Regrouping, Carrying, Borrowing: Why the Vocabulary Changed


Many parents feel frustration when their children come home using arithmetic methods and tools different from what they learned. “Why can’t they just [add/subtract/multiply]?” is a common refrain for caregivers of children in middle elementary. Then, finally, the children are taught the same method the parents know, but… the language has changed! In math, carrying and borrowing are both now called regrouping. Why, oh why, can’t they just teach it the same way with the same words?

Regrouping is a general term that includes both carrying and borrowing. This change of language more accurately describes the math structure, keeps vocabulary consistent all the way through math education, and agrees with how we use the words in standard English.

We can think of almost any whole number as a “part + part = whole.” Regrouping in math can mean either looking at the whole instead of the parts, or changing up the parts that make up the same whole. It is a lot like making change: you might want to turn a \$10 bill into ten \$1 bills, or you might want to turn five \$1 bills into a \$5 bill. In adding multi-digit numbers, we sometimes turn ten 1’s into one 10 (or ten 10’s into one 100, etc.) We used to call this carrying. In subtracting numbers, we sometimes turn one 10 into ten 1’s (or one 100 into ten 10’s, etc.) We used to call this borrowing. Regrouping refers to choosing a different grouping, a different set of “parts”, for the same whole total amount.

An image of four groupings of US currency, each group totaling $10. One group is made up of ten $1 bills. Another group is made up of two $5 bills. The third group is made up of five $1 bills and one $5 bill. The fourth group is made up of one $10 bill.
\$10 can be grouped or regrouped as ten \$1 bills, two \$5 bills, one \$10 bill, or even five \$1 bills and one \$5 bill. Choosing a different grouping of the same total, a grouping that better serves our needs, is called “regrouping” in math.

But why did the language change? Read on to learn more.

More accurately describes what’s going on

When we “carried the one,” we actually took ten of the place value we just added and turned it into one of the next place value (e.g., we added 7+6 to get 13, which we then split up (regrouped) into 10+3, writing the 3 and carrying the 1.) We regrouped by changing up the grouping of the same amount, by re-combining some of the parts. When we “borrowed,” we actually took one of the next place value and turned it into ten of the current place value (e.g., we took one away from the top of the tens column and turned it into an extra ten 1s instead.) We regrouped by changing up the group of the same total amount, by breaking it into parts.

“Carrying” is a type of regrouping where one denomination is “traded in” for equivalent units of the next larger denomination.
The word "borrowing" across the top followed by two rows of US money separated by an arrow pointing to the next row. The first row shows 3 $10 bills plus 3 $1 bills. The second row shows 2 $10 bills and 13 $1 bills.
“Borrowing” is a type of regrouping where one denomination is “traded in” for equivalent units of the next smaller denomination.

Consistency within Math of similar language for similar actions

Where else in math have we ever used the words “borrow” or “carry.” Regrouping, however, will come back into play when students learn about multiplication, fractions, decimals, and even algebra. As just a few examples:

  • When your child learns about factoring, we can use the same grouping language: 24 of something might be grouped as
    • one group of 24 (factors: 1 and 24)
    • two groups of 12 (factors: 2 and 12)
    • three groups of 8 (factors: 3 and 8)
    • four groups of 6 (factors: 4 and 6)
    • six groups of 4 (another view of factors 4 and 6)
    • eight groups of 3 (another view of factors 3 and 8)
    • twelves groups of 2 (another view of factors 2 and 12)
    • twenty-four groups of 1 (another view of factors 1 and 24)
24 pink flowers in animated gif, each scene showing the same 24 flowers grouped in each way found in the bulleted list
Each scene shows the same 24 flowers in various groupings.
  • One of the fractions models your student will be introduced to involves regrouping a whole into 5 parts (called fifths) or 7 parts (called sevenths) or similar.
  • In decimals, regrouping will come up as a combination of the regrouping ideas from addition/subtraction and fractions.
  • In algebra, your student will revisit the idea of factoring, this time with more “generic” and less specific numbers, known as polynomials. You may even have learned a strategy of “factoring by grouping” in algebra. As it turns out, this use of the word grouping is closely related to the same grouping terminology your student is learning now.

And these are only a few of the examples of how this term will be used throughout your child’s math education. A consistent language woven throughout allows students to see the connections and more deeply understand how each new skill is related to and expands upon earlier ideas.

Consistency with standard English usage

When we “borrow from the tens place,” do we ever really intend to give it back? What about carrying? We wrote the one up above the other numbers, but did we really carry it there? This type of disconnect between mathematical language and everyday language has contributed to the view that math rules are arbitrary (they’re not) and that they are not connected to things we do every day (they are). These beliefs have challenged math learners for generations, leading to misunderstandings, frustration, and miscommunications all along the way.

If you’d like a visual demonstration of regrouping, check out the video below from the Math Teacher Barbie YouTube channel. For a broader view of grouping and regrouping, check out the post Grouping and Regrouping in Math.

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Barbra

Barbie has taught math, supported students, and volunteered in classrooms for over 20 years. Her daughter is currently learning math in a Common Core state.

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