Number - Addition and Subtraction with Regrouping

Place Value Alignment: Before adding or subtracting, digits must be lined up vertically according to their place value ($Hundreds, Tens, Ones$). Visually, imagine vertical grid lines separating each column to ensure that $Ones$ are stacked over $Ones$ and $Tens$ are stacked over $Tens$.

Regrouping in Addition (Carrying): When the sum of digits in a column is $10$ or more, you must regroup. For example, if the $Ones$ column sum is $13$, you write the $3$ in the $Ones$ place and carry the $1$ (which represents $10$) over to the $Tens$ column. Visually, this is represented by writing a small $+1$ above the $Tens$ digit.

Regrouping in Subtraction (Borrowing): If the top digit in a column is smaller than the bottom digit, you must borrow from the next place value. Visually, this looks like crossing out the digit in the left column, reducing it by $1$, and placing a small $1$ in front of the digit in your current column to turn it into a 'teen' number (e.g., $2$ becomes $12$).

Subtracting Across Zeros: When you need to borrow but the next column is a $0$, you must continue to the next place value (like the $Hundreds$) to borrow. Visually, this creates a chain reaction where the $Hundreds$ digit decreases, the $0$ in the $Tens$ place becomes a $9$, and the $Ones$ place finally receives the $10$ it needs.

The Inverse Relationship: Addition and subtraction are opposite operations. You can use addition to check a subtraction problem and vice versa. If $a - b = c$, then $c + b$ must equal $a$. This is a helpful 'loop' visual to verify if your calculations are correct.

Base-10 Visuals: Addition and subtraction can be modeled using Base-10 blocks. A large square 'flat' represents $100$, a long 'rod' represents $10$, and a tiny 'cube' represents $1$. Regrouping is simply swapping $10$ cubes for $1$ rod, or breaking $1$ rod into $10$ cubes.