Transcript Grade 2 Algebraic Thinking

Operations and
Algebraic Thinking
Second Grade
Cindy Foubert, Jeannette Robinson, Haley Snyder,
Kristin Finney and Andrelyn Alexander
Operations and Algebraic Thinking
Cluster Headings

Cluster 1
 Represent
and Solve Problems Involving Addition and
Subtraction

Cluster 2
 Add

and Subtract within 20
Cluster 3 & 4
 Work
With Equal Groups of Objects to Gain Foundations for
Multiplication
Critical Areas Relating to Operations and
Algebraic Thinking

Fluency with addition and subtraction to 100

Solve problems within 1000

Use models for addition and subtraction

Use efficient accurate and generalizable methods to
compute sums and differences of whole numbers in base
ten notation

Accurately apply appropriate methods for the context

Mentally calculate sums and differences with 10s and 100s
Future Learning Affects
What if a second grade student did not grasp the
standard 2.0A.1
Which future standards would be affected?
Look at the Zimba Chart!
Represent and Solve Problems Involving
Addition and Subtraction

CCSS: 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 number to represent the problem.
Examples

Results Unknown


Change Unknown


There are 29 animals at the pet store. The pet store receives 18 more. How many are
there now?
There are 29 animals at the pet store. Some more animals are brought to the pet store.
There are now 47 animals. How many animals were brought to the store?
Start Unknown

There are some animals at the pet store. Then 18 more animals were brought to the
store. There are now 47 animals. How many animals were at the store at the beginning?
http://www.youtube.com/watch?v=rTjCCwRn0ps
Progression
CCSS: 2.OA.A.1
Kindergarten
Solve addition
and subtraction
word problems,
and add and
subtract within
10, e.g., by using
objects or
drawings to
represent the
problem.
First Grade
Use addition and
subtraction within 20 to
solve word problems
involving situations of
adding to, taking from,
putting together, taking
apart, and comparing, with
unknowns in all positions,
e.g., by using objects,
drawings, and equations
with a symbol for the
unknown number to
represent the problem.
Solve word problems
that call for addition of
three whole numbers
whose sum is less than or
equal to 20, e.g., by using
objects, drawings, and
equations with a symbol
for the unknown number to
represent the problem.
Second Grade
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.
Third Grade
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.
Misconceptions


May end their solution to a two-step word problem after they complete the first step.

Example: In the morning there are 25 students in the cafeteria. 18 more students come in. After a
few minutes, some students leave. If there are 14 students still in the cafeteria , how many
students left the cafeteria?

Number grabbing – 25 + 18 + 14 =
Misunderstanding the equal sign as the “answer in coming up.”

Overgeneralization if equal sign is always on the left

“They may be predisposed to think of equality in terms of calculating answers rather than as a
relation because it is easier for young children to carry out steps to find an answer than to identify
relationship among quantities.” (2nd Grade Common Core Flip Chart)

Students must, “conceptualize the equal sign as a balance.” (Van de Walle, 263)

“Students can not make sense of algebraic questions without having meaningful instruction on two
very important (and poorly understood) topics: the equal sign and variables.” (Van de Walle, 262)
Misconceptions Continued

Example: Cindy took the 5 stickers she no longer wanted and gave them to
Haley. Now Cindy has 10 stickers left. How many stickers did Debbie have to
begin with?

Relying on keywords

May think the word “left” always means to subtract to find the solution.
“Students need to solve problems where key words are contrary to such thinking.”
(Common Core Flip Book)
Teaching Approaches

Place value blocks (Base 10 blocks)

Number line

Hundreds chart

Schematic representations (drawing pictures)

Color tiles

Counters

Ten frames

Have students write word problems for each other

Teacher Modeling

Teacher Read Alouds

Whole class discussions

To show, solve and explain their reasoning

Construct viable arguments and critique the reasoning of others
Add and Subtract Within 20

CCSS: 2.OA.2

Fluently add and subtract within 20 using mental math
strategies. By the end of Grade 2 know from memory all
sums of two one-digit numbers.

Strongly connected to all standards in this domain,
focuses on students being able to fluently add and
subtract within 20.

