Operations and Algebraic Thinking Second Grade

1. Operations and Algebraic ThinkingSecond Grade Cindy Foubert, Jeannette Robinson, Haley Snyder, Kristin Finney and Andrelyn Alexander

2. Operations and Algebraic ThinkingCluster 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

3. 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

4. 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!

5. 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.

6. 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

7. ProgressionCCSS: 2.OA.A.1 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. 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. Kindergarten Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

8. 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)

9. 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)

10. 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

11. 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

12. ProgressionCCSS: 2.OA.2 • 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 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. • 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. Second Grade Use place value understanding and properties of operations to add and subtract within 20

13. 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

14. 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

15. 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.

16. ProgressionsCCSS: 2.OA.C.3 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. • First Grade • Relate counting to addition and subtraction (e.g. by counting on 2 to add 2) 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 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.

17. 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

18. 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

19. 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

20. 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).

21. 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

22. 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

23. 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

24. 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?

25. 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

26. 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

27. Questions • Were the objectives and expectations clear? • How would these two lessons support your grade level’s standards? • Do you have any suggestions or additions to the lessons? • Are there any other tools, strategies or methods that could be used? • Could you modify the exit ticket for your own classroom?