Chapter 11

Developing Strategies for Addition and Subtraction Computation

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Direct Modeling

Direct modeling involves using physical objects or drawings to represent and solve mathematical problems. This approach is often used by younger students or those at an early stage of understanding mathematical concepts.

• Definition: Direct modeling is the process by which students use concrete objects or pictorial representations to illustrate and solve problems. This method is grounded in their tangible and visual understanding of numbers and operations.
• Purpose: It helps students develop a concrete understanding of mathematical concepts before moving on to more abstract representations. It builds a strong foundation by making connections between the physical world and mathematical symbols.
• Example: For an addition problem like 5 + 3, students might use counters or draw circles to represent each number and then combine them to find the sum.

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Invented Strategies

Invented strategies, sometimes referred to as student-generated strategies, are the methods that students create on their own to solve mathematical problems. These strategies are typically more meaningful and understandable to the student than traditional algorithms because they are based on the student's own understanding and reasoning.

• Definition: Invented strategies are non-standard methods created by students to solve mathematical problems. They are often more flexible and intuitive than standard algorithms.
• Characteristics:
• They may vary widely among students.
• They are typically based on a student's understanding of number relationships and properties.
• They often involve breaking numbers apart and recombining them in ways that make sense to the student.
• Example: For the problem 47 + 38, a student might decompose the numbers into parts they find easier to work with, such as 40 + 30 and 7 + 8, then combine these sums to get 70 + 15, and finally combine 70 and 15 to get 85.
• Advantages: These strategies encourage deeper understanding and flexibility in thinking, allowing students to develop a more robust number sense.

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Standard Algorithms

Standard algorithms are the conventional step-by-step procedures that are traditionally taught in schools for performing arithmetic operations. They are efficient and reliable methods for obtaining correct answers but may not always promote deep understanding.

• Definition: Standard algorithms are formal, procedural methods for solving mathematical problems, usually involving a set sequence of steps.
• Characteristics:
• They are consistent and widely recognized.
• They often require memorization and practice to master.
• They are typically introduced after students have a solid understanding of the underlying concepts through direct modeling and invented strategies.
• Example: For addition, the standard algorithm involves aligning the numbers by place value, adding each column starting from the rightmost digit, carrying over any extra value to the next column as needed.
• Importance: While standard algorithms are efficient for computation, it is crucial for students to understand the reasoning behind these procedures. Teaching should focus on ensuring that students grasp the conceptual foundations before moving on to these algorithms.

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Toward Computational Fluency

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Role of Teachers

Teachers play a crucial role in developing computational fluency. Van de Walle emphasizes the importance of:

• Assessing Students' Understanding: Regularly assessing students' computational strategies and understanding to provide targeted instruction.
• Providing Rich Mathematical Tasks: Designing tasks that encourage exploration and the use of multiple strategies.
• Fostering a Growth Mindset: Encouraging students to view mistakes as learning opportunities and to persist in finding solutions.

Integration with Other Mathematical Strands

Computational fluency should not be taught in isolation. It should be integrated with other strands of mathematics, such as number sense, algebra, and geometry, to provide a comprehensive mathematical education.

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Reasoning Strategies

Counting On and Counting Back:

• Addition: Start with the larger number and count on the smaller number.
• Subtraction: Start with the number being subtracted and count back the smaller number.

Making Tens:

• Addition: Decompose one of the numbers to make a ten first. For example, to add 8 + 5, think of 8 + 2 + 3.
• Subtraction: Use tens to simplify the problem. For instance, to subtract 14 - 6, think of 14 - 4 - 2.

Doubles and Near Doubles:

• Addition: Use known doubles facts (e.g., 4 + 4) to find near doubles (e.g., 4 + 5 can be thought of as 4 + 4 + 1).
• Subtraction: Relate to doubles facts to find differences (e.g., if 6 + 6 = 12, then 12 - 6 = 6).