Adding and subtracting fluently refers to:
 Knowledge
of the procedures
 Knowledge
of when and how to use appropriately
 Skill
in performing them with flexibility, accurately
and efficiently
Progression
CCSS: 2.OA.2
Kindergarten
First Grade
Second Grade
Third Grade
K.OA.A.1
Represent
addition and
subtraction with
objects, fingers,
mental images,
drawings1,
sounds (e.g.,
claps), acting out
situations, verbal
explanations,
expressions, or
equations.
K.OA.A.5
Fluently add and
subtract within
5.
Add and subtract
within 20,
demonstrating fluency
for addition and
subtraction within 10.
Use strategies such as
counting on; making
ten; decomposing a
number leading to a
ten; using the
relationship between
addition and
subtraction; and
creating equivalent but
easier or known sums
Use place
value
understanding
and properties of
operations to add
and subtract
within 20
Fluently multiply
and divide within
100, using
strategies such as
the relationship
between
multiplication and
division or
properties of
operations. By the
end of Grade 3,
know from
memory all
products of two
one-digit numbers.
Add and Subtract Within 20
Teaching Approaches


Give varied experiences making 10, breaking numbers apart and working on mental strategies,
rather than repetitive timed tests

Provide time for students to develop and share their methods of mental computation

Study how numbers relate to the anchor numbers (5 & 10)

Provide activities for students to apply the commutative and associative properties

Provide simple word problems
Strategies Students Use

Counting on

Making tens (9 + 7 = 10 + 6)

Decomposing a number leading to a ten (14 – 6 = 14 – 4 – 2 = 10 – 2 = 8

Fact Families (8 + 5 = 13 is the same as 13 – 8 = 5)

Doubles

Doubles plus one (7 + 8 = 7 + 7 + 1)

http://www.youtube.com/watch?v=xAtC-7_Ijqg
Add and Subtract Within 20
Misconceptions
 Assume
that answers to addition problems must be
greater
 Apply
strategies to a mixed up set of problems
 Being
able to explain strategies used
Work With Equal Groups of Objects to
Gain Foundations for Multiplication

CCSS: 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.
Teaching Approaches

Students explore odd and even numbers in a variety of ways including:

Investigating if a number is odd or even by determining if the number of objects can be divided
into two equal sets, arranged into pairs, or counted by twos.

Observing that the sum of two even numbers is even, that the sum of two odd numbers is even,
or that the sum of an even and an odd number is always odd. (similar statements can be made
about multiplication)

+2 Approach (even and odd numbers can have 2 added to them to get to the next even or odd
number)

Have students separate no more than 20 objects into two equal groups.

Then, the students will be asked to find those pairs of numbers that will have some objects
remaining when separating the piles into two groups, and then those numbers that will have no
objects remaining after separating the collections into two groups.

If a number of objects is odd and you split it into two, there will be a leftover.

If you do this with the second odd number, it will have a leftover also. So, if you put these two
numbers together, the two leftovers will go together so there won’t be a leftover in the sum.

If a number of objects is even and you split it into two, there will be no objects left over.
Progressions
CCSS: 2.OA.C.3
Kindergarten
Understand the
relationship
between numbers
and quantities;
connect counting to
cardinality.
Identify whether
the number of
objects in one
group is greater
than, less than, or
equal to the
number of objects
in another group
e.g. using matching
and counting
strategies
First Grade
Second Grade
Relate counting to
addition and
subtraction (e.g. by
counting on 2 to add 2)
Relates doubles
additions up to 20
to the concept of
odd and even
numbers and to
counting by 2s
(the easiest
count-by in Level
2) by pairing and
counting by 2s the
things in each
addend.
Third Grade
Use multiplication
and division within
100 to solve word
problems in
situations
involving equal,
arrays, and
measurement
quantities, e.g. by
using drawings and
equations with a
symbol for the
unknown number
to represent the
problem.
Work With Equal Groups of Objects to
Gain Foundations for Multiplication

