Dividing decimals

I feel students need to get grounded conceptually in this topic. So many times, all they learn about decimal division are the rules of how to go about decimal division when using long division, and it becomes an "empty" skill - a skill that lacks the conceptual foundation.

So for starters, we can do two different kinds of mental math division problems.

1. Division by a whole number - using mental math
   
   Here it is easy to think, "So much is divided between so many persons".
   
   0.9 ÷ 3 is like "You have nine tenths and you divide it between three people. How much does each one get?" The answer is quite easy; each person "gets" 0.3 or three tenths.
   
   And... remember ALWAYS that you can check division problems by multiplication. Since 3 × 0.3 = 0.9, we know the answer was right.
   
   0.4 ÷ 100 turns out to be an easy problem if you write 0.4 as 0.400:
   0.400 ÷ 100 is like "You have 400 thousandths and you divide it between 100 people; how much does each one get?" The answer is of course 4 thousandths, or 0.004. Check: 100 × 0.004 which is 100 × 4/1000 = 400/1000 or 0.400 = 0.4.
   
   Here are some more similar ones:
   
   0.27 ÷ 9
   
   0.505 ÷ 5
   
   0.99 ÷ 11
   ...and you can make more, just think of the multiplication tables.
   



2. Division where the quotient (answer) is a whole number
   
   This time it helps to think, "How many times does the divisor go into the dividend?" In these types of mental math problems, the answer ends up being a whole number. (Of course the teacher has to plan these problems just right.)
   
   For example, 0.4 ÷ 0.2. Ask, "How many times does 0.2 fit into 0.4?" The answer is, 2 times. So 0.4 ÷ 0.2 = 2. Again, we can check it by multiplying: 2 × 0.2 = 0.4.
   
   Other similar division problems to solve mentally:
   
   1 ÷ 0.5
   
   3 ÷ 0.5
   
   0.09 ÷ 0.03
   
   0.9 ÷ 0.1
   
   2 ÷ 0.4
   
   1 ÷ 0.01
   
   ...and so on.
   

This decimal division lesson taken from my Decimals 2 book illustrates these two kinds of mental division problems.

Towards the general case

After the student is familiar with the two special cases above, we can go forward and study decimal division problems in general. Even here, we will divide the problems into two classes, depending on whether the divisor is a whole number or not.

1. The divisor is a whole number.
   
   For example, 3.589 ÷ 4 or 0.1938 ÷ 83. These can simply be solved by long division as they are. Just put the decimal point in the same place in the quotient as where it is in the dividend.
   
   The "stumbling block" may come when the division is not even (this also leads into the study of repeating decimals). Generally, you can continue the division indefinitely by tagging zeros to the dividend, such as making 3.589 to be 3.589000. Then when you've continued the division as long as you wish (or as long as the book tells you to do it), cut the decimal off at a desired accuracy and round it.
   
   Typical problem in a textbook would say, "Do 2.494 ÷ 3 and give your answer with 3 decimal digits." For this, you need to do the long division until the fourth decimal digit - so as to be able to round to 3 decimal digits. Since 2.494 does not have four decimal digits, you tag a zero to it to make it have so (2.4940).
   
   Fortunately, this process is not generally difficult. It's the second case that's more of a problem.
   
   



2. The divisor is not a whole number.
   
   Here, we do something quite special before dividing, and turn the problem into one where the divisor is a whole number. Then, the actual division is done like explained above.
   
   I say this is special, because this special thing that we do is based on a very important general principle of arithmetic:
   
   If you multiply both the dividend and the divisor by some same number, the quotient won't change.
   
   Let's see it in action with some easy numbers:
   
   1000 ÷ 200 = 5
   
   100 ÷ 20 = 5
   
   10 ÷ 2 = 5
   
   Each time both the dividend and the divisor change by a factor of ten, but the quotient does not change.
   
   We can also try it using a factor of 3 (or any other number):
   
   8 ÷ 2 = 4
   24 ÷ 6 = 4
   72 ÷ 18 = 4
   
   Let's try one more time, with a factor of 2:
   
   30 ÷ 6 = 5
   
   15 ÷ 3 = 5
   
   7.5 ÷ 1.5 = 5
   
   3.75 ÷ 0.75 = 5
   
   H hopefully by now you have convinced the student(s) of this principle. Now we can apply it to those pesky decimal division problems.
   
   
   
   
   This image shows how the decimal division problem 0.644 ÷ 0.023 can be changed into another problem, with a whole number divisor, and with the same answer.
   
