Subtraction with regrouping is often the nemesis of math teachers; it is one of those topics that many students just don’t “get” or remember. But keeping the focus on number sense, instead of procedure, will help students make sense of regrouping and help them apply the concepts in new situations.
When working with whole numbers, keep in mind that the standard algorithm of lining the numbers up vertically and regrouping is only one way to subtract. Encourage students to try another strategy as well, or to check their work with another method. For example, a student might think about 23 – 8 by saying I need 2 more to get to 10, then 10 more to get to 20, then 3 more to get to 23. So the number I need to add to 8 to get to 23 is 2 + 10 + 3 = 15. Or they might know that 20 – 8 is 12, so they need another 3. Or they may break up the 8 to 3 + 5, take the 3 away from 23, and then take the 5 away. The number line and the hundreds chart help foster this kind of number sense with subtraction. Ask students to subtract a different way and explain how they did it – you might be surprised by their creativity!
When deciding whether regrouping is needed, emphasize the grouping that took place that enabled us to express the quantity in writing using place value. If I were doing 23 – 8, I would say to students, let’s look at the ones in 23. How many are there? The correct answer is that there are 23 ones in 23. There are 3 in the ones place because the other 20 have been regrouped to the tens place. Remind students that when you added, if you had more than 9 in the ones place, you made a ten and regrouped those ten ones to be a one in the tens place. So next, do we have enough ones to subract 8? Of course we do, we have 23 ones. But 20 of them have been grouped to make tens and the 20 are now represented by 2 tens in the tens place. Since 3 ones are not enough to take 8 away from, we need to go to the tens place and break up one of those tens that we composed, moving ten ones to the ones place. So now we are expressing 23, instead of as 20 + 3 (two tens and three ones), as 10 + 13 (one ten and 13 ones). With 13 in the ones place, we can now subtract the 8 ones. Go through this type of reasoning with every problem until all students can explain it on their own.
Research shows that young students struggle to understand that a ten is simultaneously one ten and ten ones. Teachers cannot assume that students understand fully what adults take for granted. By keeping the emphasis on the place values, composing and decomposing a ten, and connecting to regrouping in addition, we can avoid having the standard algorithm become just a procedure that students will memorize and then forget.
Here is a great virtual manipulative that helps connect the place value concepts to the standard algorithm.
The concept of renaming numbers in different ways in order to facilitate the operation of subtraction resurfaces when working with rational numbers. In subtracting 2 and 7/8 from 5, think about a delivery of 5 pizzas from the store where they forgot to slice up the pies. Can you give someone 7/8 of a pie if they haven’t been sliced? No. What should you do? Slice up the pie! Into how many pieces? Well, if they want 7/8, which is a collection of 7 parts of a whole that has been partitioned into eight equal parts, then slice it into 8 pieces. Once it is sliced, instead of calling it one whole pie, we can call it 8/8 pie. But then we have one less whole pie that hasn’t been sliced. So, we rename 5 (five whole unsliced pies) as 4 and 8/8 (four whole unsliced pies and one whole that has been partitioned into 8 equal parts). Now we can subtract. The next step is to add a special offer from the store – an extra slice (1/8 of a pie) when you purchase 5 whole pies. Now the 5 and 1/8 can be renamed as 4 whole unsliced pies plus 8/8 pie plus the extra 1/8 of a pie. The concept of breaking up a whole and giving it a different name should not be difficult for students who understood that a ten can be broken up and renamed as ten ones.
Number sense is really all about composing and decomposing numbers. A focus on number sense will make even the most challenging procedures easier for students to understand and apply from year to year.