Breaking Apart Numbers (Decomposition):

• Addition: Break numbers into smaller, more manageable parts. For example, 25 + 47 can be broken into (20 + 40) + (5 + 7).
• Subtraction: Decompose numbers to simplify the process. For instance, 53 - 27 can be broken into (53 - 20) - 7

Reasoning Strategies continued…

Using Compatible Numbers:

• Addition: Adjust numbers to make them easier to add. For example, with 39 + 56, think of 40 + 55.
• Subtraction: Adjust numbers to make them easier to subtract. For instance, with 52 - 38, think of 52 - 40 + 2.

Compensation:

• Addition: Adjust one number to make a ten, then compensate by adjusting the other number. For example, 29 + 38 can be adjusted to (30 + 38) - 1.
• Subtraction: Adjust both numbers up or down to simplify the problem, then compensate. For instance, 53 - 29 can be adjusted to (54 - 30) + 1.

Using Number Lines:

• Addition: Visualize the addition by jumping forward on the number line. For example, for 7 + 5, start at 7 and make 5 jumps forward.
• Subtraction: Visualize the subtraction by jumping backward on the number line. For instance, for 12 - 4, start at 12 and make 4 jumps backward.

Using Place Value Strategies:

• Addition: Add tens and ones separately, then combine. For example, 34 + 27 can be thought of as (30 + 20) + (4 + 7).
• Subtraction: Subtract tens and ones separately, then combine. For instance, 72 - 45 can be thought of as (70 - 40) + (2 - 5).

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Algorithm for Addition

Counting On Strategy:

• This strategy involves starting with the larger number and counting on the smaller number.
• For example, to solve 7+57 + 57+5, you would start with 7 and count on five more: 8, 9, 10, 11, 12. So, 7+5=127 + 5 = 127+5=12.

Decomposing Numbers (Making Tens):

• This strategy involves breaking numbers into parts to make a ten, which simplifies addition.
• For example, to solve 8+68 + 68+6, you can decompose 6 into 2 and 4. Then, 8+2=108 + 2 = 108+2=10, and 10+4=1410 + 4 = 1410+4=14. So, 8+6=148 + 6 = 148+6=14.
• This method leverages students' understanding of place value and the base-ten system.

Using Number Lines:

• A number line can be an effective visual tool for understanding addition.
• For example, to solve 5+75 + 75+7, you would start at 5 on the number line and make a jump of 7 units forward to land on 12.
• This helps students visualize the process of addition as a movement along a continuum.

Connecting Addition & Subtraction to Place Value

Importance of Place Value in Addition and Subtraction

Van de Walle emphasizes that a solid grasp of place value is crucial for students to perform addition and subtraction effectively. Understanding place value helps students to:

1. Recognize the Value of Digits: Understand that the value of a digit depends on its position within a number (units, tens, hundreds, etc.).
2. Decompose Numbers: Break down numbers into smaller, more manageable parts based on their place values.

Strategies for Connecting Addition and Subtraction to Place Value

1. Using Base-Ten Blocks: Manipulatives such as base-ten blocks help students visualize and understand how numbers are composed and decomposed. This hands-on approach reinforces the concept of place value.
2. Expanded Form: Teaching students to write numbers in expanded form (e.g., 345 as 300 + 40 + 5) helps them see the contribution of each digit according to its place value. This understanding aids in performing addition and subtraction.
3. Regrouping (Carrying and Borrowing): Van de Walle highlights the importance of teaching regrouping in the context of place value. Students learn to "carry" or "borrow" by redistributing value among different place values (e.g., regrouping ten ones into one ten).

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Connecting Addition & Subtraction to Place Value

Understanding Numeration and Place Value

1. Foundation of Numeration:
• Numeration is the system of using numbers to represent quantities. It includes understanding the structure and organization of the number system, such as the base-10 system.
• Place value is a critical component of numeration, where the position of a digit in a number determines its value (e.g., in the number 345, the digit 4 represents 40 because it is in the tens place).
2. Building Conceptual Understanding:
• Van De Walle emphasizes the importance of students grasping the concept of place value to make sense of larger numbers. This understanding is crucial for performing operations like addition and subtraction with two- and three-digit numbers.
• Activities that highlight the base-10 system, such as using manipulatives (e.g., base-10 blocks), help students visualize and internalize the concept of place value.

