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Is it regrouping? Upon googling it seems regrouping is borrowing or carrying over collectively. What's the word for not borrowing and carrying over? It's supposedly to train mental computation.

Example 1: When evaluating $21-9$, one can perform borrowing or do the following:

$$21-9$$ $$=10+11-9$$ $$=10+2$$ $$=12$$

Apparently the students who do this first build up mastery of addition and then subtraction facts like 11-9 so the idea is to teach students that they can reduce 21-9 to 11-9 to train mental computation instead of borrowing 10 from 20.

Example 2: For $12+9=21$, one can perform carrying over or do the following:

$$12+9$$ $$=12+8+1$$ $$=20+1$$ $$=21$$

The idea is that students are supposed to have mastered different ways to make 10 (1+9,2+8,3+7, etc) and then to make 20 (11+9,12+8,13+7,etc) so students would want to extract from 9 that which they can use to make 20 with 12, which is 8 to train mental computation instead of carrying over 10 from 2+9=11.

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    $\begingroup$ Are you asking what this is called? It seems to be called "Bridging 10s" or "Making 10s" a la helpingwithmath.com/by_subject/addition/making-10-1oa6.htm $\endgroup$
    – Opal E
    Commented Aug 11, 2017 at 19:41
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    $\begingroup$ Mathematically, it's all just a direct application of the associative property. $\endgroup$
    – rnrstopstraffic
    Commented Aug 12, 2017 at 4:55
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    $\begingroup$ From your examples: decomposing and regrouping. Regarding the above comments, I favor the term friendly numbers (comes up in compensation strategies, too) and, just as associativity can be phrased as regrouping, so, too, can commutativity be phrased as reordering. $\endgroup$
    – Benjamin Dickman
    Commented Aug 13, 2017 at 4:34
  • $\begingroup$ @OpalE Thanks! Post as answer? Do you know where I can read more about this? Perhaps some debates or articles about which method is better or something? $\endgroup$
    – BCLC
    Commented Aug 16, 2017 at 9:57

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This topic seems to be called "Bridging 10s" or "Making 10s": http://www.helpingwithmath.com/by_subject/addition/making-10-1oa6.htm

I'm not sure that you will find many discussions about the utility of this in particular, but many of the debates around "Common Core mathematics" include disagreements on whether it is beneficial for students to learn alternate methods of computing the same sum.

I have found a description of a number of different addition strategies here: https://www.whatihavelearnedteaching.com/models-strategies-for-two-digit-addition-subtraction/

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  • $\begingroup$ thanks Opal E! i have no idea why i upvoted but didn't accept your answer already $\endgroup$
    – BCLC
    Commented Oct 10, 2021 at 18:27
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When I studied those things, in Spanish it's was called "suma algebraica", i. e., algebraic sum. Numbers in a term could be positive or negative.

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  • $\begingroup$ Welcome to the site! Unfortunately, you don't address the true question of the original post (OP) which is about rewriting a difference to include a sum, specifically to prime the concept of "borrowing". It appears that "suma algebraica" simply refers to any combination of addition and subtraction: matematicabasica-cdl.blogspot.com/2011/10/suma-algebraica.html $\endgroup$
    – Brendan W. Sullivan
    Commented Aug 17, 2017 at 23:36

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