   In each step, we multiply both the dividend and the divisor by 10. This, of course, is the same process as moving the decimal point.
   
   Many textbooks only show the student the "trick" of moving the decimal point... but don't show him what that idea is based on.
   
   An example
   
   To solve 13.29 ÷ 5.19, we need to first change the problem so that the divisor 5.19 is a whole number. We multiply both the dividend and the divisor by 10 as many times as needful to accomplish that:
   
   13.29 ÷ 5.19
   = 132.9 ÷ 51.9
   = 1329 ÷ 519, and now off you go to do long division... I'm not saying it's the easiest long division problem in the world, since the divisor is 519. Let's try an easier one.
   
   
   2,916 ÷ 0.02
   = 29,160 ÷ 0.2
   = 291,600 ÷ 2 and now you can do the long division.
   
   Of course, in reality you can also multiply by 100 instead of taking two steps of multiplying by 10. But students can start out by multiplying by 10 as many times as needed.
   

Please also see the lesson on dividing decimals by decimals, from my Math Mammoth Decimals 2 book.

Off-Road Algebra

Here's an algebra resource that should interest (at least some) boys: Off-Road Algebra is a unit-study that revolves around the world of off-road motorcycle racing. The lessons fit pre-algebra and algebra, or approximately 9th grade.

For each of the 30 lessons, you view a video, then solve a problem. You'll also get printable explanations and solutions to all problems.

Glancing over the problems, they seem to cover a wide array of topics, such as miles per gallon, velocity, slope of ramps, GPS coordinates, decibels, acceleration, turn angles, lap times, and so on.

Off-Road Algebra

And it's all free (sponsored by Learning.com and Aha!Math) - so thanks, HotChalk and Learning.com!

Decimal multiplication

This is a tough topic... in a sense. It is not difficult at all, if you just follow the rule given in your math textbook, because the rule is pretty straightforward:

• To multiply decimal numbers, multiply them as if there were no decimal points, and then put as many decimal digits in the answer as there are total in the factors.

The difficulty is only if you try to understand why we have such a rule - where does it come from?

Understanding the rule for decimal multiplication is actually fairly simple, because it comes from fraction multiplication. But, I will propose here a little different way of explaining all this.

First, look over this decimal multiplication lesson that is taken from Math Mammoth Decimals 2 book.

It talks about how 0.4 × 45 is like taking 4/10 part of 45. The same applies if you have 0.4 × 0.9 - you can think of it as taking 4/10 part of 0.9.

Can you see now why the answer to 0.4 × 0.9 has to be smaller than 0.9?

Or, turn it around: 0.9 × 0.4 is taking 9/10 of 0.4, and so the answer has to be smaller than 0.4 (slightly smaller).

Thinking this way, it shouldn't be a big surprise that 0.9 × 0.4 equals 0.36. (The student needs to have a solid grasp of decimal place value prior to this so he can immediately see that 0.36 is smaller than 0.4.)

Now, once your student is comfortable with this idea (as explained in the lesson), then you can proceed on with the explanation based on fraction multiplication. See, we're taking it one step at a time!


Comparing fraction multiplication and decimal multiplication
(I have not yet written a lesson about this for my books, but will do so for the Light Blue 5-B.)

Remember, decimals are fractions.

Let's take an easy example first.

0.5 × 0.7 is solved with fractions like this:

(5/10) × (7/10) = 35/100 = 0.35

Notice the denominators 10 and 10 got multiplied to produce the denominator 100 for the answer, and so the answer written as a decimal has two decimal digits.

Another example:

0.384 × 2.91

= (384/1000) × (291/100)

= (384 × 291) / (1000 × 100)

= 111744 / 100000

= 1.11744

The denominators 1000 and 100 have as many zeros as as you have decimal digits in the number. The denominator of the answer is 100,000 — with 5 zeros — so the answer as a decimal has five decimal digits.

One more time:

0.45 × 1.3

= (45/100) × (13/10)

= (45 × 13) / (100 × 10)

= 585 / 1000

= 0.585

So... when you write decimals as fractions, the denominators are powers of ten that have as many zeros as there are decimal digits in the decimal number. When you multiply, those denominators get multiplied, and you get another power of ten that has as many zeros as there were in the factors. That, in turn, translates being a decimal number with as many decimal digits as there were decimal digits in the factors.

(In case you don't know: powers of ten are the numbers 10¹, 10², 10³, 10⁴, 10⁵, and so on. Written without the exponential notation these are 10; 100; 1000; 10,000; 100,000; and so on.)