Blending Numeration and Computation

1. Integrated Instruction:
• Instead of teaching place value and computation as separate topics, Van De Walle advocates for an integrated approach where students apply their understanding of place value to computational tasks.
• For example, when adding two-digit numbers, students should think in terms of combining tens and ones, rather than just following a procedural algorithm.
2. Developing Strategies:
• The focus shifts from rote memorization of algorithms to developing flexible strategies for computation. These strategies are grounded in students' understanding of place value and numeration.
• Strategies such as breaking apart numbers (e.g., decomposing 47 + 36 into 40 + 30 and 7 + 6) or using known facts (e.g., recognizing that 50 + 30 is easier to compute) are encouraged.
3. Encouraging Mental Math and Estimation:
• By blending numeration and computation, students become adept at using mental math and estimation. These skills are essential for efficient and accurate computation.
• For instance, students might estimate the sum of 78 + 47 by rounding to the nearest ten (80 + 50) and then adjusting the estimate.

Connecting Addition & Subtraction to Place Value Continued…

Advantages of Blended Instruction

1. Deepened Understanding:
• Students develop a deeper understanding of the number system and the operations they perform. This understanding leads to greater mathematical fluency and confidence.
2. Transferable Skills:
• The strategies learned through blended instruction are transferable to more complex mathematical tasks. Students learn to approach problems flexibly and creatively.
3. Reduced Reliance on Algorithms:
• Students are less dependent on memorized algorithms, which can often lead to errors if not understood conceptually. Instead, they develop a toolkit of strategies that can be adapted to various problems.

Implementation in the Classroom

1. Use of Manipulatives and Visuals:
• Teachers should use manipulatives (e.g., base-10 blocks, number lines) and visual aids to help students concretize their understanding of numeration and place value.
2. Rich Problem-Solving Tasks:
• Providing rich, real-world problem-solving tasks that require the application of numeration and computational strategies helps students see the relevance of what they are learning.
3. Discussion and Reflection:
• Encouraging students to discuss their strategies and reflect on their thought processes fosters a collaborative learning environment where conceptual understanding is prioritized.

Three Types of Computational Strategies: Direct Modeling

1. Direct Modeling: This developmental step comes right before invented strategies. It is the use of manipulatives and/or drawings along with counting to represent word problems and the meaning of operations (addition and subtraction).This strategy helps students understand the base ten model. Once students have grasped the base ten model, teachers should “fade out” direct modeling by:
1. Recording students’ verbal explanations
2. Have students, who have solved a problem with models, think about they can solve the problem mentally (or without the use of manipulatives)
3. Have students to write a numeric record that accompany their models.

Three Types of Computational Strategies: Invented Strategies

2. Invented Strategies: This strategy refers to anything other than the standard algorithm and does not use manipulatives or drawings. During this developmental step, student use their knowledge about the base ten model to support their understanding of addition and subtraction. It can include mental math and writing down intermediate steps to work through problems. Benefits of invented strategies include:

• Students make fewer errors.
• Students develop number sense.
• It is the basis for mental computation and estimation.
• It generates procedural proficiency.

Three Types of Computational Strategies: The Standard Algorithm

3. The Standard Algorithm: This developmental step contains three main components:

• Knowing and performing the step by step procedures
• Knowing how the algorithm works and how to apply it
• Knowing when an algorithm should be used and comparing the value and usefulness of different algorithms

Standard algorithm should not be “just” memorized. Students should understand the “why” for mathematical concepts. (For example, regrouping in addition and subtraction). Standard algorithm can be different across cultures. Therefore, teachers should be knowledgeable about these cultural differences when working with diverse/multicultural student populations.

Development of Invented Strategies in Addition & Subtraction

Creating a Supportive Environment for Invented Strategies

Teachers should create a supportive environment for invented strategies. When creating a supportive environment for invented strategies, teachers should promote curiosity and encourage students to try various ways to solve a problem. Teachers should avoid identifying the correct answer too early into the process. Teachers should use discourse to discuss various ideas. Teachers should also show examples of student work or have students present the strategy they used to solve a problem.

