Operations and Algebraic Thinking in First Grade

1. Operations and Algebraic Thinking in First Grade Amanda Maydeck ElaeniaSerdehely Pamela Henning

2. What is Algebra? • http://0-ediv.alexanderstreet.com.innopac.library.unr.edu/view/1737930

3. Algebra in First Grade • Teaching just computation and arithmetic, “is an inadequate benchmark, because a lot of interesting mathematics in primary school does not depend on the ability of students to successfully apply specific arithmetic algorithms” (Lopez and Estrella, 2011)

4. Algebra in First Grade • First Graders learn the meaning of +1, +2, +5 is the same as moving up this many spaces on the number line. They learn the inverse of this meaning that -1, -2, -5 is the same as moving back this many spaces on the number line. Students do not need to count both numbers to solve the problem. • First graders start to discover patterns in numbers and use this to solve addition and subtraction problems. First graders first learn their doubles facts and use this to solve doubles +1. • First Graders also discover how to make groups of ten. This helps build fluency in addition and subtraction problems. • First Graders learn the meaning of the equal sign to be the same as. Students look at problems and determine if the statement is true or false. This is the beginning to learning how to balance an equation

5. Represent and solve problems involving addition and subtraction • 1.OA.1: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.2 • 1.OA.2: 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.

6. What does it look like? • Teachers pose questions that relate to familiar contexts for students that result in the use of addition or subtraction. • Problems can consist of problem structure in which the • Result is unknown • Change is unknown • Start is unknown • Students must use their understanding of addition and subtraction to work through context based problems.

7. What does it look like? • http://www.teachertube.com:8809/viewVideo.php?video_id=44503

8. Example Problem 1.OA.1 9 boys and 8 girls were in the class. How many children were in the class in all? Result unknown Possible algebraic equations: 9+8= ? 8+9= ? ? = 9+8 ? = 8+9 Samples provided by illustrative mathematics

9. 1.OA.1 17 children were in the class. 9 were boys and the rest were girls. How many girls were in the class? Addend Unknown (Change) Possible algebraic equations: 17=9+ ? 17= ? + 9 9+ ? =17 ? +9=17 17–9= ? Samples provided by illustrative mathematics

10. 1.OA.1 17 children were in the class. There were some boys and 8 girls. How many boys were in the class? Addend Unknown (Start) Possible algebraic equations: 17=8+ ? 17= ? + 8 8+ ? =17 ? +8=17 17–8= ? Samples provided by illustrative mathematics

11. 1.OA.2 • Jasmine has eight daisies and three vases - one large, one medium-sized and one small. She puts 5 daisies in the large vase, 2 in the medium vase and 1 in the small vase. • Can you find another way to put daisies so that there are the most in the large vase and least in the small vase? • Try to find as many ways as you can put the daisies in the vases with the most in the large vase and the least in the smallest vase. If you think you have found them all, explain how you know those are all the possibilities. Samples provided by illustrative mathematics

12. What does it look like?

13. Misconceptions • Students may have difficulty identifying the logical structure of the word problem. • Wording may make it difficult for students to decipher between join or separate problems with start unknown being the most difficult. • Language in compare problems might be challenging for ELL students based on confusing vocabulary. • Equal sign can be misinterpreted as operational rather than relational. • When faced with a representational problem with three addends, students may leave out the third addend or lack organization in their problem.

14. Progression • In order for first grade students to be able to represent and solve problems involving addition and subtraction: • K.OA.1: Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. • K.OA.2: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

15. Progression • Following first grade students will use their understanding of representing addition and subtraction problems algebraically in order to: • 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.

16. Progression • In Kindergarten students learn their numbers by associating objects with numbers. They use this knowledge to solve addition and subtraction problems within 10. This leads in to students being able to solve addition and subtraction problems within 20 without using objects. By the end of first grade students can solve addition and subtraction problems by counting up. This leads into second grade where students learn how to solve addition and subtraction problems within 100. Students know addition and subtraction facts with fluency in single digit problems.

17. Understand and apply properties of operations and the relationship between addition and subtraction • 1.OA.3:Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) • 1.OA.4:Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

18. What does it look like? • Students should be able to differentiate between and use the: • Communicative property of addition (3+2=2+3) • Associative property of addition (4+1=3+2) • Students will use what they know about unknown addend problems to relate subtraction to addition (Fact Families).

19. Example Problems Communicative Property • Build a two-color train using less than ten connecting cubes. Place cubes of the same color together. • Draw your train and write a number sentence to describe it from left to right. • Flip your train, draw it again and write the turn around fact. • Repeat with other two color trains. 2 + 3 = 5 3 + 2 = 5 Example provided by K-5MathTeachingResources.com

20. Example Associative Property

21. 1.OA.4 Choose a domino. Draw the domino. Write a fact family for the domino you chose. Repeat with other dominoes. 6 + 3 = 9 3 + 6 = 9 9 - 3 = 6 9 - 6 = 3 Example provided by K-5MathTeachingResources.com

22. Misconceptions • Students might be confused about the meaning of the equals sign. In the problem 2+4=3+3 students may want to substitute first 3 in the second problem with a 6. This is because they may see the equal sign as a symbol indicating that the answer should follow. • Students be unsure if each property (associative or communicative) works consistently. • Teacher must provide many examples to deter this misconception • Students many times confuse the order of numbers in a fact family resulting in an incorrect number sentence.

