( with a division or long division symbol this expression is read “12 divided by 3 equals 4.” Notice here, though, that you have to start with what is underneath the symbol. The blue box below summarizes the terminology and common ways to represent division.ġ2 ÷ 3 = 4 ( with a division symbol this equation is read “12 divided by 3 equals 4.”)
The answer to a division problem is called the quotient. The number that is dividing the dividend is called the divisor. In the work in this topic, this number will be the larger number, but that is not always true in mathematics. The number that is being divided up, that is the total, is called the dividend. We showed this division written as 15 ÷ 5 = 3, but it can also be written two other ways:Įach part of a division problem has a name. So, 15 divided by 5 is equal to 3.Īs with multiplication, division can be written using a few different symbols. Then keep adding rows until you get to 15 small squares. Ask yourself, if you were to make a rectangle that contained 15 squares with 5 squares in a row, how many rows would there be in the rectangle? You can start by making a row of 5: This repeated subtraction can be represented by the equation: 15 ÷ 5 = 3.įinally, consider how an area model can show this division. This is like subtracting 5 from 15 three times. Notice that there are 3 jumps that you make when you skip count by 5 from 15 back to 0 on the number line. Consider how many jumps you take by 5s as you move from 15 back to 0 on the number line. Just as you can think of multiplication as repeated addition, you can think of division as repeated subtraction.
You could also use a number line to model this division. You could represent this situation with the equation: Consider the picture below.ġ5 cookies split evenly across 5 plates results in 3 cookies on each plate. If there are 15 cookies to be shared among five people, you could divide 15 by 5 to find the “fair share” that each person would get.
For example, one might use division to determine how to share a plate of cookies evenly among a group. Where he went wrong, unsurprisingly, was in how he interpreted the remainder.Division is splitting into equal parts or groups. He was right to get a common denominator: using repeated addition or subtraction forces that choice. Had Jacob realized/remembered that, he’d have been fine. We always want the remainder to be either a whole number, as we get with something like 10 divided by 7 = 1 R 3, a decimal, or the remainder divided by the original divisor. Hence there is enough for 3 and (3/10) / (4/10) = 3 3/4, and that’s what we calculate more efficiently with the standard division of fractions algorithm. Having converted halves and fifths to tenths (which is absolutely necessary if he’s going to solve this by repeated subtraction), Jacob forgets what the remainder means: 3/10 of a cup as a fraction of a whole needed for a batch. Remainders in this context are what’s left over as a fraction of the DIVISOR. But he’s left with a remainder (3/10 of a cup) and simply appends that to the 3 complete batches.
Jacob does repeated subtraction correctly and ascertains that there are 3 full batches possible with 1 1/2 cups of flour and 2/5 cups of flour per batch.
Similar errors occur with decimal fractions when students have to interpret remainders. Just gave this a more careful look, since new comments have made it pop up in my email again.Įverything Jacob does is correct except for handling the remainder, and I’m wondering if others realize why his answer, which is correctly calculated via repeated subtraction, doesn’t provide the same answer as with standard division of fractions.