Subtracting with Base Ten Representations

7/28/2019 Subtracting with Base Ten Representations

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In this section, students explore subtracting with the base ten representationsof number.They begin with a group of blocks (containing blocks-of-100,blocks-of-10, and ones) and take away another group of blocks,unpacking if

necessary.Students then determine the number of blocks that remain.The

activities in this section emphasize modeling subtraction,and these

concepts will be formalized in the next section.

Subtracting with the Counter

Invite students to join you for a small-group or whole-class demonstrationactivity. Make available single blocks, empty holders, blocks-of-10 and aCounter. Cover the dials. Present a story problem about the blocks. Forexample:

Haley has 274 blocks.

She lets Liam use 121 of them.

How many blocks does Haley have left?

After reading the problem, ask the students,

On the Counter, how can you show all of Haley's blocks?

Have a volunteer load the blocks on the Counter and record the number onthe whiteboard. Then ask,

How many blocks did Haley let Liam use?

Have a student record this number on the whiteboard below the number thatshows the total. Ask,

What can you do to show the blocks she gave to him?

Invite a volunteer to remove 121 from the Counter, place these blocks on thetable in front of the counter, and set the dials to show the number of blocksthat Haley has left. As with addition, it is fine if students begin with theblocks-of-100 instead of ones. Ask,

How can we write a number sentence to show what we did with the blocks?

You might also ask,

When Liam gives his blocks back to Haley, how many blocks will she have?

Have a volunteer clear the blocks and cover the dials to prepare for some

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Focus Subtracting with the Counter or with the

Place mats


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The Counter mat serves this purpose when working with the Counter. Whenworking with the Place mat, students should remove the blocks and placethem just below the appropriate columns.

Next present an example that requires regrouping, such as 216 125. Askingquestions while students work helps them to think more explicitly about theprocess. For example, ask,

Why are you opening that block-of-100?

Where do you put that block-of-100? Why?

Traditionally, students experience greater diff iculty with subtraction whenthey must regroup and there are no tens, for example in 402 286. Withblocks inside blocks, this process is more obvious to students. They f irstrepresent 402 on the Place mat. To get more single blocks, they must takea block-of-100, unpack it, and then further unpack one of the blocks-of-10,taking care to put each block in the appropriate place. To make sure thatstudents understand that this is an equivalent representation ask,

What number does 3 blocks-of-100, 9 blocks-of-10, and 12 ones represent?

Students can then remove 286 and set the Digit Flip Cards to show the answer.

Students should have many experiences using the blocks to solve subtractionstory problems (including comparisons and how many more as well astake-away situations) and to find differences for examples written in verticaland horizontal forms. As they work with the blocks, they will develop theirown methods for finding the answers. During this time, continue to ask ques-tions to help students become more conscious of the steps they are taking,and why. For example,

Do you think you will have to unpack this time?

What did you do when you did not have enough ones?

How could you check your work?

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When subtracting with the Place mat,students remove blocks to a position just below the mat andset the Digit Flip Cards to show the amount remaining.


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Students should also communicate their work using drawings, numbers,and words. These questions and representations help students to make theconnection to a written algorithm when the time is appropriate.

Connecting Two Ways to Subtract

Previously, working with single blocks and number lines, students countedback or removed a quantity of single blocks to find the difference. Thisapproach to subtraction is based on the intuitive view of number as a group orline of single objects. In this section, students have been using base ten repre-sentations of number. Its important for students to realize that they get thesame results whether they use the number line or the Place mat. Students willalso recognize that as numbers get larger, working with just the single blocks

becomes very difficult. Yet, subtracting with the Place mat does not get muchharder with larger numbers.

To connect and compare these two models of subtraction, start with anexample that can be done readily either way, such as 89 46. Have thestudents find the difference first with number lines and then with packedrepresentations on the Place mat. Ask,

How are these two ways to subtract the same? How are they different?

Next, ask the students to imagine a long number line and present them with aproblem such as 873 212. Ask,

How would you use a long number line to find 873 212? Would you rather use a Place

mat or a number line to find the difference? Why?

Practicing Key Ideas

Race to Zero

Students work in pairs, each with their own Place mat (or two students may play as

a team against another team). Play begins with 499 blocks on the mat. Students take

turns rolling two dice of different colors to find how many blocks to remove from the

mat. One color tells the number of single blocks; the other color, the number ofblocks-of-10. At the end of each turn, students set the Digit Flip Cards to show the

amount that remains on the Place mat.The first student (or team) to have no blocks

remaining on the mat is the winner.

Students can write a subtraction sentence or example to record each turn, or keep a

list of the remaining amounts.

The Race to Zero could also be played on Counters instead of Place mats.

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How Much Is Gone?

Students work in pairs.One student creates a collection of blocks of different sizes.

Another student forms a collection that represents a smaller number.The students

then work together to find how many blocks they need to take from the first collec-

tion in order to make it equal to the second collection.Students can record theirfindings in any way they choose.

Assessing Learning

1. Have the student show 342 on the Place mat and say,

Show me how to take away 128 of these blocks and find the number that is left.

Please tell me what youre thinking as you work.Does the student model the process correctly?

set the Digit Flip Cards to show the correct difference? clearly explain his or her thinking?

2. Present a written example such as 232 151 in vertical form and say,

Show me how to use the Place mat to find the difference.Once the student finds the difference, ask,

If you combine the blocks again, how many will there be? What number sentence can

you write to record this action?Does the student model the process correctly?

set the Digit Flip Cards to show the correct difference? immediately recognize the total or combine the blocks to answer? write a correct number sentence?

3. With blocks and a Place mat available, present a story problem such asthe following.

Kelly has ridden her bicycle for 104 miles this month.

Marcos has ridden his bicycle for 59 miles this month.

How many more miles has Kelly ridden than Marcos?Does the student

use the Place mat? model the problem correctly? find the correct difference?

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Subtracting with Base Ten Representations