The student knew there was borrowing, aka renaming, involved, because the top number ended in two 0's, and neither 1 nor 5 can be subtracted from 0. However, his method of renaming made me suspect he was unclear on just what was being renamed or borrowed from.
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| This was the student's solution. Can you spot two errors? |
Here we see the student has attempted to cover his bases by renaming the digits all the way across the top. He ends up with an incorrect answer, however, because the pattern he has learned is to turn zeroes into tens and subtract one from the digits on the left.
Using a whiteboard with colored markers, the student works through the problem. When he gets to the end, I call his attention to the hundreds column: did he need to borrow anything to subtract 3 from 7? No? Okay, so let's erase the 6. In fact, let's start over.
This time, I guide the student through the "borrowing" one step at a time:
1. Can 1 be subtracted from 0? No. We need to borrow from the left column to turn 0 into 10.
2. We can't borrow 1 from 0, but we CAN go one more column to the left and borrow 1 from 80. what is 80-1? It is 79. So we mark out the 80 and write 79 above it, then put the 1 above the last 0.
3. Now we start to subtract: 10-1= 9, 9-5= 4, 7-3= 4, 7- 0 = 7. Final answer: 7,449.
After a few times practicing with the whiteboard, it's time for a change of pace. So we build another problem with 3D letters. In this photo, we are using magnetic numbers on a paper-lined cookie sheet, and the paper has columns marked on it so we can line up the numbers correctly. (Try asking the student why the numbers need to be lined up carefully. What could happen if the top and bottom numbers were misaligned?)
This time, I guide the student through the "borrowing" one step at a time:
1. Can 1 be subtracted from 0? No. We need to borrow from the left column to turn 0 into 10.
2. We can't borrow 1 from 0, but we CAN go one more column to the left and borrow 1 from 80. what is 80-1? It is 79. So we mark out the 80 and write 79 above it, then put the 1 above the last 0.
3. Now we start to subtract: 10-1= 9, 9-5= 4, 7-3= 4, 7- 0 = 7. Final answer: 7,449.
After a few times practicing with the whiteboard, it's time for a change of pace. So we build another problem with 3D letters. In this photo, we are using magnetic numbers on a paper-lined cookie sheet, and the paper has columns marked on it so we can line up the numbers correctly. (Try asking the student why the numbers need to be lined up carefully. What could happen if the top and bottom numbers were misaligned?)
This time, we guide the student through the process:
1. In the ones column, you have 3 - 6. Is that doable? No, you must borrow from the tens column on the left.
2. The tens column has a zero! No tens to borrow; go left again.
3. The hundreds column has a zero! No hundreds to borrow; go left again.
4. The thousands column has a 5. Change it to 499+1 (4990 + 10), use the 10 to turn the 3 into a 13. Now subtraction can proceed.
By keeping all the digits within a column the same color, it becomes clear where the "1" is being borrowed from.
If the concept is still unclear, it may help to have the student perform the following exercise:
You'll need a box of 500-1000 craft sticks, a bag of 50 or more rubber bands, a large sheet of paper, a regular lined paper, and markers Prepare the sticks by bundling about 45 sets of ten sticks. Leave the rest loose. Then bundle 3 sets of 100 sticks by putting 10 sets of the 10-stick bundles together for each set of 100.
Prepare the large sheet of paper by folding it into three columns, and labeling the columns left to right, "hundreds," "tens," and "ones."
1. Give your student 364 sticks as pictured (3 hundreds, 6 tens, 4 singles.) Show her that each small bundle is 10 and the large bundles are 100.
2. Have her place the sticks on the sheet of paper in the appropriate columns.
3. Ask her how many sticks there are altogether. Have her write the number down on a separate sheet of lined paper (or a white board.)
4. Ask the student to write below 364, "-295."
5. Ask the student to take 295 sticks away from her collection of 364 to see what the result is.
If she starts by taking the 2 hundred bundles away, she'll be unable to complete the subtraction. She has to start with the ones.
When she starts by trying to take away 5 ones, she will not have enough ones. What is the solution? Borrow a 10 bundle. Now the 4 is 14. If she breaks apart the 10 bundle, she can take 5 from 14 and have 9 single sticks left over.
She now has 5 bundles of ten (50), but needs to subtract 9 (90). Solution? Borrow a hundreds bundle. Now she has 15 bundles of ten (150). When she separates the hundreds bundle, she can subtract 9 of them (90) and have 6 (60) left over.
She now has two hundreds bundles left over. When she subtracts those, there are none left. What she does have left is 6 bundles of ten and 9 singles, equalling 69.
5. Have the student work the problem on the lined paper, this time writing in the borrowed digits and marking out/changing the required donor digits. Her answer should again be 69.
With enough practice, the student should see the correlation between borrowing from the next highest bundle of sticks and borrowing from the digit on the left.
By keeping all the digits within a column the same color, it becomes clear where the "1" is being borrowed from.
If the concept is still unclear, it may help to have the student perform the following exercise:
You'll need a box of 500-1000 craft sticks, a bag of 50 or more rubber bands, a large sheet of paper, a regular lined paper, and markers Prepare the sticks by bundling about 45 sets of ten sticks. Leave the rest loose. Then bundle 3 sets of 100 sticks by putting 10 sets of the 10-stick bundles together for each set of 100.
Prepare the large sheet of paper by folding it into three columns, and labeling the columns left to right, "hundreds," "tens," and "ones."
1. Give your student 364 sticks as pictured (3 hundreds, 6 tens, 4 singles.) Show her that each small bundle is 10 and the large bundles are 100.
2. Have her place the sticks on the sheet of paper in the appropriate columns.
4. Ask the student to write below 364, "-295."
5. Ask the student to take 295 sticks away from her collection of 364 to see what the result is.
If she starts by taking the 2 hundred bundles away, she'll be unable to complete the subtraction. She has to start with the ones.
When she starts by trying to take away 5 ones, she will not have enough ones. What is the solution? Borrow a 10 bundle. Now the 4 is 14. If she breaks apart the 10 bundle, she can take 5 from 14 and have 9 single sticks left over.
She now has 5 bundles of ten (50), but needs to subtract 9 (90). Solution? Borrow a hundreds bundle. Now she has 15 bundles of ten (150). When she separates the hundreds bundle, she can subtract 9 of them (90) and have 6 (60) left over.
She now has two hundreds bundles left over. When she subtracts those, there are none left. What she does have left is 6 bundles of ten and 9 singles, equalling 69.
5. Have the student work the problem on the lined paper, this time writing in the borrowed digits and marking out/changing the required donor digits. Her answer should again be 69.
With enough practice, the student should see the correlation between borrowing from the next highest bundle of sticks and borrowing from the digit on the left.
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