Lesson 9Using the Partial Quotients Method

Let’s divide whole numbers.

Learning Targets:

I can use the partial quotients method to find a quotient of two whole numbers when the quotient is a whole number.

9.1 Using Base-Ten Diagrams to Calculate Quotients

Elena used base-ten diagrams to find . She started by representing 372.

a diagram breaking up 372 by hundreds, tens, and ones

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 .

Discuss with a partner:

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

9.2 Using the Partial Quotients Method to Calculate Quotients

Andre calculated 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?
How might Andre calculate ? Explain or show your reasoning.

9.3 What’s the Quotient?

Find the quotient of using one of the methods you have seen so far. Show your reasoning.
Find each quotient and show your reasoning. Use the partial quotients method at least once.

Lesson 9 Summary

We can find the quotient 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.

a diagram breaking up 345 by hundreds, tens, and ones

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 . 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 () 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 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 , then the base-ten diagrams will be time-consuming.

Lesson 9 Practice Problems

Here is one way to find using partial quotients.

Show a different way of using partial quotients to divide 2,105 by 5.
Andre and Jada both found using the partial quotients method, but they did the calculations differently, as shown here.
How is Jada's work similar to and different from Andre’s work?

Explain why they have the same answer.
Which might be a better way to evaluate : drawing base-ten diagrams or using the partial quotients method? Explain your reasoning.
Here is an incomplete calculation of .

Write the missing numbers (marked with “?”) that would make the calculation complete.
Use the partial quotients method to find .
Which of the polygons has the greatest area?
1. A rectangle that is 3.25 inches wide and 6.1 inches long.
2. A square with side length of 4.6 inches.
3. A parallelogram with a base of 5.875 inches and a height of 3.5 inches.
4. A triangle with a base of 7.18 inches and a height of 5.4 inches.
One micrometer is a millionth of a meter. A certain spider web is 4 micrometers thick. A fiber in a shirt is 1 hundred-thousandth of a meter thick.
1. Which is wider, the spider web or the fiber? Explain your reasoning.
2. How many meters wider?