 and 2.NBT.B.5
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.
Standards in bold are specifically targeted within instructional materials.
Domains:
Operations and Algebraic Thinking
Clusters:
Clusters outlined in bold should drive the learning for this period of instruction.
2.OA.C Work with equal groups of objects to gain
foundations for multiplication.
2.OA.C.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 2’s; write an equation to express an even
number as a sum of two equal addends.
2.OA.C.4 Use addition to find the total number of
objects arranged in rectangular arrays with up to 5
rows and up to 5 columns; write an equation to
express the total as a sum of equal addends.
Revised 7/2016
2.OA.B Add and subtract within 20.
Standards:
2.OA.B.2 Fluently add and subtract within 20
using mental strategies. By end of Grade 2, know
from memory all sums of two one-digit numbers.
Property of MPS
2.OA.A Represent and solve problems involving
addition and subtraction.
2.OA.A.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 to
represent the problem.
Page 1 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2
Foundational Learning
1.OA.A
1.OA.B
1.OA.C
1.OA.D
2.OA.B
Future Learning
2.OA.A.1
2.NBT.B.5
2.NBT.B.9
3.OA.A.1
3.OA.A.3
Key Student Understandings
 Students will understand that when a number can be decomposed (broken apart) into two
equal addends (e.g., 10 = 5 + 5), the number is even.
 Ability to use concrete materials to model the meaning of odd and even numbers.
 Knowledge that writing an equation to express an even number as the sum of two equal
addends is the same as using doubles (e.g., 4 + 4 = 8, 7+7=14).
 Students will understand that rectangular arrays can be represented with repeated
addition equations.
 Students flexibly apply commutative and associative properties of addition to calculate
sums of numbers mentally and in writing.
 Students develop the concept that even numbers are named when they skip count by 2’s.
Assessments

Formative Assessment Strategies

Evidence for Standards-Based Grading
Common Misconceptions/Challenges
2.OA.C Work with equal groups of objects to gain foundations for multiplication.
 Students may count by ones instead of counting by groups. These students may have limited understanding of cardinality and have not developed a
unitized system for counting by a group.
 Students need repeated exposure to and practice with counting sequences in order to become fluent with skip-counting. Counting routines provide
opportunities for students to create the anchors they need for solving problems efficiently.
 Knowing that even numbers end in 0, 2, 4, 6, 8 and odd numbers end in 1, 3, 5, 7, 9 does not ensure that students understand the meaning of evenness.
An example of this: a child may say that 358 is odd because you can pair 3 and 5 and 8 is leftover.
 Students may confuse the terms row and columns and interchange them when writing a repeated addition sentence. The focus should be on the
repeated addition of the representation.
2.OA.B Add and subtract within 20.
 Students count by ones instead of applying a strategy based on number relationships. Use dot patterns or tens frames to help students practice “seeing”
smaller numbers inside larger numbers. Then help students record their thinking in an equation form so they can make connections from the concrete
representation to the abstract notation.
 Just because students have memorized the basic facts does not ensure that they see how numbers relate to each other. It is important that students
understand the meaning of the facts. This understanding plays a major role in their number sense and mental computation.
Revised 7/2016
Property of MPS
Page 2 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2

Students tend to learn their addition facts first. When learning subtraction facts, focus on “think addition” and how the numbers are related.
2.OA.A Represent and solve problems involving addition and subtraction.
 Some students end their solution to a two-step problem after they complete the first step. Students need to check their work to see if their answer
makes sense in terms of the problem’s context. Kids need many opportunities to solve a variety of two-step problems and develop the habit of reviewing
their solution after they think they have finished.
 Many children have misconceptions about the equal sign. The equal sign means “is the same as”; however, many primary students think that the equal
sign means “give an answer”. Students need to see examples of number sentences with an operation to the right of the equal sign and the answer on the
left, so they do not overgeneralize from those limited examples.
 Students might rely on a key word or phrase in a problem to suggest an operation that will lead to an incorrect solution. Students need to solve problems
where key words are contrary to such thinking: Debbie took the 8 stickers she no longer wanted and gave them to Anna. Now Debbie has 11 stickers left.
How many stickers did Debbie have to begin with? (the word left in this example does not indicate subtraction as a viable solution method) It is
important that students avoid using key words to solve problems. The goal is for students to make sense of the problem and understand what it is asking
them to do, rather than search for “tricks” and/or guess at the operation needed to solve the problem.
Instructional Practices
Domain: 2.OA
Cluster: 2.OA.C Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.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.

