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 & 4Work 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 AffectsWhat 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

There are 29 animals at the pet store. The pet store receives 18 more. How many are there now?

Change Unknown 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

ProgressionCCSS: 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 proceduresKnowledge of when and how to use appropriatelySkill in performing them with flexibility, accurately

and efficiently

Kindergarten

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.

First Grade

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

Second Grade

Use place value understanding and properties of operations to add and subtract within 20

Third Grade

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.

ProgressionCCSS: 2.OA.2

Add and Subtract Within 20Teaching 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 20Misconceptions

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.

ProgressionsCCSS: 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

Relate counting to addition and

subtraction (e.g. by counting on 2 to add

2)

Second Grade

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

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

Work With Equal Groups of Objects to Gain Foundations for

Multiplication

Progressions 2.OA.C.4:

Kindergarten

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

First Grade

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 http://www.engageny.org 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

Lesson 1Engage New York-

Make Number Bonds of Ten & Twentywith 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?