dividing by two-digit numbers

7/28/2019 Dividing by Two-Digit Numbers

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After working through the previous sections, students can divide anynumber by a single-digit number. The next step is to extend theirthinking to dividing by multidigit numbers. Students will find this step

easy when they use the blocks. The physical process is exactly the same as

for single-digit divisors. The recording scheme is also the same, and again

assumes an understanding of reading the code to any place (section 1-7).

For example, in dividing 836 by 13, the student should recognize that the

number 836 contains 83 tens.

Before proceeding with general two-digit divisors, you may want to make

sure that students have the number sense and mental computation skills to

divide by multiples of 10. Specifically, students should be able to extend their

use of basic facts to find the answers to examples such as 560 70. This ability

allows students to estimate and to check the reasonableness of answers to

examples such as 563 69.

Dividing by Ten

Students may already have ways of understanding division by 10. For exam-

ple, they may recognize the relationship between multiplication and divisionand know that since 10 6 = 60, then 60 10 = 6. At this point, studentsshould also recognize that the number of tens in 60 is 6, from reading thecode to the tens place. Problems based on the repeated subtraction or howmany groups model of division make the relationship clear.

Have students find 320 10. Some students may represent 320 with theblocks, remove the covers from the blocks-of-100, and count all the blocks-of-10. Others might make 10 equal groups. This provides a good opportunityto remind students that the answer is the same whether they make groups of10 or 10 groups. Ask,

Which way is easier? Why?

If no one suggests reading the code to the tens place, provide additionalexamples and have students record the related number sentences. Then ask,

Do you see any patterns that can help you to find the answers without using the blocks?

Encourage students to form a generalization about how to divide by 10, statedin their own words and from their own discovery, to this effect: When anynumber is divided by 10, the result is a number with the same exact digits, only shifted

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FocusExtending ideas to two-digit divisors


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4-64-6one place to the right. Then have students extend their thinking to find 1200 10, 2540 10 and 38,460 10.

For a greater challenge, ask students to use the generalization to find 367

10. Some may recognize the remainder of 7, while others will just see the 36tens. This is fine for now. In later years they will extend their ideas to decimalnotation and recognize the answer as 36.7.

You may want to extend this idea to dividing by 100. Proceed in a similarmanner. In this case we read the code to the hundreds place, or shift the digitstwo places to the right.

Dividing by Multiples of 10

Have students find 80 2 and 80 20. Again, it is easier to make groups of 20than 20 equal groups. Once students agree that the answers are 40 and 4, ask,

What do you notice about these answers? (Once again, the digits shift one place

to the right when you divide by tens.)

Have students model a variety of other expressions such as 90 30 and 120 40. Again, students will find that they can divide as if by ones and then shiftthe digits one place to the right. Have students apply this thinking to otherexamples. Ask,

How can you find 450

50? What about 2400

30?Encourage students to state a generalization and relate it to the generalizationthey formed for multiplying by tens. For example, they might note that whenthey multiply by a multiple of 10, they multiply as if by ones and then shiftthe digits one place to the left. When they divide by a multiple of ten, they

divide as if by ones and then shift the digits one place to the right.

Challenge students to find 15,000 30 and explain their thinking.

Again, you may want to extend these ideas to multiples of 100. The next step

is learning how to divide by any two-digit number.

Modeling Division of Two-Digit Numbers

Given enough workspace, students can model any division example. It isimportant for them to recognize that the process is exactly the same fortwo-digit divisors. Present an example such as 288 12 and have studentsdistribute the blocks to form 12 equal groups. When students represent theproblem with the blocks, the process becomes clear. Ask questions such as:


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Can you give a block-of-100 to each group? How many blocks-of-10 will you have

after unpacking?

Repeat for other examples such as 468 13, 180 15, and 158 11.

Using Paper-and-Pencil Techniques

Recording schemes for examples with two-digit divisors are the same as thosefor one-digit divisors. Again, the emphasis is on keeping track of the numberof blocks in each place. Recording on graph paper can be helpful.

Have students work with the blocks as they record what they are doing. Askquestions such as,

How many hundreds can you give to each group?How many hundreds have you given out in all?How many hundreds are left? What will you do with these blocks?

If you want students to use the traditional written algorithm, be sure that theyconnect their recordings with their physical actions with the blocks.

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1 3) 8 3 6 8 hundreds cannot be shared. When unpacked, there are

7 8 83 tens, 6 tens are given to each group, using 78 tens.5 6 5 tens remain; when unpacked, there are 56 ones.

5 2 4 ones are given to each group, using 52 ones.

4 4 single blocks are left over.

While many people rely on calculators for multidigit computation, under-standing the division process helps students to check the reasonableness of theanswers. Present story problems such as the following, which asks students foran estimation of the answer:

The school has 2200 blocks.

The blocks will be shared among 12 classrooms.Are there enough blocks to give each classroom 200 blocks?

When students think about the blocks, this is quite obvious. There will be 22blocks-of-100. They can give each group (each classroom) 1 block-of-100,but not 2.

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Practicing Key Ideas

Using Division Facts

In pairs, students choose a basic division fact (for example,42 7 = 6). Then they

write a list of other division sentences they know based on this fact. (In this case, theymight write 420 7 = 60, 420 70 = 6, 4200 70 = 60, and so on.)

More Than 9

Have students work in pairs. One student picks a two-digit number. Then together

they identify three numbers that they can divide the chosen number by and get a

quotient that is greater than 9. Students then record the related number sentences.

Assessing Learning

1. Present 170 10. Ask the student to find the answer and explain his orher thinking. Does the student work abstractly or use the blocks? find the correct answer? clearly explain his or her thinking?

2. Repeat for 2800 70. Does the student work abstractly or use the blocks? find the correct answer? clearly explain his or her thinking?

3. Present 397 14. Ask the student to show you how to find the quotientusing the blocks and to explain his or her thinking. Does the student model the example correctly? find the correct answer? clearly explain his or her thinking?

4. Present a story problem that invites estimation and have the studentexplain his or her thinking. For example:

Mr. Sanchez has 1850 blocks to share with 30 students.

Does he have enough blocks to give each student 70 blocks?

Does the student find the correct answer? reason correctly? clearly explain his or her thinking?

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dividing by two-digit numbers

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