Students need to understand that a collection of objects can be one “thing” or one “group of things”, and that a group contains a given number of
objects. Investigate separating no more than 20 objects into two equal
groups. Find the numbers that will have some objects remaining and no
objects remaining after separating the collections into two equal groups.
Odd numbers will have some objects remaining while even numbers will
not. For an even number of objects in a collection, show the total as the
sum of equal addends (repeated addition).

Grade 2 students apply their work with doubles to odd and even
numbers. Students should have ample experiences exploring the
concept that if a number can be decomposed (broken apart) into two
equal addends or doubles addition (e.g., 10 = 5 + 5), then that number
(10 in this case) is an even number. Students should explore this concept
with concrete objects (e.g., counters, cubes, etc.) before moving
towards pictorial representations such as circles or arrays.
o Example: Is 8 an even number? Prove your answer. Student
responses at right:
Revised 7/2016
Property of MPS
Page 3 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2

The focus of this standard is placed on the conceptual understanding of even and odd numbers. An even number is an amount that can be made of two
equal parts with no leftovers. An odd number is one that is not even, or cannot be made of two equal parts. The number endings of 0, 2, 4, 6, and 8 are
only an interesting and useful pattern or observation and should not be used as the definition of an even number. (Van de Walle & Lovin, 2006, p. 292)

Provide the opportunity for students to write equations representing sums of two equal addends, such as: 2 + 2 = 4, 3 + 3 = 6, 5 + 5 = 10, 6 + 6 = 12, or
8 + 8 = 16. This understanding will lay the foundation for multiplication and is closely connected to 2.OA.4.
2.OA.C.4 Use addition to find the total number of objects in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the
total as a sum of equal addends.

A rectangular array is an arrangement of objects in horizontal rows and vertical columns. Arrays can be made out of any number of objects that can be
put into rows and columns. All rows contain the same number of items and all columns contain an equal number of items.

Grade 2 students use rectangular arrays to work with repeated addition, a building block for multiplication in third grade. A rectangular array is any
arrangement of things in rows and columns, such as a rectangle of square tiles. Students explore this concept with concrete objects (e.g., counters,
bears, square tiles, etc.) as well as pictorial representations on grid paper or other drawings. Due to the commutative property of multiplication, students
can add either the rows or the columns and still arrive at the same solution.
o Example: What is the total number of circles below?
Show/explore that by rotating the array 90° to form 4 rows with 3 objects in each row. Write two different equations to represent 12 as a sum of
equal addends: by rows, 12 = 3 + 3 + 3 + 3; by columns, 12 = 4 + 4 + 4. Have students discuss this statement and explain their reasoning: The two
arrays are different and yet the same.
Student A
I see 3 counters in each column and
there are 4 columns. So I added
3 + 3 + 3 + 3. That equals 12.
3 + 3 + 3 + 3 = 12
Revised 7/2016
Property of MPS
Student B
I see 4 counters in each row and there are
3 rows. So I added 4 + 4 + 4. That equals
12.
4 + 4 + 4 = 12
Page 4 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2

Have students use objects to build all the arrays possible with no more than 25 objects. Their arrays should have up to 5 rows and up to 5 columns. Ask
students to draw the arrays on grid paper and write two different equations under the arrays: one showing the total as a sum by rows and the other
showing the total as a sum by columns. Both equations will show the total as a sum of equal addends.
The equation by rows: 20 = 5 + 5 + 5 +5
The equation by columns: 20 = 4 + 4 + 4 + 4 + 4

Students may arrange any set of objects into a rectangular array. Objects can be cubes, buttons, counters, etc. Objects do not have to be square to make
an array. Geoboards can also be used to demonstrate rectangular arrays. Students then write equations that represent the total as the sum of equal
addends as shown below.
Source: https://www.engageny.org/resource/grade-2-mathematics-module-6
Domain: 2.OA
Cluster: 2.OA.B Add and subtract within 20.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Grade 2 students internalize facts and develop fluency by repeatedly using strategies that make sense to them. When students are able to demonstrate
fluency they are accurate, efficient, and flexible. Students must have efficient strategies in order to know sums from memory. Research indicates that
teachers can best support students’ memory of the sums of two one-digit numbers through varied experiences including making 10, breaking numbers
apart, and working on mental strategies. These strategies replace the use of repetitive timed tests in which students try to memorize operations as if
there were not any relationships among the various facts. When teachers teach facts for automaticity, rather than memorization, they encourage
students to think about the relationships among the facts. (Fosnot & Dolk, 2001)
Revised 7/2016
Property of MPS
Page 5 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2

