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.
![Base ten diagram representing 372. 3 large squares labeled, 3 hundreds, 7 rectangles labeled, 7 tens, and 2 small squares labeled, 2 ones.](https://cms-im.s3.amazonaws.com/51vkRyzpTUvsUYK9o4xryzkj?response-content-disposition=inline%3B%20filename%3D%226-6.5.D1.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5.D1.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141608Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=027880fa48884071b36b5e0ee28385a92fcac49d2b9a9d1cbf2a382bb90f0dcb)
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 \(372 \div 3\).
![3 groups of base-ten blocks. Each group consists of 1 large square labeled, hundreds, 2 rectangles labeled, tens, and 4 small squares labeled, ones.](https://cms-im.s3.amazonaws.com/fkkGdYrXpr2RRVYCjAgJQYhe?response-content-disposition=inline%3B%20filename%3D%226-6.5.D1.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5.D1.Image.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141608Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9647025b9196fd9860cddc6f93e917d308c4e08aee7df4df3caa224101f384ed)
- 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
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Andre calculated \(657 \div 3\) using a method that was different from Elena’s.
- 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?
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How might Andre calculate \(896 \div 4\)? Explain or show your reasoning.
9.3: What’s the Quotient?
- Find the quotient of \(1,\!332 \div 9\) using one of the methods you have seen so far. Show your reasoning.
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Find each quotient and show your reasoning. Use the partial quotients method at least once.
- \(1,\!115 \div 5\)
- \(665 \div 7\)
- \(432 \div 16\)
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.
![Base-ten diagram representing 345. 3 large squares labeled, hundreds, 4 rectangles labeled, tens, 5 small squares labeled, ones.](https://cms-im.s3.amazonaws.com/X94fJgn6G86B7T7QEGt7f4TW?response-content-disposition=inline%3B%20filename%3D%226-6.5.D1.Image.10a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5.D1.Image.10a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141608Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ac3d8b5ba415bda79376233387c00c7e08c674122d73fa5d6b5500eeb4173a03)
We can think of the division by 3 as splitting up 345 into 3 equal groups.
![3 groups of base-ten blocks. Each group contains 1 large square, labeled hundreds, 1 rectangle labeled, tens, and five small squares labeled, ones.](https://cms-im.s3.amazonaws.com/EGUEha2DoyBA8qaMqk5hNtad?response-content-disposition=inline%3B%20filename%3D%226-6.5.D1.Image.11a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5.D1.Image.11a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141609Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dec938b6699d7b227e8426fb831bee5bc2e99787a426b60c12ad5db9cd74be3e)
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.
![2 partial quotients methods of 345 divided by 3.](https://cms-im.s3.amazonaws.com/GsnGkNmMWGu9ComcpoJDfTSX?response-content-disposition=inline%3B%20filename%3D%226-6.5.D1.Image.12.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5.D1.Image.12.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141609Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=91388af547ee6fff8295a74f4b8b4338f646df471c6b280df2d497c04bab3b47)
- 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.