Misconceptions
VS


Videos

http://www.youtube.com/watch?v=_lo71AhB4GQ Grouping

http://www.youtube.com/watch?v=v3OvzQv17lA +2 Approach
Work With Equal Groups of Objects to
Gain Foundations for Multiplication

CCSS: 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 sum
of equal addends.
Misconceptions

Vocabulary

Row vs. Column

Equal; equal grouping

Array

Repeated addition

Multiplication process

Using row and column numbers interchangeably instead of using only row or only column
numbers
Work With Equal Groups of Objects to
Gain Foundations for Multiplication

CCSS: 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 sum
of equal addends.
Teaching approaches

Literature

Amanda Bean’s Amazing Dream (Neuschwander), Each Orange Had Eight Slices (Giganti), Too
Many Kangaroo Things to Do (Murphy), My Full Moon is Square (Pinczes)

One Hundred Hungary Ants by Elinor J. Pinczes

http://www.youtube.com/watch?v=kmdSUHPwJtc

Manipulatives: square tiles, cubes, teddy bear counters, two-sided counters, geoboards, etc.

Grid paper: pictures/drawings

Teacher modeling

Small group and whole class discussions

Connections between repeated addition and multiplication
Progressions
2.OA.C.4:
Kindergarten
First Grade
Identify whether
the number of
objects in one
group is greater
than, less than, or
equal to the
number of objects
in another group,
e.g., by using
matching and
counting
strategies
Add and subtract within
20, demonstrating fluency
for addition and
subtraction within 10. Use
strategies such as counting
on; making ten;
decomposing a number
leading to a ten; using the
relationship between
addition and subtraction;
and creating equivalent
but easier or known sums.
Understand the meaning of
the equal sign, and
determine if equations
involving addition and
subtraction are true or
false.
Second Grade
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.
Third Grade
Interpret products of
whole numbers,
(e.g., 5x7 as the total
number of objects in
5 groups of 7).
Use multiplication
and division within
100 to solve word
problems in situations
involving equal
groups, arrays, and
measurement
quantities.
Apply properties of
operations as
strategies to multiply
and divide (e.g., if
6x4=24 than 4x6=24).
Research

Common Core State Standards for Mathematics

Common Core Standards Flip Book by Melisa Hancock

Engage NY

Teacher Tube

YouTube

Van de Walle

NCTM

Focus on Mathematically Proficient Students (Amy Salgo)

Whole Class Mathematics Discussions-Teruni Lamberg’s Blog

Progressions

Mathforum.org
http://www.engageny.org
Lesson 1
Engage New YorkMake Number Bonds of Ten & Twenty
with a focus on an addition two step story
problem


Standards to cover

2.OA.1

2.OA.2
Background knowledge

K.OA.3

K.OA.4

K.NBT.1

1.OA.6
Review of Ten Frame

Place ten frame cards from lowest to highest number

Discussion of addition sentences with ten frame cards



Play Rainbow Buddy Game with 10 frame

Discussion of Game
Discussion of 20 Frame

Play Rainbow Buddy Game with two 10 frames for problems up to 20

Discussion of Game
Intro to story problem
Word Problem
Haley already has 10 stickers. Now her goal is to
collect 20 in all. She got 4 more on Monday and
4 again on Tuesday.
 How
many does she have in all?
 How
many more does she need to make her goal?
 How
many does she need if her goal is to collect 30
stickers?
Lesson 2- Engage New York- Module 1
Make number bonds through twenty with a
subtraction focus and apply to two-step word
problem


Standards to cover

2.OA.1

2.OA.2
Background knowledge

K.OA.3

K.OA.4

K.NBT.1

1.OA.6
Assessments

Informative- Observations

Formative- Exit Ticket

Student Summative Assessment

Teacher Rubric for Mathematical Practice
(lesson Planning)
http://mathforum.org/pcmi/hstp/resources/rubric/rubric.pdf
Questions
1.
Were the objectives and expectations clear?
2.
How would these two lessons support your grade level’s
standards?
3.
Do you have any suggestions or additions to the lessons?
4.
Are there any other tools, strategies or methods that
could be used?
5.
Could you modify the exit ticket for your own
classroom?