Instruction in basic facts should focus on making sense, using strategies for remembering facts, and seeing relationships. Students need to understand
the relationship of addition and subtraction as well as the commutative property. When students see the relationships such that 3 + 5 = 8 and that 5 + 3 =
8, they lessen the number of facts that need to be learned. Recalling basic facts takes time. Instant recall of the basic facts varies from child to child. The
use of timed tests is discouraged as a method of learning. Timed tests are only appropriate after students have indicated that they know the facts. Basic
facts allow students to be accurate and efficient in computation and mental math. The use of objects, diagrams, or interactive whiteboards, and various
strategies to help students develop fluency to add and subtract within 20 using mental strategies. https://hcpss.instructure.com/courses/106/pages/2dot-oa-dot-b-2-about-the-math-learning-targets-and-increasing-rigor

The use of mental strategies will help students during explorations and make sense of number relationships as they add and subtract within 20. The
ability to calculate mentally with efficiency is very important for all students. Mental strategies may include the following:
o Counting on
o Making tens (9 + 7 = 10 + 6)
o Decomposing a number leading to a ten ( 14 – 6 = 14 – 4 – 2 = 10 – 2 = 8)
o Fact families (8 + 5 = 13 is the same as 13 - 8 = 5)
o Doubles
o Doubles plus one (7 + 8 = 7 + 7 + 1)

Building upon their work in Grade 1, Grade 2 students use various addition and subtraction strategies in order to fluently add and subtract within 20:
It is no accident that the standard says
“know from memory” rather than
“memorize.” The first describes an outcome,
whereas the second might be seen as
describing a method of achieving that
outcome. So no, the standards are not
dictating tests. (McCallum, October 2011)
Revised 7/2016
Property of MPS
Page 6 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2
Domain: 2.OA
Cluster: 2.OA.A Represent and solve problems involving addition and subtraction.
2.OA.A.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 to represent
the problem.

Develop fluency with addition and subtraction through exploring the four different addition and subtraction situations and their relationship to the
position of the unknown.
Examples of Add-to, Take-from, and Compare problems:
o Add-to: David had $37. His grandpa gave him some money for his birthday. Now he has $63. How much money did David’s grandpa give him?
$37 + ? = $63
o Compare: David has 63 stickers. Susan has 37 stickers. How many more stickers does David have than Susan? 63 – 37 =  Even though the
modeling of the two problems above is different, the equation, 63 - 37 = ? can represent both situations (Think: How many more do I need to
make 63?)
o Take-from: David had 63 stickers. He gave 37 to Susan. How many stickers does David have now? 63 – 37 = 
o Take-from-Start Unknown: David had some stickers. He gave 37 to Susan. Now he has 26 stickers. How many stickers did David have before?
? - 37 = 26
Revised 7/2016
Property of MPS
Page 7 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2
There are fifteen distinct kinds of single-step
addition and subtraction word problem (see Table
1, left). Students must leave grade 2 with a strong
command of addition and subtraction word
problems to be prepared for future learning; these
problem situations will recur in elementary school
with fractions, and yet again in middle school with
variables. Mastering addition and subtraction
situations in a whole-number setting gives students
a resource they can draw upon for integrating first
fractions, and then variables, into their
mathematical repertoires along the way to college
readiness.
Students should master all of the darkly shaded
problems types in kindergarten and then move on
to master the lightly shaded problem types by the
end of grade one and be introduced to the
unshaded problem types, which they will continue
to master in grade two.
It is important to attend to the difficulty level of the
problem situations in relation to the position of the
unknown.
o Result Unknown problems are the least
complex for students followed by Total
Unknown and Difference Unknown.
o The next level of difficulty includes Change
Unknown, Addend Unknown, followed by
Bigger Unknown.
o The most difficult are Start Unknown, Both
Addends Unknown, and Smaller Unknown.
(CCSS-M, p. 88)
 Identify problem types by discussing the action in the problem and how students represent their thinking regardless of the level of difficulty.
Revised 7/2016
Property of MPS
Page 8 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2

One-step word problems use one operation. Two-step word problems use two operations which may include the same operation or opposite operations.

Two-Step Problems: Because Second Graders are still developing proficiency with the most difficult subtypes (see Table 1, CCSSM Glossary p. 88 or p. 3 of
this document): Add To/Start Unknown; Take From/Start Unknown; Compare/Bigger Unknown; and Compare/Smaller Unknown, two-step problems
should not involve these sub-types (Common Core Standards Writing Team, May 2011). Most two-step problems should focus on single-digit addends
since the primary focus of the standard is the problem-type.

