Number & Operations for Teachers

Copyright David & Cynthia Thomas, 2009

Modeling Addition and Subtraction with Regrouping

It is one thing to understand the concept of addition and subtraction of whole numbers and integers.  Keeping track of the complexities of these operations when working with large numbers is a different matter.  For that, we need a concept model to illustrate the place value system of numeration and an expanded algorithm to keep track of details.  Figure 2.8 shows two concept models for the operation 109 + 211 = 320.  Both models are based on base ten blocks.  One model is very literal, showing in detail the individual ones in the tens and hundreds graphics.  The other is more symbolic, without proportional scaling between the graphical icons representing ones, tens, and hundreds.  Note that in each representation, 10 ones are regrouped as 1 ten.

3(100)                                2(10)     0(1)

 3(100)

 2(10)

 0(1)

 Hundreds

 Tens

 Ones

Figure 2.8: Two Representations of a Base Ten Block Model for 109 + 211 = 320

Figure 2.9 shows pairs the second of these concept models with an expanded algorithm for the same operation.  The graphical and numerical representations both contain essentially the same information.  In discussions based on the Concept – Written Notations– Spoken Language model shown in Figure 1.3, students should develop fluency in representing and explaining arithmetic operations with whole numbers.  This is the first step in developing a genuine understanding of the logical bases for arithmetic operations.

 3(100)

 2(10)

 0(1)

 Hundreds

 Tens

 Ones

Concept Model: Base Ten Blocks

Algorithm: The expanded algorithm is a numerical representation that helps to clarify the meaning of place value.

1(10)

1(100) + 0(10) +  9(1)

+ 2(100) + 1(10) +  1(1)

3(100) + 2(10) +10(1)

3(100) + 2(10) + 0(1)

Written in standard form, the answer as 320

Figure 2.9: Concept Model and Expanded Algorithm for 109 + 211 = 320

Figure 2.10 demonstrates the use of this approach in the context of a related subtraction problem involving regrouping, 320 – 109 = 211.

 2(100)

 1(10)

 1(1)

 Hundreds

 Tens

 Ones

Concept Model: Base Ten Blocks

Algorithm: The expanded algorithm is a numerical representation that helps to keep place value.

1(10)    10(1)

3(100) + 2(10) +  0(1)

1(100) 0(10) 9(1)

2(100) + 1(10) +  1(1)

Written in standard form, the answer as 211

Figure 2.10: Concept Model and Expanded Algorithm for 320 – 109 = 211

Figure 2.11 shows the “standard” algorithms associated with the operations modeled in Figures 2.9 and 2.10.  These notations provide little conceptual support for learners.  They were designed as procedural notations for use by people who already understand the conceptual basis for computation but who must also become reliable and fast in routine computation.  Unfortunately for young learners, the very brevity of these notations is often a source of genuine confusion.  For instance, in the process of “borrowing”, a 1 is written above the ones and tens columns of the subtraction problem.  These 1’s are identical in appearance but represent different quantities.  Furthermore, the oral language used to describe arithmetic operations is often misleading.  Referring to the addition problem in Figure 2.11, a well-meaning adult might say something like “Nine plus one equals ten … write the zero … carry the one … one plus one equals two ...”, obscuring the fact that 10 ones are regrouped into 1 ten and that ten is relocated in the tens column.  Statements of this sort hide the meaning of regrouping.

 Addition with regrouping: 109 + 211 = 320         1    1 0 9 + 2 1 1    3 2 0 Subtraction with regrouping: 320 – 109 = 211                                           1    1    3 2 0 - 1 0 9    2 1 1

Figure 2.11: Standard Algorithms for Addition & Subtraction with Regrouping

Developing proficiency in the use of standard algorithms is a goal of elementary mathematics education.  But the process by which students acquire this skill should have as its foundation a deep understanding of the meaning of arithmetic algorithms.  The use of concept models and expanded algorithms is a powerful tool in the development of this foundation.

 Tech Resources Investigate  Base Blocks Addition and Base Blocks Subtraction at the National Library of Virtual Manipulatives for Interactive Mathematics