Decimals

As with whole numbers, the most advanced level of mastery of decimals is to be able to complete word problems that use decimals and a variety of operations.  Students can practice with word problems with just one operation.  Then, students who can move through word problems that involve several operations (in different combinations),you can feel confident that they fully understand decimals and how to work with them. 

When dividing by a decimal, the easiest way to deal with it is to get the decimal out of the divisor.  In math, the rule is, you can do anything to a number, as long as you do the same to every other number in the problem or equation.  So, you multiply the divisor by 10 (or 100 or 1000) or whatever you need in order to push the decimal all the way to the right.  Then you multiply the dividend by the same amount to keep the problem balanced.  

To divide with decimals, students must be good at long division, and careful about lining up place values when doing long division.  When using long division to divide decimals (with decimals in the dividend, but not the divisor) all students have to do is bring the decimal up from the dividend and into the answer row.  If the student lines up answer digits correctly, the decimal will land in the correct place in the answer!

Multiplying with decimals is almost exactly like multiplying whole numbers -- you just have to pay attention to the decimal point.  And, when you multiply numbers with decimals, after you multiply, you count the number of digits behind the decimal point in each factor, add those numbers together, and move the decimal point in the answer that many digits to the left. 

The process is very simple -- the reasons behind it are pretty simple too, although we don't always think through the process while we're doing it:

The process of subtracting numbers with decimals is exactly the same as the process of subtracting whole numbers.  The only difference is in the set-up.

And, no matter how many decimal places you have, it's the same process. With longer numbers, just be careful to line up decimals and fill in empty place values with zeros so you don't forget to borrow!

No matter what size numbers you're working with, if even one number has a decimal, you have to line the numbers up by their decimal points.  

The process of subtracting numbers with decimals is exactly the same as the process of subtracting whole numbers. The only difference is in the set-up.

When working with whole numbers, the right-most digit of every number you deal with is a ones digit, so as long as you line up numbers along the right side, all of the place values line up.

The one rule of adding with decimals applies no matter how many decimal places you're working with!

When you add with decimals, you must line up the numbers by the decimals before you add. 

If you don't line the numbers up by the decimals, you'll be adding numbers of different place values together. We all know that if you add 3 dollars and 2 dimes, you don't end up with 5 dollars.  You end up with $3.20.  If you line numbers up by the decimal, you avoid those kinds of mistakes. 

The process of adding numbers with decimals is exactly the same as the process of adding whole numbers. The only difference is in the set-up.

When working with whole numbers, the far right digit of every number you deal with is a ones digit, so as long as you line up numbers along the right side, all of the place values line up. 

Once you know place value, it’s easy to compare numbers with and without decimals.  Remember, the most important digit in any number is the digit farthest to the left. (Think about it, if you won the lottery and they offered you either \$1.9 million or \$4.1 million, you'd take the \$4.1 even though the .9 is higher than the .1, the 4 is the more important digit in that number!). So, whenever you want to compare decimals, line your numbers up along the decimal (as if you were adding up money), and compare from left to right.  

A decimal is a period in a number that separates whole numbers from fractions of numbers.  In the number $3.21$, $3$ represents 3 wholes (the portion of the number on the left side of the decimal represents whole numbers) and $.21$ represents 21 hundredths (the numbers on the right side of the decimals represent a portion of a whole).