As second grade students solve one- and two-step problems they use manipulatives such as snap cubes, place value materials (groupable and pregrouped), ten frames, etc.; create drawings of manipulatives to show their thinking; or use number lines to solve and describe their strategies. They then
relate their drawings and materials to equations. By solving a variety of addition and subtraction word problems, second grade students determine the
unknown in all positions (Result unknown, Change unknown, and Start unknown). Rather than a letter (“n”), boxes or pictures are used to represent the
unknown number. For example:

Second Graders use a range of methods, often mastering more complex strategies such as making tens and doubles and near doubles for problems
involving addition and subtraction within 20. Moving beyond counting and counting-on, second grade students apply their understanding of place value
to solve problems.

This standard focuses on developing an algebraic representation of a word problem through addition and subtraction. The intent is NOT to introduce
traditional algorithms or rules, but to “make meaning” of operations.
Revised 7/2016
Property of MPS
Page 9 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2

Second graders should work on ALL problem types regardless of the level of difficulty. Mastery is expected in second grade. Students can use interactive
whiteboard or document camera to demonstrate and justify their thinking.

Solving algebraic problems requires emphasizing the most crucial problem solving strategy—understand the situation. Students now build on their
work with one-step problems to solve two-step problems and model and represent their solutions with equations for all the situations shown in Table 1.
The problems should involve sums and differences less than or equal to 100 using the numbers 0 to 100. It is important that students develop the habit
of checking their answer to a problem to determine if it makes sense for the situation and the questions being asked.

Ask students to write word problems for their classmates to solve. Start by giving students the answer to a problem. Then tell students whether it is an
addition or subtraction problem situation. Also let them know that the sums and differences can be less than or equal to 100 using the numbers 0 to 100.
For example, ask students to write an addition word problem for their classmates to solve which requires adding four 2-digit numbers with 100 as the
answer. Students then share, discuss and compare their solution strategies after they solve the problems.
One-Step Example: Some students are in the cafeteria. 24 more students came in. Now there are 60 students in the cafeteria. How many were in the
cafeteria to start with? Use drawings and equations to show your thinking.
 Student A: I read the equation and thought about how to write it with numbers. I thought, “What and 24 makes 60?” So, my equation for the
problem is □ + 24 = 60. I used a number line to solve it. I started with 24. Then I took jumps of 10 until I got close to 60. I landed on 54. Then, I
took a jump of 6 to get to 60. So, 10 + 10 + 10 + 6 = 36. So, there were 36 students in the cafeteria to start with.

Student B: I read the equation and thought about how to write it with numbers. I thought, “There are 60 total. I know about the 24. So, what is
60 – 24?” So, my equation for the problem is 60 – 24 = □. I used place value blocks to solve it. I started with 60 and took 2 tens away. I needed
to take 4 more away. So, I broke up a ten into ten ones. Then, I took 4 away. That left me with 36. So, 36 students were in the cafeteria at the
beginning. 60 – 24 = 36.
Revised 7/2016
Property of MPS
Page 10 of 11
Mathematics 2016-2017—Grade 2
Weeks 5-6—September/October
enVisionmath2.0—Topic 2
Two-Step Example: There are 9 students in the cafeteria. 9 more students come in. After a few minutes, some students leave. There are now 14 students in
the cafeteria. How many students left the cafeteria? Use drawings and equations to show your thinking.
 Student A: I read the equation and thought about how to write it with numbers: 9 + 9 - ? = 14. I used a number line to solve it. I started at 9 and
took a jump of 9. I landed on 18. Then, I jumped back 4 to get to 14. So, overall, I took 4 jumps. 4 students left the cafeteria.

Student B: I read the equation and thought about how to write it with numbers: 9 + 9 -  = 14. I used doubles to solve it. I thought about
double 9s. 9 + 9 = 18. I knew that I only needed 14. So, I took away 4, since 4 and 4 eight. So, 4 students left the cafeteria.
Differentiation
Literacy Connections
 Academic vocabulary terms
The Common Core Approach to Differentiating Instruction (engageny How to Implement a Story of Units, p. 14-20)
Linked document includes scaffolds for English Language Learners, Students with Disabilities, Below Level Students,
and Above Level Students.

Vocabulary Strategies

Literacy Strategies
Resources
enVisionmath2.0
Developing Fluency
Grade 2 Fact Fluency Plan
Addition Thinking Strategies
Topic 2 Pacing Guide
Grade 2 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
Revised 7/2016
Property of MPS
Page 11 of 11