I never liked the way division was taught to me in grade school. Basically, it went like this. In the third grade, they taught division as simply the reverse of multiplication. That by itself was fine, but they would include problems that didn't come out evenly BEFORE teaching fractions and decimals. So what did they do? They taught "remainders," and the answers would be written as the whole number, followed by the letter R, followed by the remainder.
Having a remainder is fine when dealing with word problems involving units that cannot be divided into fractions (such as salable units or people), but not when dealing with pure math. It just made sense to me to each the use of fractions and decimals BEFORE introducing problems that don't come out evenly.
Here's an example. Let's say you have three money problems:
1) Divide five dollars between two people.
2) Divide nine dollars between four people.
3) Divide seven dollars between three people.
If we were to teach the "remainder" solution, all three questions would have the same answer: 2 R 1:
5 ÷ 2 = 2 R 1
9 ÷ 4 = 2 R 1
7 ÷ 3 = 2 R 1
However, if we write all three problems as fractions and then solve without using remainders, then we get three different answers:
5 / 2 = 2.5
9 / 4 = 2.25
7 / 3 = 2.33333333...
Of course, since we're talking about a money problem, there are indivisible units involved, so a remainder is necessary in the third problem, but not the first two. So all three solutions would look like this:
1) $5 divided by 2 people equals $2.50 per person.
2) $9 divided by 4 people equals $2.25 per person.
3) $7 divided by 3 people equals $2.33 per person with a penny left over.
Also, I think it's time to retire the division sign (÷). Just write all the problems in algebraic form, with the dividend in the numerator and the divisor in the denominator. And while we're at it, let's reinforce the idea that subtraction is just addition using a negative number. It's WAY easier than that whole "PEDMAS" left-to-right bullshit.
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