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Concrete
Use of counters: Start with a subtraction problem that does not involve regrouping. Allow the child to talk about it and possibly solve it with mental math.
There are 47 children on the playground. 24 go home. How many are left on the playground. Can you figure this out?
If the child does not present a mental math solution, ask him or her to show the number of students who started on the playground (47) with unifix cubes in tens and ones on a base-ten placemat, then ask them to take away the number of students who went home (24). If they dont know how to do this, show them and explain what you are doing: We will take away the number of students who went home. 24 went home. We will take away 10 (one stack) and 10 more (another stack), so thats 20, then well take away 4 more. Then try another problem and transfer the responsibility for doing it to the child. Provide feedback as needed until the child can do this independently.
At this point, simply write the subtraction problem horizontally, to show the problem with numerals.
Then pose a problem that requires regrouping, perhaps using the same context:
There are 53 children on the playground. 38 go home. How many are left on the playground. Can you figure this out?
Ask the child to show 53 children using unifix cubes in 10s and 1s on a base-ten placemat. As they try to use the same procedure, most will say something like I cant take 8 away from 3. If needed, show them how to break apart a rod of 10 into 10 ones. Verbalize what you are doing: How many are in this rod of ten? (Ten.) Lets take apart the rod into 10 ones and place them in the ones column on the base ten placemat. As you regroup in this way, record the procedure symbolically (53 becomes 40 and 13). Then have the child take away 38 and count the answer. Have the child write the answer.
Representational
Base-ten placemat: The base-ten placemat is used to draw the graphic representation, which is simply a recording of what students did with unifix cubes. The base-ten placemat helps them maintain the correct place value during the problem. (see next page)
Abstract
Writing number sentences and using the formal algorithm: Eventually you can remove the base ten placemat and let them write the problem symbolically, using the formal algorithm that is the exact symbolic translation of the work they did with the concrete and representational materials. Students should learn to write addition and subtraction problems horizontally, since this is the way that problems are most often recorded; then they need to learn to write them vertically to use the standard algorithm, lining up the place values correctly.
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