Estimation

Background

Terminology

Digit: A digit is a number (0,1,2,3,4,5,6,7,8,9) that can be used alone or in combination with other digits to communicate a value.

Place Value Notation/ Decimal Notation/ Base-10 Notation: Our way of writing numbers is a complex system based on groups of ten. The order in which we arrange digits determines the value the digit represents. We have many "places" and each place has it's own value, hence "place value." A digit placed directly to the left of the decimal point stands for ones, the next place to the left denotes groups of ten, the next place to the left from that denotes groups of hundreds, and so on. We can represent an infinite number of possible values using just ten digits because of our notation system. We don't have to write 200 + 30 + 9, we can just write 239 and the previous values are assumed. This allows us to write large numbers very compactly.

Place Value Components/ Very Round Numbers: This refers to the value of each digit in a multi-digit number when it is considered on its own. It is a digit times a power of 10. For example, the number 3,461 has four place value components. They include 3,000, 400, 60 and 1. ³ A very round number is a number with only one non-zero digit, e.g., 3000, 400, 60 and 1. Our base-ten place value system expresses every whole number as a sum of very round numbers. For example, the number 3,461 is the sum 3000 + 400 + 60 + 1. When very round numbers are combined like this to make some number, they are called the place value components, or very round components, of that number.

Order of Magnitude: This term refers to the number of zeros used to write a very round number. ⁴ For an arbitrary whole number, the order of magnitude is one less than the number of digits.

Composing/Decomposing Numbers

Chinese equivalents for the terms "compose" and "decompose" are prevalent in Chinese mathematics learning. ⁵ However, these are relatively unfamiliar terms to early childhood math teachers in the United States. ⁶ If we begin by focusing on what these words mean, we will help ourselves and our students understand how the terms relate to mathematics. Compose is a verb that means "to make or form by combining things, parts, or elements." ⁷ In mathematics, when we compose a number, we are making a number by combining two or more smaller values. Essentially, we are looking for ways to group the possible parts of a number. Decompose is a verb that means "to separate or resolve into constituent parts or elements." ⁸ In mathematics, when we decompose a number, we are pulling that number apart into smaller values. There may be a variety of ways to compose or decompose a number, depending on how many digits it contains. We can compose 3 in a few ways. (1 + 1 + 1) or (2 + 1) or (1 + 2) or (0 + 3) are all ways of making 3. We can decompose 9 in a numerous ways. (There are 30!) Some examples are: 9 can be broken into (2 and 7), (5 and 4), (1 and 8) or (2 + 3 + 4). This concept is integral to early childhood mathematics education, and I believe we should introduce not only these terms, but also these concepts, with young children as they begin to build concepts of number. We can use "make" or "combine" as synonyms for compose, and we can use "unmake" or "break apart" as other ways of explaining "decompose," but we should use these terms interchangeably.

Expanded Form

Expanded form expresses a number as the sum of its place value components. Expanded form helps us visualize a number by looking at its place value components separately. Each digit is considered on its own, as a separate value. When we consider each digit separately, its order of magnitude becomes more apparent. Then we see that none of the digits can represent the same order of magnitude. ⁹ Examples of expanded form:

If we take the time to explicitly teach children to break apart numbers in this way, they will be able to utilize this strategy when doing operations. The study of numbers in expanded form should be introduced after students have broken numbers into tens and ones first. They need to be guided to see each digit as a certain number of tens and ones (or thousands and hundreds) first. Then you can begin to use expanded form.

Order of Magnitude

Once you understand expanded notation of multi-digit numbers, you can begin to explore the concept of order of magnitude. When you break a given number into expanded form, you can see each digit represents a different power of ten. We can refer to the pieces of a number shown in expanded form (these multiples of 10) as "place value components." None of the place value components can represent the same place value. I can write 1,111 in expanded form like this, 1,000 + 100 + 10 + 1, and we will see that it has 4 place value components. Then we can further analyze these components.

This chart helps us begin to name each order of magnitude. 1,000 is order of magnitude 3 because it is 1 x 10 ³. Any 4-digit number is order of magnitude 3, which includes numbers from 1,000-9,999. 100 is order of magnitude 2 because it is 1 x 10 ². Any 3-digit number from 100-999 is order of magnitude 2. 10 is order of magnitude 1 because it's 1 x 10 ¹. Any 2-digit number, 10-99, is order of magnitude 1. 1 is order of magnitude 0 because it is 1x 10 ⁰. Any other 1-digit number is also order of magnitude 0.

The chart also helps us to see that with each successive digit to the left, the value of the digit is ten times bigger. 10 is ten times bigger than 1. 100 is ten times bigger than 10. 1,000 is ten times bigger than 100. If you want a deeper understanding of the implications of order of magnitude in estimation, I would point you to pages 27-30 in R. Howe's Taking Place Value Seriously. ¹⁰

Understanding order of magnitude helps us more accurately conceive the difference between very large numbers that can often be abstract. It is hard for us to know just how significant the difference is between 1 million and 1 billion. However, if we begin to analyze the decimal notation system in term of order of magnitude, we should at least know that 1 billion is a thousand times bigger than 1 million: ten groups of a million is ten million; ten groups of ten million is one hundred million; and ten groups of one hundred million make a billion. In all, this is 1,000 groups of a million. Then you begin to realize that that is a huge difference, and it is this realization that can help us more accurately estimate.

Second graders do not need to have an advanced understanding of order of magnitude. We needn't try to explain the role of exponents in noting order of magnitude. However, we do want to build a foundation by emphasizing that 10 is ten times bigger than one and 100 is ten times bigger than 10. This foundation will help students grasp the significance between each of the places in our decimal notation system. It will also help them be able to estimate more readily by giving them experience with the difference in size between values that are different order of magnitude.