Estimation

Order of Magnitude

A clear understanding of the concept of order of magnitude will increase the effectiveness of teaching this unit in the classroom. This level of information is too complex for the early elementary grades but having this background concept knowledge follows the philosophy of truly understanding before teaching. Because students are at different levels, being prepared to further explain this concept is empowering the child to understand more fully.

Orders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of aroundhttp://en.wikipedia.org/wiki/100_%28number%29 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value.

For example, let's say the average height of a human being is about 1.7 meters (about 5'7"). For the sake of simplicity, let's round off 1.7 meters to the nearest power of 10, which is 10 ⁰ m (or 1 m). This does not mean the average height of a person is a mere 1 meter, but rather the average height is closer to 1 meter (or 10 ⁰ meters) than it is to 10 meters (or 10 ¹ meters). Similarly, rounding the height of an ant, which is about 8 x 10 ^{- 4} meters, to the nearest power of ten results in 10 ^{- 3} meters. Another way of saying this is that the order of magnitude of the height of an ant is 10 ^{- 3} meters. Now, if we compare the height of a human being (10 ⁰ meters) with the height of an ant (10 ^{- 3} meters), we come up with the ratio human height/ant height = 10 ⁰/10 ^{- 3} = 10 ^{0 - ( - 3 )} = 10 ³ = 1000. A human being is roughly 1000 times (or 10 ³ times) taller than an ant. In other words, a human being is 3 orders of magnitude (3 powers of 10) taller than an ant.

Our common decimal system is a highly sophisticated method for writing whole numbers efficiently. It uses only 10 symbols (the digits: 0,1,2,3,4,5,6,7,8,9), arranged in carefully structured groups, to express any whole number. Further, it does so with impressive economy. To express the total human population of the world would require only a ten-digit number, needing just a few seconds to write.

This introduction and explanation from Dr. Howe's essay on place value gives an example of the order of magnitude concept. As he clarifies in this essay, the first thing to know about decimal notation is that the base ten expression of a number implicitly breaks the number into a sum of numbers of a very special type.

7,452 = 7,000 + 400 + 50 + 2

This is called expanded form. Although this is not a first grade skill, it is important to build the conceptual knowledge of the order of magnitude. Limiting subject matter knowledge to a narrow scope restricts a teacher's capacity to promote conceptual learning to their students. Knowing the importance of this foundation skill will improve the ability of the students to understand why instead of just how.

The summands in the expanded form of a number the place value components of the number. Each place value component is a digit (possibly 0) times a power of 10:

7,000 = 7 × 1000 = 7 × (10 × 10 × 10) = 7 × 10 ³

400 = 4 × 100 = 4 × (10 × 10) = 4 × 10 ²

50 = 5 × 10 = 5 × 10 ¹

2 = 2 × 1 = 2 × 10 ⁰

A single place number is a digit times a denomination.

The order of magnitude also may be described as the number of zero digits used to write the denomination, or one less than the total number of digits used. By the order of magnitude of a (non-zero) place value number, we mean the order of magnitude of its denomination. Finally, the order of magnitude of a base ten number is defined to be the order of magnitude of its largest (non-zero) place value component. Thus, 7,452 has order of magnitude 3, the same as its largest decimal component, 7000; and the 400, the 50 and the 2 have orders of magnitude 2, 1 and 0, respectively. ⁷

Within this unit of working with two-digit numbers, the important principle to remember is that a two digit number is so many 10s and so many 1s, e.g., 47 is four 10s and seven 1s. Then the adding of the 10s and the 1s should become natural with guidance and discussion. Again logical sequence is important. Before two-digit addition is presented, it is important to establish the "make a ten" principle. Using the model of Singapore math, addition facts are not just a list to memorize, but an opportunity to begin understanding the structure of the number system. The emphasis is placed on the process of forming ten ones into a 10 when the sum of the digits is greater than 10, and of decomposing a 10 back into ones in the corresponding subtraction problems.

And as stated earlier, subtraction should be linked with addition until the students think of it as undoing addition. Keeping in mind that fact families (or in this curriculum, number bonds) are not limited to one-digit numbers. This process of linking subtraction problems to addition problems helps the students to make sense of subtraction.