Using Long Division
Elena used base-ten diagrams to find \(372 \div 3\). She started by representing 372.
She made 3 groups, each with 1 hundred. Then, she put the tens and ones in each of the 3 groups. Here is her diagram for \(372 \div 3\).
Andre calculated \(657 \div 3\) using a method that was different from Elena’s.
How might Andre calculate \(896 \div 4\)? Explain or show your reasoning.
Lin has a method of calculating quotients that is different from Elena’s method and Andre’s method. Here is how she found the quotient of \(657\div 3\):
Discuss with your partner how Lin’s method is similar to and different from drawing base-ten diagrams or using the partial quotients method.
Lin’s method is called long division. Use this method to find the following quotients. Check your answer by multiplying it by the divisor.
Find each quotient.
Here is Priya’s calculation of \(906 \div 3\).
We can find the quotient \(345\div 3\) in different ways.
One way is to use a base-ten diagram to represent the hundreds, tens, and ones and to create equal-sized groups.
We can think of the division by 3 as splitting up 345 into 3 equal groups.
Each group has 1 hundred, 1 ten, and 5 ones, so \(345 \div 3 = 115\). Notice that in order to split 345 into 3 equal groups, one of the tens had to be unbundled or decomposed into 10 ones.
Long division is another method for calculating quotients. It relies on place value to perform and record the division.
When we use long division, we work from left to right and with one digit at a time, starting with the leftmost digit of the dividend. We remove the largest group possible each time, using the placement of the digit to indicate the size of each group. Here is an example of how to find \(345 \div 3\) using long division.
We start by dividing 3 hundreds into 3 groups, which means 1 hundred in each group. Instead of writing 100, we simply write 1 in the hundreds place, knowing that it means 1 hundred.
There are no remaining hundreds, so we work with the tens. We can make 3 groups of 1 ten in 4 tens, so we write 1 in the tens place above the 4 of 345. Subtracting 3 tens from 4 tens, we have a remainder of 1 ten.
We know that 1 ten is 10 ones. Combining these with the 5 ones from 345, we have 15 ones. We can make 3 groups of 5, so we write 5 in the ones place.
In total, there are 3 groups of 1 hundred, 1 ten, and 5 ones in 345, so \(345 \div 3 = 115\).
Long division is an algorithm for finding the quotient of two numbers expressed in decimal form. It works by building up the quotient one digit at a time, from left to right. Each time you get a new digit, you multiply the divisor by the corresponding base ten value and subtract that from the dividend.
Using long division we see that \(513 \div 4 = 128 \frac14\). We can also write this as \(513 = 128 \times 4 + 1\).
\(\displaystyle \require{enclose} \begin{array}{r} 128 \\[-3pt] 4 \enclose{longdiv}{513}\kern-.2ex \\[-3pt] \underline{4{00}} \\[-3pt] 113 \\[-3pt] \underline{\phantom{0}8{0}} \\[-3pt] \phantom{0}33 \\[-3pt] \underline{\phantom{0}32} \\[-3pt] \phantom{00}1 \end{array} \)