Supporting Rigorous Mathematics Teaching and Learning

Transcription

1 Supporting Rigorous Mathematics Teaching and Learning Mathematical Structures: Addition and Subtraction Word Problem Types PARTICIPANT HANDOUT Tennessee Department of Education Elementary School Mathematics, Grade 1 December 6, 2012 learning research and development center institute for learning

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. Common Core State Standards The Standards include 2 types of standards: Standards for Mathematical Content. Standards for Mathematical Practice UNIVERSITY OF PITTSBURGH 2

3 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. 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 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? 2012 UNIVERSITY OF PITTSBURGH 3

4 The CCSS for Mathematical Content: Grade 1 Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. 1.OA.1 1.OA.2 1.OA 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. 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 = 11 is known, then = 11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 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. Add and subtract within 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., = = = 14); decomposing a number leading to a ten (e.g., 13 4 = = 10 1 = 9); using the relationship between addition and subtraction (e.g., knowing that = 12, one knows 12 8 = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13). Work with addition and subtraction equations. 1.OA.7 1.OA.8 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, = 2 + 5, = 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, =?. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 2012 UNIVERSITY OF PITTSBURGH 4

5 The CCSS for Mathematical Content: Grade 2 Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. 2.OA.1 2.OA 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 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. Number and Operations in Base Ten Use place value understanding and properties of operations to add and subtract. 2.NBT.5 2.NBT.6 2.NBT.7 2.NBT Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Add up to four two-digit numbers using strategies based on place value and properties of operations. 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 , and mentally subtract 10 or 100 from a given number 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 2012 UNIVERSITY OF PITTSBURGH 5

6 Table 1: Common Addition and Subtraction Situations 2012 UNIVERSITY OF PITTSBURGH 6

7 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 UNIVERSITY OF PITTSBURGH 7

8 Mapping Devices Study the four mapping devices. What is the benefit of each mapping device? How do the mapping devices differ from each other? 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 2012 UNIVERSITY OF PITTSBURGH 8

9 Part-Part Whole Mapping Device 2012 UNIVERSITY OF PITTSBURGH 9

10 Ten Frame 2012 UNIVERSITY OF PITTSBURGH 10

11 Twenty Frame 2012 UNIVERSITY OF PITTSBURGH 11

12 Number Line 2012 UNIVERSITY OF PITTSBURGH 12

13 Hundreds Chart UNIVERSITY OF PITTSBURGH 13

14 Linking to Research/Literature Connections Between Representations Pictures Manipulative Models Written Symbols Realworld Situations Oral Language Adapted from Lesh, Post, & Behr, UNIVERSITY OF PITTSBURGH 14

15 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 UNIVERSITY OF PITTSBURGH 15

16 Recording Sheet: One-Digit Addition and Subtraction Problem- Solving Strategies (First Grade) Student Describe and Name the Student s Problem- Solving Strategy Identify the Mathematical Practice Standards used by the student A B C D E F G H I 2012 UNIVERSITY OF PITTSBURGH 16

17 One-Digit Addition and Subtraction Problem-Solving Strategies (First Grade) Connie had 13 marbles. She gave 8 to Juan. How many marbles does Connie have left? Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? 2012 UNIVERSITY OF PITTSBURGH 17

18 One-Digit Addition and Subtraction Problem-Solving Strategies (First Grade) cont. Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan? 2012 UNIVERSITY OF PITTSBURGH 18

19 One-Digit Addition and Subtraction Problem-Solving Strategies (First Grade) cont. Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether? Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether? 2012 UNIVERSITY OF PITTSBURGH 19

20 Problem Solving Strategies 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 = OR = ). Use doubles and compensate (7 + 6 = OR = ). Decompose an addend and add to make a friendly number (8 + 6 = (8 + 2) + 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) Count up from the known added, solve a missing addend problem (14 6 solve 6 + = 14) UNIVERSITY OF PITTSBURGH 20

21 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 UNIVERSITY OF PITTSBURGH 21

22 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 UNIVERSITY OF PITTSBURGH 22

23 Possible Solution Paths: Second Grade Solution Path A Connie had 28 marbles. Juan gave her 17 more marbles. How many marbles does Connie have altogether? Solution Path B Connie had 28 marbles. Juan gave her 17 more marbles. How many marbles does Connie have altogether? Solution Path C Connie had 45 marbles. She gave 28 to Juan. How many marbles does Connie have left? 2012 UNIVERSITY OF PITTSBURGH 23

24 Analyzing Teaching and Learning: Addition and Subtraction Lesson Watch the addition and subtraction lesson. Use the recording sheet below 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. Analyzing Teaching and Learning: Addition and Subtraction Lesson Recording Sheet Observations Line Numbers How is student learning supported? Wondering 2012 UNIVERSITY OF PITTSBURGH 24

25 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 UNIVERSITY OF PITTSBURGH 25