237 Chapter 34: Multiplying and Dividing by Decimals by Powers of ...

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The famous short-cut for multiplying and dividing decimals by powers of ten ... First of all, a power of ten is what we get when we multiply the number 1 by some.
Chapter 34: Multiplying and Dividing by Decimals by Powers of Ten The famous short-cut for multiplying and dividing decimals by powers of ten can be easily explained using base-ten column notation. First of all, a power of ten is what we get when we multiply the number 1 by some number of tens. This could be 1 itself, or 1 × 10 = 10, or 1 × 10 × 10 = 100, or 1000, or so on. The reason these are called powers is because they can be easily represented with exponential powers. For example, one-thousand is called "ten to the third power" because 1000 = 10 × 10 × 10 = 103. Similarly, when we divide 1 by a number of tens, we get a negative power of 10. For 1 example, 1 y 10 y 10 10  2 . The pattern of exponents can make a lot of sense when 100 illustrated on base-ten number columns:

This is only to explain the use of the phrase "powers of ten" and its connection to the base-ten numbering system. Children will not need to concern themselves with exponents, especially negative exponents, until they learn algebra in middle or high school. For the rest of this chapter, we will use "a power of ten" to only mean a positive power of ten, a number of tens multiplied together. We saw earlier that when a whole number is multiplied by 10, each of its digits moves to the left. This is because ones become tens, tens become hundreds, and so on when multiplied by 10. For example, a number like 504:

represents 500 + 4. When multiplied by 10, this becomes (500 + 4) × 10 = 500 × 10 + 4 × 10 = 5000 + 40. It is as if each of the digits moved to the left one space.

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The short-cut we discussed earlier for this was called "add a zero to the end of the number." For example, 67 × 10 = 670, as if a single zero were just tacked onto the end of the 67. The extended short-cut was called "add a zero at the end for every zero in the power of ten." When we multiply by 1000, for example, we are really multiplying by 1000 = 10 × 10 × 10. Each of these tens "adds a zero," so multiplying by 1000 involves "adding three zeros to the end," just as the number 1000 itself has 3 zeros. For example, 67 × 10,000 = 670000. In general, it is a good idea to add all the zeros first, and then worry putting in a comma after every third digit: 67 × 10,000 = 670,000. With decimal numbers, we have to go back to the "shift all the digits over" concept rather than the "add zeros to the end." For example, let us multiply 54.29 × 10. We begin by representing 54.29 on a base-ten number chart:

2 9 + . When we multiply by 10, each of these parts must 10 100 2 + be multiplied by 10 (because of the distributive property). Thus 54.29 × 10 = (50 + 4 + 10 9 2 9 9 ) = 50 × 10 + 4 × 10 + × 10 + × 10 = 500 + 40 + 2 + : 100 10 100 10

This means that 54.29 = 50 + 4 +

Just as before, this number consists of the exact same digits as before, each shifted one column to the left. Without the number columns, however, things look a bit different. When we multiply 54.29 × 10 = 542.9, it does not look as though anything has moved to the left. Instead, it looks as though the decimal point has moved to the right:

This, of course, is the exact same thing.

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Similarly, when we multiply by a larger power of ten, we have as many shifts as there are zeros in the power of ten. For example, look at 3.125 × 100. The number 3.125 looks like:

This represents 3.125 = 3 +

1 2 5 + + . 10 100 1000

It follows by distribution that 3.125 × 100 = (3 +

1 2 5 + + ) × 100 10 100 1000

1 2 5 × 100 + × 100 + × 100 10 100 1000 5 . This looks like: = 300 + 10 + 2 + 10

= 3 × 100 +

As before, we can look at this as each digit moving over two columns to the left. We can also look at 3.125 × 100 = 312.5 and imagine that the decimal point has moved two spaces to the right:

In either case, the "two columns" or "two spaces" comes from the fact that 100 is formed by two 10's being multiplied together (100 = 10 × 10), thus multiplying by 100 is the same as multiplying by 10 twice. Similarly, multiplying by 1000 is the same as multiplying by 10 three times because 1000 = 10 × 10 × 10. Thus, for example, 128.91065 × 1000 = 128,910.65. We'll show this with the "moving decimal point" notation because it takes less space. Of course, there is no need for a child to write out the number three times. The "moving decimal" can be drawn right on the starting number. It would be nice, however, for the final answer to be written out separately.