23. Progression • First they develop understanding of number concepts. • Using understanding of number concepts, students are able to add and subtract within 10. • Following the mastery of addition and subtraction within 10, students progress to addition and facts between 11-20 and then 20-100. • During this process students are taught addition and subtraction together to help relate one to the other.

24. Progression • In order for first grade students to be able to apply properties of operations and find relationships between addition and subtraction they must first: • K.OA.1: Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. • K.OA.2: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

25. Progression • Following first grade students will use their understanding of properties of operations and find relationships between addition and subtraction they will: • 2.NBT.9: Explain why addition and subtraction strategies work, using place value and the properties of operations.

26. Add and subtract within 20 • 1.OA.5: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). • 1.OA.6:Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

27. What does it look like? • Students use many different strategies to add and subtract. Strategies can include: • Counting on • Using the number line • Using counters to represent addition and subtraction problems • Acting out problems • Using models • Drawing pictures • And many more…

28. What does it look like? • http://www.brainpopjr.com/math/additionandsubtraction/countingon/

29. What does it look like? 5+4=

30. Misconceptions • Students may struggle with the use of their manipulatives. • Double hopping on the number line • Losing track of counters • Sloppy drawings • Etc… • Students might confuse the + and – sign. • Many times students are more apt to add rather than subtract. Because of this they will not look at the sign and assume addition.

31. Progression • In order for first grade students to be able add and subtract within 20 they must first be able to: • K.CC.4c: Understand that each successive number name refers to a quantity that is one larger. • K.OA.1: Represent addition and subtraction with objects, fingers, mental images, drawings2, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. • K.OA.2:Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. • K.OA.3:Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

32. Progression • Following first grade students will use their understanding of adding and subtracting within 20 to: • 2.OA.2:Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.

33. Work With Addition and Subtraction Equations • 1.OA.7:Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. • 1.OA.8:Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = 􀃍 – 3, 6 + 6 = 􀃍.

34. What does it look like? 1.OA.7 Compare the number of circles in each box. If they are equal, write a number sentence. 4+3=5+1+1 Example from illustrative mathematics

35. 1.OA.8 Find the missing number in each of the following equations: 9−3=□ 8+□=15 16−□=5  □=7−2 13=□+7 6=14−□  Example from Illustrative Mathematics

36. Misconceptions • http://www.youtube.com/watch?v=eO3OQI9Jwts 2:24-end

37. Progression • In order for first grade students to be able to work with addition and subtraction equations they must be able to: • 1.OA.1: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 • 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

38. Progression • Following first grade students will use their understanding of working with addition and subtraction equations they must be able to: • 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. • 2.OA.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.

39. Assessment • http://www.solonschools.org/MR/AS.asp (last video) • http://www.solonschools.org/MR/ASVid.asp (right)

40. Resources • University of Arizona Progression Documents • Zimba Chart • Illustrative Mathematics • NCTM Illuminations • NCTM • CCSS • Algebra and the Elementary Classroom by Maria L. Blanton • Math Workstations by Debbie Diller

41. Lesson 1 • You will need: a recording sheet, and a set of cards • Find your shoulder buddy. • With your partner use the deck of cards to make as many sets of 12 as you can. • Write the different ways you found on the paper. • Be ready to share. 1.OA.3: Apply properties of operations as strategies to add and subtract.

42. Now try 3 cards. • How many ways can you make 12 with 3 cards.

43. What was a pair that someone found? • Did anyone find a different pair? • What strategies did you use to find the pairs? • Did anyone use a different strategy? • Did you notice any patterns when finding pairs of numbers?

44. What prior knowledge would the students need in order to be successful in this lesson? • How was this lesson cognitively demanding? • How does this lesson apply to algebraic thinking/reasoning?

45. Differentiation options: • During lesson the sums can be altered to make it easier or to add a challenge for students • If students are able to master the three card pairs easily, you may add a fourth card and find patterns within the addition sentences with four addends. • Students can create story problems to go along with the number sentences that they created.

46. Lesson 2 • You will need: 1 game mat for you and your partner, and 1 set of cards. • Each player will use one side of the game mat (player 1 or player 2). • Take turns turning over a card. • Decide if the number on the card will fit in one of the blank spaces in the number problems on your side of the game mat. • If it does, write the number in the blank space that it belongs. • If it doesn’t, put the card face up in the discard pile. The next player can choose this card if they want or pick a new card. • Keep going until one player fills in every answer on their side of the game mat. 1.OA.8: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.