Background
Place Value
The place value system is decimal notation. This allows a digit (i.e. one of the numbers 0,1,2,3,4,5,6,7,8, or 9) to positionally indicate powers of ten involved in the "base ten" decomposition of a number determined by its location. To illustrate with a simple example, in this system the notation "123" represents the number "one hundred twenty three". In this system, a particular place value is ten times the value of the place to its right, but only one-tenth the value of that to its left. The system's beauty of compacting numbers allows for reasonable representations of extremely large numbers in a space that is workable, but leads to misconceptions of actual values in the representation. The close proximity of one place directly next to the place to its right suggests relative closeness in value when in actuality it is ten times the value. As an example, consider the digit 7 located in the "ones" place (the rightmost place). Moving this digit 7 one place value to the left into the "tens" place gives a number with value of 70, or 7 x10, or 7 tens, or ten times greater than 7. The next place to the left is the "hundreds" place, and moving the digit 7 there creates a number with a value 700, or 7 x 10 x10, or 70 x 10. 700 is ten 70s or is 100 times the original 7 in the ones place. That is, these numbers look similar, but are far from each other in value.
The values 700, 70, and 7 are examples of related values in the base ten system. The genius of this system creates confusion of the system. My students can name place values and show the value of a digit in a place, yet they lack the understanding of the relationship between the values after the tens place and struggle with how the representation of digits is a combination of values. Students are often confused by the values represented in the compacted form of the number. For example, if a student is asked what numbers are represented in 123 the response is often one, two and three. The correct response is 100 + 20 + 3. The error here is due to lack of a deeper understanding of place value. A student providing the correct answer indicates that they have deep understanding of place value and positional notation.
How can we overcome misunderstandings in place value? In Roger Howe and Susanna S. Epp's essay "Taking Place Value Seriously: Arithmetic, Estimation, and Algebra" 2 the authors support explicit instruction of place value to students to create a more unified and conceptual understanding of the place value system with emphasis on the structure and systematic organization. The place value system is a complex one in an efficient condensed form. The idea to take the compressed number structure (i.e. 123) and intentionally unpack the content (i.e. one hundred plus twenty plus three) emphasizes the place value concepts. This expanded notation not only allows for an increase in conceptual understanding, but also allows for more computational flexibility due to deeper understanding. Students have computational flexibility when they can chose an algorithm or strategies that works best for the information presented. This unit will provide examples of different algorithms for addition, subtraction and multiplication.
Expanded form
The compacted number becomes more representative of value when shown unpacked in a variety of expanded form notation. If the number 7,892 is presented in compact form, its true value can be confusing for students. To clearly show students the parts of a compacted number, expanding the form shows each digit is connected to place value. For example 7,892 becomes 7,000 + 800 + 90 + 2. This form allows each number to be shown at its actual value. Now students can see the 8 in 7,892 as the 800 it represents. In another slightly more advanced expanded form, the number 7,892 is (7 x 1,000) + (8 x 100) + (9 x 10) + (2 x10). In this form, the digit is now the single place digit multiplied by the place value. In this form the place value is separated from the digit to show the number of 1,000s, 100s, and so on. The 8 again now shows 8 hundreds or 8 x 100. A third and even slightly more advanced expanded form additionally depicts increasing powers of ten. For example, 7,892 is 7 x (10 x 10 x 10) + 8 x (10 x 10) + 9 x (10) + 2 x (1). This form shows the single place digit, the place value and also how place value is constructed using powers of ten. Students see the 800 both as total value, single digit times the place value, and as a relationship between place values. The extraction of the single digit defines the digit and how it relates to positional place value. Expanded form explicitly shows digits to the left will always represent greater values than any earlier values because its place alone is worth more. The complexity of the number is better represented with expanded form notation.
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