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As usual, the most kind and educational thing you can do is to not point out the "moving decimal point" trick at first. Instead, do a number of problems out with base-ten column notation and distribution. Hopefully, one of your students will notice the pattern and announce it to the rest of the class. In this manner, the student will get all the honor and glory associated with "making math easy for everyone." As always, this trick will be named after the student, both to praise that person and to help the others recall (ask, for example: "remember Samantha's trick for these problems?"). In an ideal class, the students keep their eyes peeled for tricks, patterns, and short-cuts. This is so much more fun and educational than telling the students what to do and penalizing them for deviations. If no one in the class notices the short-cut, however, you should slowly prompt them toward it, and ultimately just tell them. One useful trick is to line up a whole series of examples, to focus the class on pattern recognition. For example, show them: 3.17 × 10 = 31.7 23.905 × 10 = 239.05 282.096 × 10 = 2820.96 3.17 × 100 = 317 23.905 × 100 = 2,390.5 282.096 × 100 = 28,209.6 etc. Be sure to include examples where you multiply by more tens than you have decimal places. For example, look at 29.8 × 1000. The 29.8 has only one decimal place, but it is being multiplied by 1000 = 10 × 10 × 10. With base-ten columns, this involves moving each digit to the left three times:

This number is 20,000 + 9,000 + 800, which can only be written with two more zeros at the end: 29,800. With the "moving the decimal place" notation, we use low, swinging, curvy steps to move the decimal place in order to suggest places where zeros ought to go:

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Because division is the inverse operation to multiplication, the process of dividing by a power of ten is the exact opposite of multiplying. Rather than moving each digit to the left in base-ten columns, the digits move to the right. Rather than the decimal point moving to the right, it moves to the left. The number of spaces moved is still the number of zeros in the power of ten. For example, to divide 38.97 ÷ 10, we start with 38.97:

9 7 + , when we divide, we have to divide each part separately: 10 100 9 7 38.97 ÷ 10 = (30 + 8 + + ) ÷ 10 10 100 9 7 ÷ 10 + ÷ 10 = 30 ÷ 10 + 8 ÷ 10 + 10 100 8 9 7 =3+ + + = 3.897 10 100 1000

Because 38.97 = 30 + 8 +

We can either imagine that each digit has moved to the right one column or that the decimal point has moved to the left one space:

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As before, we can run into some trouble if the decimal point runs out past the digits. For example, consider 8.125 ÷ 1000. If we try to move the decimal point to the left three places, we will run out of number:

The trick, as before, is to put another zero above the crook of each step of the decimal place's movement:

This can be verified by distributing the division of the 1000. To evaluate 8.125 ÷ 1000, 1 2 5 we split up the 8.125 into 8 + + + and divide each part by 1000: 10 100 1000 1 2 5 8 ÷ 1000 + ÷ 1000 + ÷ 1000 + ÷ 1000 10 100 1000 8 1 2 5 = + + + = .008125. 1000 10000 100000 1000000 We could also write 8.125 with base-ten columns, then move each digit to the left three spaces:

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Questions: (1) Show with factoring and cancellation why

(2) Show that

3 y 10 100

1 u 10 100

1 . 10

3 by treating the problem as one of dividing fractions. 1000

(3) Show 98.164 × 10 using (a) the distribution of the 10. (b) base-ten column notation. (c) the "move the decimal point" short-cut. (4) Show 18.7 ÷ 1000 using (a) the distribution of the ÷ 1000 (b) base-ten column notation (c) the "move the decimal place" short-cut

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