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Mathematical Structures: Addition and Subtraction Word Problem Types

Supporting Rigorous Mathematics Teaching and Learning. Mathematical Structures: Addition and Subtraction Word Problem Types. Tennessee Department of Education Elementary School Mathematics, Grade 1 December 6, 2012. Session Goals . Participants will learn about:

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Mathematical Structures: Addition and Subtraction Word Problem Types

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  1. Supporting Rigorous Mathematics Teaching and Learning Mathematical Structures: Addition and Subtraction Word Problem Types Tennessee Department of Education Elementary School Mathematics, Grade 1 December 6, 2012

  2. 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.

  3. Common Core State Standards The Standards include 2 types of standards: • Standards for Mathematical Content. • Standards for Mathematical Practice.

  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? • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

  5. 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.

  6. 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?

  7. Making Sense of the Mathematical Content Standards

  8. 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.

  9. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  10. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  11. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  12. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  13. Common Core State Standards for Mathematics: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

  14. Common Core State Standards for Mathematics: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

  15. 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?

  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

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

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

  19. 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?

  20. 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.

  21. Mapping Devices and Problem-Solving Strategies

  22. Mapping Devices Study the four mapping devices. • What is the benefit of each mapping device? • How do the mapping devices differ from each other?

  23. Part-Part Whole Mapping Device

  24. Ten Frame Twenty Frame

  25. Number Line

  26. Hundreds Chart

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

  28. Identifying Students’ Problem-Solving Strategies

  29. Analyzing Single-Digit Problem-Solving Strategies • Describe and name each student’s problem-solving strategy. • Identify the Mathematical Practice Standards used by the student.

  30. Student 1A Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

  31. Student 1B Connie had 13 marbles. She gave 8 to Juan. How many marbles does Connie have left?

  32. Student 1C Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left?

  33. Student 1D Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

  34. Student 1E Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left?

  35. Student 1F Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan?

  36. Student 1G • Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

  37. Student 1H Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

  38. Student 1I • Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

  39. Problem-Solving Strategies • Addition Strategies • Counting All • Counting On or Up • 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 • Counting back • 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. • Count up from the known added, solve a missing addend problem • (14 – 6 solve 6 + __ = 14).

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

  41. Two-Digit Addition and Subtraction Situations (Second Grade) • Connie had 28 marbles. Juan gave her 17 more marbles. How many marbles does Connie have altogether? • Connie has 28 marbles. How many more marbles does she need to have 45 marbles altogether? • Connie had 45 marbles. She gave 28 to Juan. How many marbles does Connie have left? • Connie had 45 marbles. She gave some to Juan. Now she has 28 marbles left. How many marbles did Connie give to Juan? • Connie has some marbles. Juan gave her 28 more. Now she has 45 marbles. How many marbles did Connie have to start with? • Connie has 28 red marbles and 17 blue marbles. How many marbles does she have altogether? • Connie has 45 marbles. Juan has 28 marbles. How many more marbles does Connie have than Juan? • Juan has 17 marbles. Connie has 28 more than Juan. How many marbles does Connie have? Carpenter, Fennema, Franke, Levi, & Empson, 1999, p. 12

  42. 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?

  43. 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?

  44. Second Grade: Possible Solution Paths Solution Path C Connie had 45 marbles. She gave 28 to Juan. How many marbles does Connie have left?

  45. Analyzing Teaching and Learning

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

  47. 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? Cite specific line numbers. How is student learning supported by the teacher? Be prepared to make noticings and wonderings.

  48. 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.

  49. 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?

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

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