Lesson 9: Using the Partial Quotients Method

Let’s divide whole numbers.

9.1: Using Base-Ten Diagrams to Calculate Quotients

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 $372 \div 3$.

Discuss with a partner:

  • Elena’s diagram for 372 has 7 tens. The one for $372 \div 3$ has only 6 tens. Why?
  • Where did the extra ones (small squares) come from?

9.2: Using the Partial Quotients Method to Calculate Quotients

  1. Andre calculated $657 \div 3$ using a method that was different from Elena’s.

    Discuss the following questions with a partner:

    • Andre subtracted 600 from 657. What does the 600 represent?
    • Andre wrote 10 above the 200, and then subtracted 30 from 57. How is the 30 related to the 10?
    • What do the numbers 200, 10, and 9 represent?
    • What is the meaning of the 0 at the bottom of Andre’s work?

  2. How might Andre calculate $896 \div 4$? Explain or show your reasoning.

9.3: What’s the Quotient?

  1. Find the quotient of $1,\!332 \div 9$ using one of the methods you have seen so far. Show your reasoning.

  2. Find each quotient and show your reasoning. Use the partial quotients method at least once.

    1. $1,\!115 \div 5$
    1. $665 \div 7$
    1. $432 \div 16$

Summary

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.

Another way to divide 345 by 3 is by using the partial quotients method, in which we keep subtracting 3 groups of some amount from 345.  

  • In the calculation on the left, first we subtract 3 groups of 100, then 3 groups of 10, and then 3 groups of 5. Adding up the partial quotients ($100+10+5$) gives us 115.
  • The calculation on the right shows a different amount per group subtracted each time (3 groups of 15, 3 groups of 50, and 3 more groups of 50), but the total amount in each of the 3 groups is still 115. There are other ways of calculating $345 \div 3$ using the partial quotients method. 

Both the base-ten diagrams and partial quotients methods are effective. If, however, the dividend and divisor are large, as in $1,\!248 \div 26$, then the base-ten diagrams will be time-consuming.

Practice Problems ▶