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# To regroup or not to regroup

## Task

Sometimes when we subtract one number from another number we "regroup," and sometimes we don't. For example, if we subtract 38 from 375, we can "regroup" by converting a ten to 10 ones:

Find a 3-digit number to subtract from 375 so that:

- You don't have to use regrouping.
- You would naturally use regrouping from the tens to the ones place.
- You would naturally use regrouping from the hundreds place to the tens place.
- You would naturally use regrouping in all places.

In each case, explain how you chose your numbers and complete the problem.

## IM Commentary

In a typical subtraction task where students have learned and are using the standard US algorithm, we present a variety of problems that require regrouping in different places and ask students to solve them. This task presents an incomplete problem and asks students to choose numbers to subtract (subtrahends) so that the resulting problem requires different types of regrouping. This way students have to recognize the pattern and not just follow a memorized algorithm--in other words, they have to think about what happens in the subtraction process when we regroup. This task is appropriate to use after students have learned the standard US algorithm.

This is an instructional task; students would benefit from working in groups where they can each try different numbers and compare what they find. Once they have found numbers that work, they can explain to each other how they found them and why their choices require regrouping. Manipulatives like base ten blocks can be used to visualize and act out the regrouping process.

Giving a complete explanation is significantly harder than finding numbers that work in each case. But even if fourth graders shouldn't be expected to give as concise and complete an explanation as is listed in the solution, they can explore which digits in which place require regrouping and describe the general pattern.

## Solution

- There are many possible answers. The important part is that the digit in the hundreds place is less than or equal to 3, the digit in the tens place is less than or equal to 7, and the digit in the ones place is less than or equal to 5. For example, 111 would work. Writing the problem in expanded form can help us see what is going on: $$\begin{align} 375-111 =& 300 + 70 + 5 - 100 - 10 - 1\\ =& 300-100 + 70 - 10 + 5 -1\\ =& 200 + 60 + 4\\ =& 264 \end{align}$$
- There are many possible answers. The important part is that the digit in the ones place is greater than 5 while the digit in the hundreds place is less than or equal to 3 and the digit in the tens place is less than 7 (if it equals 7, then we will have to regroup from the hundreds place to the tens place after we regroup from the tens to the ones). For example, 266 would work. Writing the problem in expanded form can help us see what is going on: $$\begin{align} 375-266 =& 300 + 70 + 5 - 200 - 60 - 6\\ =& 300-200 + 70 - 60 + 5 - 6\\ =& 300-200 + 60 - 60 + 15 - 6\\ =& 100 + 9\\ =& 109 \end{align}$$
- There are many possible answers. The important part is that the digit in the tens place is greater than 7 while the digit in the hundreds place is less than 3 and the digit in the ones place is less than or equal to 5. For example, 293 would work.
- There are many possible answers. The important part is that the digit in the ones place is greater than 5 and the digit in the tens place is greater than or equal to 7 while the digit in the hundreds place is less than 3 (if it equals 3, the difference will be negative, and students don't study signed numbers until 7th grade). For example, 296 would work.

We already mentioned how we had to choose the number as part of the explanations above. To summarize and generalize (assuming we write the numbers stacked as shown to make the explanation easier):

- If all digits in the bottom number are less than or equal to the corresponding digits in the top number, we don't have to regroup to complete the problem.
- If the digit in the ones place on the bottom is greater than the digit in the ones place on the top, we will regroup from the tens place to the ones place.
- Similarly, if the digit in the tens place on the bottom is greater than the digit in the tens place on the top, we will regroup from the hundreds place to the tens place.
- To ensure regrouping in both places, the digit in the bottom number in the ones place has to be greater than the corresponding digit in the top number. The digit in the tens place of the bottom number has to be greater than or equal to the corresponding digit on the top.

## To regroup or not to regroup

Sometimes when we subtract one number from another number we "regroup," and sometimes we don't. For example, if we subtract 38 from 375, we can "regroup" by converting a ten to 10 ones:

Find a 3-digit number to subtract from 375 so that:

- You don't have to use regrouping.
- You would naturally use regrouping from the tens to the ones place.
- You would naturally use regrouping from the hundreds place to the tens place.
- You would naturally use regrouping in all places.

In each case, explain how you chose your numbers and complete the problem.

## Comments

Log in to comment## Howard Phillips says:

about 5 yearsIt seems to me that the regrouping thing would be "obviously' necessary if one were to rewrite the calculation in full as

Now clearly the 8 cannot be subtracted from the 5 and so the 70 needs to be rewritten as 60 + 10 Adding the 10 to the 5 is optional, but doing it we get

and returning to the place value representation, 367

The layout shown in the problem formulation is a mess.

Who remembers borrow and payback???????

Also, the layout of my comment has been chewed up by the system.

## Kristin says:

about 5 yearsI agree that writing the numbers in expanded form makes it easier to see this. Many students have only seen the representation where you stack the numbers, and this task is written for them. If students haven't seen expanded form, then it would be good for them to see it at some point--perhaps a task like this would be a good place to introduce it.

And yes, the formatting can be tricky because everything is rendered in HTML using markdown, which is not the same as, for example, a word processing program.

## Kristin says:

about 5 yearsOK--I've added expanded form into the solution of the first two parts. I hope someone will chime in about whether this is an improvement or not.

## Aaron says:

about 5 yearsThe expanded form makes sense and is an important strategy for kids to work with, but I'd like to see it as a different task. Adding it to the solution might not do it justice and doesn't seem to add all that much value to this particular problem.

## Kristin says:

about 5 yearsI agree it should be addressed in another task. We are working on some more illustrations for the 4.NBT standards and will be sure to cover expanded form.

## aor says:

almost 6 yearsI'm surprised to see this task, since the standard 3.NBT.A.2 calls for using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. The comment with the task states that it is appropriate to use after students have learned the standard US algorithm. That doesn't happen until fourth grade.

In addition, this is a poor example for regrouping. To solve 375 - 8 = doesn't it make more sense to subtract the 8 in parts? 375 - 5 = 370. 370 - 3 = 367

## Kristin says:

almost 6 yearsThanks for catching that issue with the alignment (not sure how that slipped by us). I've fixed the standard alignment. I agree that we hope students will not use the standard algorithm to subtract 375-8, but to illustrate regrouping I think it is ok.

## Aaron says:

about 5 yearsI agree that 375-8 is perfectly fine for illustrating regrouping, however I worry about students who will want to use the standard algorithm for everything - whether it is appropriate or not. Providing an example showing a non-strategic choice of solution method sends the (unintended) message that it is ok to use the standard algorithm on problems that don't need it. Fast forward to high school when students use the Law of Sines for every right triangle they encounter even though a simple trig ratio is probably the better strategic choice.

I think the problems we choose as examples (and exercises and assessments) should be consistent with the over all mathematical messages that we want our kids to internalize. In this case, coming up with an example where regrouping is a good strategic choice might be worth a second thought.

## Kristin says:

about 5 yearsYou make a good case. I changed it to 375-48--let me know if you think there is a better example than that.

## Aaron says:

about 5 yearsAt the risk of sounding difficult...I might use 375-38 (or 376-48) instead, only because I wouldn't want my kids to regroup something for which a different strategy might be the better choice i.e., 375-50, then plus 2. I don't know, maybe I'm nit-picking here. In any case, I like 375-48 better than 375-8.

## Kristin says:

about 5 yearsThat's an excellent point--I've made that change.