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Development of Invented Strategies in Addition & Subtraction

Models to Support Invented Strategies

There are four common types to solve addition and subtraction. They include the following:

• Break Apart Strategy
• Split Strategy
• Jump Strategy
• Shortcut Strategy

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Development of Invented Strategies in Addition & Subtraction

Adding and Subtracting Single Digit Numbers

Teachers should try to extend the knowledge of basic facts. Teachers should challenges students to use mental math to solve problems and encourage mathematical dialogue.

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Development of Invented Strategies in Addition & Subtraction

Adding and Subtracting Multi Digit Numbers

This method begins in first grade with adding ten more.

Example: 20 + 10 = 30.

In second and third grade, students begin to add double digit numbers.

Example: 15 + 15 = 30

In fourth grade, students begin to add three-digit numbers and beyond.

Example: 115 + 215 = 330

Development of Invented Strategies in Addition & Subtraction

Subtraction As “Think Addition”

Students would learn how to use the procedure of counting up and finding the missing part to solve subtraction problems.

Example: 25 - 15 = ?

How much do you have to add to 15 to get to 25?

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Development of Invented Strategies in Addition & Subtraction

• Take Tens from the Tens, then Subtract Ones

70 - 40 —> 30 - 6

24 + 3 —> 27

• Take Away Tens, Then Ones

73 - 40 —> 33 - 3

30 - 3 —> 27

• Take Extra Tens, Then Add Back

73 - 50 —> 23 + 4 = 27

• Add to the Whole If Necessary

73 - 46 —> 76 - 46 —> 30 - 3 = 27

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Take Away Subtraction

Example Problem: 73 - 46

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Development of Invented Strategies in Addition & Subtraction

Extensions and Challenges

Crossing a Ten (or More)

• This procedure involves regrouping (“a strategy that involves rearranging numbers into groups of ten when performing addition or subtraction operations”).

Larger Numbers

• The use of different strategies when adding and subtracting larger numbers. For example, “chunking off”.

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Standard Algorithms for Addition and Subtraction

Big Ideas:

• Decomposition (taking apart) and composition (combining) of numbers in various ways are flexible methods of addition and subtraction. Addition and subtraction strategies that build on decomposing and composing numbers in flexible ways contribute to students’ overall number sense.
• Multi-digit numbers can be built up or taken apart in many ways to make the numbers easier to work with. These parts can be used to estimate answers in calculations rather than using the exact numbers involved.
• Computational estimation involves using easier-to-handle parts of numbers or substituting difficult-to-handle numbers with closely compatible numbers so that the resulting computations can be done mentally.
• Reasoning strategies provide flexible methods of computing that vary with the numbers and the situation.
• Flexible methods for computation require a deep understanding of the operations and properties of the operations including the inverse operation of addition and subtraction.
• Standard algorithms are step-by-step processes based on place value.

Standard Algorithms for Addition and Subtraction

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• Notes on significant differences:
• Kamii and Dominick (1998) suggest that standard algorithms “unteach” place value due to focus on digits rather than the number itself.
• Reasoning strategies often begin with the largest units (leftmost digits) because they focus on the entire number. In contrast, the standard algorithm starts with the ones units, which is also counter to how students write. By starting on the right with a digit orientation, the solution is hidden until the end.
• NOTE: Exception is the standard algorithm for division.
• Reasoning strategies are a range of flexible options rather than one right way. Reasoning strategies depend on the numbers involved, so students can be flexible and make the computation easier.

Standard Algorithms for Addition and Subtraction

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• A bridge to the standard algorithms for addition and subtraction is to use partial sums or differences. The partial-sum method begins with the largest place value, records that partial sum below the problem, and then continues toward the smaller place values.
• Example:
• 358 + 276 = ?
• 300 + 200 = 500
• 50+70=120
• 8+6=14
• 500+120+14= 634
• If all the digits of the minuend (first number) are larger than the digits in the subtrahend (second number), it is straightforward. However, if that is not the case, the subtraction must be notated to illustrate the negative value.
• Example:
• 645-328= ?
• 600-300 = +300
• 40-20 = +20
• 5-8 = -3
• 300+20-3= 317

Standard Algorithms for Addition and Subtraction

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• Standard Algorithm: Model then Record
• Note: Always ensure student standard algorithm is ONE method/possibility - not the ONLY; it can be beneficial for some problems while other strategies may be better choices in other situations.

