1. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Mathematical Structures: Addition and Subtraction Word Problem Types Tennessee Department of Education Elementary School Mathematics, Grade 1 December 6, 2012 Supporting Rigorous Mathematics Teaching and Learning

2. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Session Goals Participants will learn about: Common Core Content Standards and the Standards for Mathematical Practice. Types of situational word problems. Mapping devices and how they can scaffold student learning. Students’ addition and subtraction problem-solving strategies. Characteristics of assessing and advancing questions. 2

3. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Common Core State Standards The Standards include 2 types of standards: Standards for Mathematical Content. Standards for Mathematical Practice. 3

4. Common Core Standards for Mathematical Practice What would have to happen in order for students to have opportunities to make use of the CCSS for Mathematical Practice? 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. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 4

5. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Standards for Mathematical Practice Work in groups of 8; count off by 8. Each person reads one of the CCSS for Mathematical Practice. Read your assigned Mathematical Practice. Be prepared to share the “gist” of the Mathematical Practice. Each person has 2 minutes to share. Others listen for similarities and differences between the Mathematical Practice Standards. 5

6. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Discussing the Standards for Mathematical Practice What do you understand better about the Standards for Mathematical Practice now? What would you like more clarity about related to the Standards? 6

7. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Making Sense of the Mathematical Content Standards 7

8. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Common Core State Standards (Private Work) Study the first grade Operations and Algebraic Thinking Standards. Underline aspects of the standards that are familiar and circle aspects of the standards that you have not seen mentioned at the first grade level. 8

9. Common Core State Standards for Mathematics: Grade 1 9 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO Operations and Algebraic Thinking 1.OA 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. 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.

10. Common Core State Standards for Mathematics: Grade 1 10 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO Operations and Algebraic Thinking 1.OA 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.

11. Common Core State Standards for Mathematics: Grade 1 11 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO Operations and Algebraic Thinking 1.OA 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).

12. Common Core State Standards for Mathematics: Grade 1 12 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO Operations and Algebraic Thinking 1.OA 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 = ?.

13. Common Core State Standards for Mathematics: Grade 2 13 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO Operations and Algebraic Thinking 2.OA Represent and solve problems involving addition and subtraction. 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. Add and subtract within 20. 2.OA.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.

14. Common Core State Standards for Mathematics: Grade 2 14 Number and Operations in Base Ten 2.NBT Use place value understanding and properties of operations to add and subtract. 2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations. 2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. 2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations. Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

15. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Common Core State Standards (Group Discussion) What did you recognize from you current work with first grade students in the standards? What was familiar? What aspects of the standards surprised you? What have you not worked with students to understand in the past and now see included in the standards? How do the second grade standards differ from the first grade standards? 15

16. Rationale Perhaps the major conceptual achievement of the early school years is the interpretation of numbers in terms of part and whole relationships. With the application of a part-whole schema to quantity, it becomes possible for children to think about numbers as compositions of other numbers. This enrichment of number understanding permits forms of mathematical problem solving and interpretation that are not available to younger children. Resnick, L. B., 1983 In this session, teachers will learn about three types of situational word problems that can provide students with an understanding of the structure of problems and foundation that links directly to a set of key mathematical understandings that can be built directly from students’ solution paths. There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students. Warfield, 2001 16

17. Table 1: Common Addition and Subtraction Situations 17 Common Core State Standards, 2010, p. 88, NGA Center/CCSSO

18. One-Digit Addition and Subtraction Situations (First Grade) 1.Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? 2.Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether? 3.Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? 4.Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan? 5.Connie had some marbles. Juan gave her 5 more. Now she has 13 marbles. How many marbles did Connie have to start with? 6.Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether? 7.Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan? 8.Juan has 5 marbles. Connie has 8 more than Juan. How many marbles does Connie have? Carpenter, Fennema, Franke, Levi, & Empson, 1999, p. 12 18

19. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Focusing Our Discussion: Comparing Situational Word Problems 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. How do the word problem differ from each other? Which word problems might be easiest for students? Why? Which word problems might be hardest for students? Why? 19

20. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH A Developmental Sequence of Word Problems Order the types of word problems in the way in which you think students should study them. Explain why you ordered them this way. 20

21. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Mapping Devices and Problem-Solving Strategies 21

22. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Mapping Devices Study the four mapping devices. What is the benefit of each mapping device? How do the mapping devices differ from each other? 22 Mapping Device What is the benefit of each mapping device? How do the mapping devices differ from each other? Part-Part Whole Ten or Twenty Frame Number Line Hundreds Chart

23. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Part-Part Whole Mapping Device 23

24. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 24 Ten Frame Twenty Frame

25. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 25 Number Line

26. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Hundreds Chart 26 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100

27. Linking to Research/Literature Connections Between Representations 27 Pictures Written Symbols Manipulative Models Real-world Situations Oral Language Adapted from Lesh, Post, & Behr, 1987

28. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Identifying Students’ Problem- Solving Strategies 28

29. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing Single-Digit Problem-Solving Strategies Describe and name each student’s problem- solving strategy. Identify the Mathematical Practice Standards used by the student. 29

30. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1A Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? 30

31. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1B Connie had 13 marbles. She gave 8 to Juan. How many marbles does Connie have left? 31

32. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1C Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? 32

33. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1D Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? 33

34. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1E Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? 34

35. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1F Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan? 35

36. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1G 36 Connie has 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

37. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1H 37 Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

38. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Student 1I 38 Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

39. Problem-Solving Strategies 39 Addition Strategies 1.Counting All 2.Counting On or Up 3.Fact Strategies Round to 10 and then compensate by adding or subtracting (9 + 6 = 10 + 5 OR 9 + 6 = 10 + 6 – 1). Use doubles and compensate (7 + 6 = 6 + 6 + 1 OR 7 + 6 = 7 + 7 – 1). Decompose an addend and add to make a friendly number (8 + 6 = (8 + 2) + 4 = 10 + 4. 4. Known Facts Subtraction Strategies 1.Counting back 2.Fact Strategies Subtract a portion of one addend to arrive at a friendly number (10) and then subtract the remaining portion of the addend (14 – 8 = (14 – 4) – 4. 3.Count up from the known added, solve a missing addend problem (14 – 6 solve 6 + __ = 14).

40. One-Digit Addition and Subtraction Situations (Kindergarten) 1.Connie had 5 marbles. Juan gave her 3 more marbles. How many marbles does Connie have altogether? 2.Connie has 5 marbles. How many more marbles does she need to have 8 marbles altogether? 3.Connie had 8 marbles. She gave 5 to Juan. How many marbles does Connie have left? 4.Connie had 8 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan? 5.Connie has some marbles. Juan gave her 5 more. Now she has 8 marbles. How many marbles did Connie have to start with? 6.Connie has 5 red marbles and 3 blue marbles. How many marbles does she have altogether? 7.Connie has 8 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan? 8.Juan has 5 marbles. Connie has 3 more than Juan. How many marbles does Connie have? Carpenter, Fennema, Franke, Levi, & Empson, 1999, p. 12 40

41. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Two-Digit Addition and Subtraction Situations (Second Grade) 1.Connie had 28 marbles. Juan gave her 17 more marbles. How many marbles does Connie have altogether? 2.Connie has 28 marbles. How many more marbles does she need to have 45 marbles altogether? 3.Connie had 45 marbles. She gave 28 to Juan. How many marbles does Connie have left? 4.Connie had 45 marbles. She gave some to Juan. Now she has 28 marbles left. How many marbles did Connie give to Juan? 5.Connie has some marbles. Juan gave her 28 more. Now she has 45 marbles. How many marbles did Connie have to start with? 6.Connie has 28 red marbles and 17 blue marbles. How many marbles does she have altogether? 7.Connie has 45 marbles. Juan has 28 marbles. How many more marbles does Connie have than Juan? 8.Juan has 17 marbles. Connie has 28 more than Juan. How many marbles does Connie have? Carpenter, Fennema, Franke, Levi, & Empson, 1999, p. 12 41

42. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Second Grade: Possible Solution Paths Solution Path A Connie had 28 marbles. Juan gave her 17 more marbles. How many marbles does Connie have altogether? 42

43. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Second Grade: Possible Solution Paths Solution Path B Connie had 28 marbles. Juan gave her 17 more marbles. How many marbles does Connie have altogether? 43

44. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Second Grade: Possible Solution Paths Solution Path C Connie had 45 marbles. She gave 28 to Juan. How many marbles does Connie have left? 44

45. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing Teaching and Learning 45

46. The Mathematical Task Framework 46 TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000

47. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing Teaching and Learning Watch the addition and subtraction lesson. Use the recording sheet in your handout to record your observations. What are students learning? How is student learning supported by the teacher? Be prepared to make noticings and wonderings. 47

48. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Addition and Subtraction Lesson: Context Teacher: Rob Crowley District: Prince George’s School District Grade Level: First The district is using a curriculum unit developed by the Institute for Learning. The students are working on the third week of lessons. The students are working with the part-part whole mapping device. They regularly have access to manipulatives. Daily, students discuss word problems. The teacher tells students a word problem, the students explore with partners, while the teacher circulates asking assessing and advancing questions. Finally the students engage in a Share, Discuss, and Analyze discussion as a class. At this time the teacher presses students to talk about their problem-solving strategies and mathematical ideas. 48

49. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Situational Problems The students are solving the Pockets Task. I have some gum in each pocket. Altogether I have 12 pieces of gum. I have some gum in my left pocket and I have 9 pieces of gum in my right pocket. How many pieces of gum are in my left pocket? I have 13 pieces of gum. How many are in my left pocket if there are 9 in my other pocket? 49

50. Linking to Research/Literature Connections Between Representations 50 Pictures Written Symbols Manipulative Models Real-world Situations Oral Language Adapted from Lesh, Post, & Behr, 1987

51. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Bridge to Practice Over the next few weeks, engage students in solving and discussing the different types of addition and subtraction word problems. Give students many opportunities to use manipulatives and mapping devices when solving the problems. Keep track of the strategies that students use when explaining how they solved a problem. Prior to attending the next session, ask students to solve the simple word problems identified by the Common Core State Standards. Students may use manipulatives or mapping devices when they solve the problems. Bring 3 pieces of student work that show different types of thinking that students use when solving the problems. 51