Grades 3–5 · Mathematics
Why does regrouping tens and ones keep the same value?
A number can have more than one equivalent representation. Equivalent means equal in value even when the pieces are organized differently. In base ten, one ten has exactly the same value as ten single ones. This equality makes regrouping possible.
In the diagramRegrouping changes form, not valueBefore exchangeAfter exchange2 tens + 14 ones3 tens + 4 onesBoth representations have value 34.Ten ones are exchanged for one ten.
The picture begins with two tens and fourteen ones. Their combined value is twenty plus fourteen, or thirty-four. Ten of the loose ones can become another ten. After this exchange there are three tens and four ones: thirty plus four is still thirty-four. The exchange changes the counts of tens and ones together.
Nothing has been created, discarded or shared among recipients. The same whole is represented with different units. This is the reason a written calculation can move ten ones into the tens column without changing the amount. The written digits summarize the grouping; the equality of ten ones and one ten supplies the explanation.
About this explanation
Starting knowledge: Recognize tens and ones and add small whole-number counts.
Sources
- Common Core State Standards Initiative — Grade 3: Place-value reasoning in arithmetic