• For addition: Regrouping is exchanging 10 in one place-value position for 1 in the position to the left—or the reverse, exchanging 1 for 10 in the position to the right. The terms carrying and borrowing are conceptually misleading and therefore obsolete. The word regroup may initially have little meaning for young students, so start with the term trade. Ten ones are traded for a ten. A hundred is traded for 10 tens.

Standard Algorithms for Addition and Subtraction

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• Begin instruction with concrete models; then, make explicit connections to the symbolic process of combining and regrouping.
• For example, use place value mats and base-ten blocks to model numbers.
• Next, develop a written record.
• For example, use Addition and Subtraction recording pages to help students record numerals in columns as they model each step of the procedure they carry out as they combine (add) the base-ten materials.

Standard Algorithms for Addition and Subtraction

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• For subtraction, the general approach is similar to addition.
• Start by having students treat the subtraction problem as a take-away situation. With this meaning of subtraction, they model only the top number in a subtraction problem on the top half of their place-value mats, using the top double ten-frame for any ones. For the amount to be subtracted, have students write each digit and place the numerals near the bottom of their mats in the respective columns.
• To reduce errors, suggest carrying out all regrouping first. That way, the full amount can be taken off immediately. Also, explain to students that they are to begin working with the ones column first, as they did with addition.
• The process of recording each step as it is done is the same as was suggested for addition using the Addition and Subtraction recording pages. When students can explain the use of symbols involved in the recording process, they can decide to transition away from the use of the concrete materials and move toward the use of symbols.

Standard Algorithms for Addition and Subtraction

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• NOTE: Problems in which internal zeros are involved cause special difficulties.
• The common errors that emerge when students regroup across zero are best addressed at the modeling stage. For example, in 403-138 students must make a double trade: regrouping a hundreds piece for 10 tens and then a tens piece for 10 ones. After students have experience with making these double regroupings, using base-ten materials, use the following activity before giving students any hints about how they might approach regrouping across an internal zero.
• ACTIVITY: Pose a problem that requires regrouping across a zero, such as 203-78. Students work in pairs, using base-ten materials and place-value mats. Once they have identified an answer, they now check their answer, using the algorithm. If they did not get the same answer with the base-ten materials and the algorithm, encourage them to determine why. Follow up with a discussion that starts with students sharing their ideas.

Standard Algorithms for Addition and Subtraction

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• Activity: Picking Your Strategy
• Tell students you are going to show them a problem, but they are not to solve it—instead, they should decide which method from their reasoning strategies and standard algorithms they would choose and why. Projecting a list of possible strategies and thinking tools for addition and subtraction problems will support students with special needs in math and multilingual learners. After students make their selection, call out each method and have them raise their hand to indicate which strategy or thinking tool they chose. Then they use their selected method to solve the problem. When finished, they raise their thumb and hold it to their chest. Share students’ solutions for each strategy used. Then ask the class which method seemed to work best for each problem and why. Use a variety of problems whose numbers lend themselves to different strategies.

Computational Estimation Strategies

Front-End Methods

Rounding Methods

Use this method for adding and subtraction when all or most of the number have the same number of digits. For young student practice using the front digit because it does not require rounding or changing numbers

When adding and subtracting problems involving only two numbers. One strategy is to round only one of the two numbers.

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For example: Round only the subtracted number 6724-1863 becomes 6724-2000 = 4724. If you adjust when subtract a bigger number the results must be too small. The adjust will be 4800.

Compatible Number

Look for two more three compatible numbers that can be grouped to equal benchmark values. Ex: 10, 100, 500. If they can be adjusted to equal amounts it will make finding an estimate easier. In the second imagine if adjusted to make 100, 100, 100 it will make finding an estimate.