Dividing By One-Digit Numbers

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4-5 Dividing by One-Digit Numbers4-5

Students have learned how to divide any three-digit number by a single-digit number by using the blocks.This is an important achievement in andof itself. It gives students a concrete understanding of the division process.

In this section students build on that understanding, now predicting what

will happen before they share the blocks. This focus on prediction helps

students to be more conscious of their work and to develop mental compu-

tation and paper-and-pencil techniques. To internalize the process, students

need many opportunities to divide collections of blocks and should do sountil they can consistently predict the result of any division before physically

performing the task.

In division, students will need to predict two types of outcomes. For example,

consider 3 children sharing 714 blocks.

Starting with the leftmost place, students first predict the number of

blocks-of-100 each child will get. Knowing that 7 3 = 2 R1, students can

predict that each child will get 2 such blocks, and they will need to

unpack the remaining one.

Students then need to predict the number in the next place after blocks

are unpacked and added to it. In this case,they must predict that there

will be 11 ones after the block-of-100 is unpacked.

In division, these predictions are cyclical. The student now returns to predict-

ing the number of blocks-of-10 each child will receive.

The ability to predict the number in each column when blocks are unpacked

relies on knowing for example, that there are 11 tens in the number 114.This

concept of reading a number to any place was investigated in section 1-7;

you can review this section with students as needed.

Making Predictions

Present an example such as 87 3 and have students represent the 87. Ask,

How many do you think there will be in each group?

Focus Predicting the outcome when separating groups,

and finding quotients without the blocks


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Have students discuss their predictions and then distribute the blocks tocheck. Next ask,

How can you predict the number of single blocks there will be when you unpack the left-

over tens and add the new singles to the ones place?

When representing the original number on the Place mat, students mayfind it helpful to record the total number of ones at the top of the mat inthe ones column.

When blocks need to be unpacked immediately, students can predict thenumber of blocks of the next size just by reading the number. Have studentsinvestigate 243 3 and again predict before unpacking. Have students discusstheir prediction techniques. If no one mentions looking at the writtennumber to predict the number of tens ask,

How does the number code for 243 help you to predict the number of blocks-of-10?(If necessary, review section 1-7 with students.)

Now present the example 496 3 and have students represent 496 on theirPlace mats. For each place, have students predict how many there will be ineach group. Continue asking questions to encourage students to reflect ontheir actions. Help them to realize that making a mistake is fine and that theywill get better at making predictions with time. Further, they can always usethe blocks and self-correct when necessary.

Have students explore a variety of examples and related story problems overseveral days. After students gain confidence in their ability to predict one placeat a time, present the example 252 4. Have students represent 252 and thistime challenge them to predict the answer before dividing the blocks.Encourage students to explain how they made their predictions and then usethe blocks to check their thinking.

Provide other division examples and story problems, encouraging students torepresent the original quantity and then make predictions before using theblocks to model the division process. This is a good time to review the factthat a division problem such as 213 3 yields the same answer whether the 3

represents the number of groups or the number in each group. Consider aproblem such as this:

There are 213 tennis balls.

There are 3 tennis balls in each can.

How many cans of tennis balls are there?

Have students discuss whether it is easier to make groups of 3 or 3 equalgroups. Help them recognize why they can choose whichever way is easier.

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Developing Recording Steps for Division

Students should represent division concretely over and over again until theyare able to consistently predict exact outcomes at all stages of the process

when the blocks are in view. Encourage students to develop recordingschemes for their work. You can either guide them to record in a conventionalmanner, or allow them to develop their own recording methods. In eithercase, emphasize the importance of keeping track of the number of blocks ineach place. Graph paper or lined paper turned vertically can help students tokeep numbers aligned correctly.

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

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 the blocks-of-100 that are left?

If you want students to use the traditional written algorithm, be sure that theycan connect their recordings with their physical actions. For example:

1 7 4 R 2

3) 5 2 4

3 1 block-of-100 is given to each group, using 3 of them.

2 2 2 blocks-of-100 remain; when unpacked, there are 22 blocks-of-10. 2 1 7 blocks-of-10 are given to each group, using 21 of them.

1 4 1 block-of-10 remains; when unpacked, there are 14 ones.

1 2 4 single blocks are given to each group, using 12 of them.

2 2 single blocks are left over.

Following is a recording scheme that differs from the traditional form and mayhelp students to relate their work with the blocks to the symbolic recording. Bythis method, they keep track of the number of blocks in each place, crossing out

a number when it no longer represents the number of blocks in that column.

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4-54-5Missing Number

Present a division example with the dividend missing. Students first predict the

number and then use the blocks to check.

6 3 12 R 23) 9)

Encourage students to discuss their strategies for finding the missing number.

Assessing Learning

1. Present 264 6. Ask the student to represent 264 and then predict theanswer before sharing the blocks. Does the student predict correctly? self-correct if necessary? find the correct answer? clearly explain his or her thinking?

2. Present the example 803 8. Ask the student to find the answer withoutusing the blocks and explain his or her thinking while working. Doesthe student find the correct answer? clearly explain his or her thinking?

3. Present the following story problem:

Uncle Tino has 433 baseball cards.He wants his 4 nephews to share the cards.

How many cards will each child get?Does the student use paper-and-pencil strategies or model the situation with the blocks? answer correctly? interpret the remainder in a reasonable way?

4. Present the division example 629 8 and ask,

Will there be more than 100 in each group? less than 70? Why do you think so?

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

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Dividing By One-Digit Numbers