Chapter 12

Developing Strategies for Addition and Subtraction Computation

 

Big Ideas

1.        Flexible methods of addition and subtraction computation involve taking apart and combining numbers in a wide variety of ways. Most of the decomposing of numbers is based on place value or “compatible” numbers- number pairs that work easily together, such as 25 and 75.

2.        “Invented’ strategies are flexible methods of computing that vary with the numbers and the situation. Successful use of the strategies requires that they be understood by one who is using them- hence the term invented.

3.        Flexible methods for computation require a strong understanding of the operations and properties of the operations, especially the commutative property and the associative property. How addition and subtraction are related as inverse operations is also an important ingredient.

4.        The standard algorithms are elegant strategies for computing that have been developed over time. Each is based on performing the operation on one place value at a time with transitions to an adjacent position. Standard algorithms tend to make us think in terms of digits rather than the composite number that the digits make up. These algorithms work for all numbers but are often not the most efficient of useful methods of computing.

5.        Multidigit numbers can be built up or taken apart in a variety of ways. When the parts of numbers are easier to work with, these parts can be used to estimate answers in calculations rather than using the exact numbers involved. For example, 36 is 30 and 6 or 25 and 10 and 1. Also, 483 can be thought of us as 500-20+3.

6.        Nearly all computational estimations involve using easier-to-handle parts of numbers or substituting difficult-to-handle numbers with close “compatible” numbers so that the resulting computations can be done mentally.

Benefits of Student-Invented Strategies

·         Students make fewer errors

·         Less reteaching is required

·         Students develop number sense

·         Invented strategies are the basis for mental computation and estimation

·         Flexible methods are often faster than standard algorithms

·         Algorithm invention is itself a significantly important process of “doing mathematics”

Creating an Environment for Inventing Strategies

·         Avoid immediately identifying the right answer when a student states it. Give other students a chance to consider whether they think it is correct.

·         Expect and encourage student-to-student interactions, questions, discussions, and conjectures.

·         Encourage students to clarify previous knowledge and make attempts to construct new ideas.

·         Promote curiosity and openness to trying new things.

·         Talk about both right and wrong ideas in a nonevaluative or nonthreatening way.

·         Move unsophisticated ideas to more sophisticated thinking through coaxing, coaching, and strategic questioning.

·         Use familiar contexts and story problems to build background and connect to students’ experiences.

Chapter 11

Developing Whole-Number Place-Value Concepts

 

Role of Counting

1.        Counting by ones.

2.        Counting by groups and singles.

3.        Counting by tens and ones.

Developing Base-Ten Concepts

·         Grouping Activities

·         The Strangeness of Ones, Tens, & Hundreds

·         Grouping Tens to Make 100

·         Equivalent Representations

Oral and Written Names for Numbers

·         Two-Digit Number Names

·         Three-Digit Number Names

·         Written Symbols

·         Assessing Place-Value Concepts

Patterns and Relationships with Multidigit Numbers

·         The Hundreds Chart

·         Relationship with Landmark Numbers

·         Connecting Place Value to Addition and Subtraction

·         Connections to Real-World Ideas

Big Ideas:

1.        Sets of 10 (and tens of tens) can be perceived as single entities or units. For example, three sets of 10 and two singles is a base-ten method of describing 32 single objects. This is the major principle of base-ten numerations.

2.        The positon of digits in numbers determine what they represent and which size group they count. This is the major organization for developing number sense.

3.        There are patterns to the way that numbers are formed. For example, each decade has a symbolic pattern reflective of the 0-to-9 sequence

4.        The groupings of ones, tens, and hundreds can be taken apart in different but equivalent ways. For example, beyond the typical way to decompose 256 of 2 hundreds, 5 tens, and 6 ones, if can be represented as 1 hundred, 14 tens, and 16 ones but also 250 and 6. Decomposing and composing multidigit numbers in flexible ways is a necessary foundation for computational estimation and exact computation.

5.        “Really big” numbers are best understood in terms of familiar real-world references. I tis difficult to conceptualize quantities as large as 1000 or more. However, the number of people who will fill the local sport arena is, for example, a meaningful referent for those who have experienced that crowd.