mathematical structures: addition and subtraction word problem types

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


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.

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Common Core State Standards

The Standards include 2 types of standards:

• Standards for Mathematical Content.

• Standards for Mathematical Practice.

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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/CCSSO4


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.

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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?

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Making Sense of the Mathematical Content Standards

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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.

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Common Core State Standards for Mathematics: Grade 1

9Common 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.




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.




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).


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Common Core State Standards, 2010, p. 15, NGA Center/CCSSO


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 = ?.


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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.


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Number and Operations in Base Ten 2.NBTUse 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(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?

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RationalePerhaps 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

Table 1: Common Addition and Subtraction Situations


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

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Focusing Our Discussion: Comparing Situational Word Problems1.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?

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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.

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Mapping Devices and Problem-Solving Strategies

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Mapping DevicesStudy the four mapping devices.

• What is the benefit of each mapping device?

• How do the mapping devices differ from each other?

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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


Part-Part Whole Mapping Device

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LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 24

Ten Frame Twenty Frame

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 25

Number Line


Hundreds Chart

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1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Linking to Research/LiteratureConnections Between Representations

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Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987


Identifying Students’ Problem-Solving Strategies

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Analyzing Single-Digit Problem-Solving Strategies

• Describe and name each student’s problem-solving strategy.

• Identify the Mathematical Practice Standards used by the student.

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

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Student 1B

Connie had 13 marbles. She gave 8 to Juan. How manymarbles does Connie have left?

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Student 1CConnie had 13 marbles. She gave 5 to Juan. How manymarbles does Connie have left?

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

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Student 1EConnie had 13 marbles. She gave 5 to Juan. How manymarbles does Connie have left?

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Student 1FConnie has 13 marbles. Juan has 5 marbles. How many moremarbles does Connie have than Juan?

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Student 1G

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Connie has 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?


Student 1H

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


Student 1I

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Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

Problem-Solving Strategies

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Addition Strategies1. Counting All2. Counting On or Up3. 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 Strategies1. Counting back2. 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).

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?




5. Connie has some marbles. Juan gave her 5 more. Now she has 8 marbles. How many marbles did Connie have to start with?



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

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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?




5. Connie has some marbles. Juan gave her 28 more. Now she has 45 marbles. How many marbles did Connie have to start with?



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

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Second Grade: Possible Solution PathsSolution Path A


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Second Grade: Possible Solution PathsSolution Path B


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Second Grade: Possible Solution PathsSolution Path C

Connie had 45 marbles. She gave 28 to Juan. How many marbles does Connie have left?

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Analyzing Teaching and Learning

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The Mathematical Task Framework

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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


Analyzing Teaching and LearningWatch 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.

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Addition and Subtraction Lesson: ContextTeacher: Rob CrowleyDistrict: Prince George’s School DistrictGrade 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.

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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?

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Linking to Research/LiteratureConnections Between Representations

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Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987


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.

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mathematical structures: addition and subtraction word problem types

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