Toolbox

A formula sheet plus reference tools for calculator, equation solver, units, elements, graphs and molecules.

Toolbox · Column Arithmetic (+, −, ×, ÷)

Column Arithmetic (+, −, ×, ÷)

One box per digit; tap a box to blank it and build a worksheet

Formula

\begin{aligned}&a=a_n10^n+\cdots+a_1\cdot10+a_0\\[2pt]&a+b=\sum_i(a_i+b_i+c_i)\,10^i\\[2pt]&a-b=\sum_i(a_i-b_i-c_i)\,10^i\\[2pt]&a\times b=\sum_j(a\times b_j)\,10^j\\[2pt]&a=bq+r,\quad 0\le r<b\end{aligned}

One box per digit: line up equal places, carry ten to the left, borrow ten when short; division takes the highest place first.

Edit

Boxes 9

ExpressionAddition · 342 + 289

BlanksTap a digit box to blank it: the preview keeps an empty box for the student to fill. Tap again to undo.

Presets

Preview

The preview is the worksheet: blanked boxes stay empty, deleted boxes leave no gap.

Result631

Expression342 + 289

Carries2

Result digits3

342 + 289 = 631Boxes9Blanked0Deleted0

Place-value breakdownEach number is split by place value; the bar length is what that digit is really worth.

hundreds tens units

Steps

5 / 5

1Line up
$342+289$
Line up equal place values (units under units) and start from the units column.
2Add column
$2+9=11$
units: 2 + 9 = 11, write 1 and carry 1 to the tens.
3Add column
$4+8+1=13$
tens: 4 + 8 + 1 = 13, write 3 and carry 1 to the hundreds.
4Add column
$3+2+1=6$
hundreds: 3 + 2 + 1 = 6, write 6.
5Result
$342+289=631$
So 342 + 289 = 631.

Concept

A column layout puts place value on show: once the boxes line up, every add, subtract and multiply stays inside its own column, with carries and borrows noted in small digits next door.

Addition and subtraction must line up equal places, or 3 tens and 3 units get confused; in multiplication each partial product ends in the place of its multiplier digit; division starts from the highest place and writes the quotient above the matching digit.

One box per digit pins every step to a square: blank a box and the student has to supply exactly that digit, using the same layout as the answer key.

Going further

Why division starts on the left: Division is fair sharing: what is left in a high place must be merged with the next digit before sharing again, so it has to run from large to small. Addition and multiplication start on the right because a carry or borrow only reaches one place to the left.
Why partial products shift: Multiplying 123 by the tens digit 4 really gives 4 tens copies of 123, that is 4920, so the last digit lands in the tens column. The shift is just place value at work.
Predict the number of digits: An n-digit times an m-digit number has at most n + m digits; an n-digit dividend divided by an m-digit divisor gives at most n − m + 1 quotient digits. Counting digits first catches most slips.
Three ways to check: Check addition with subtraction, subtraction with addition, and division with divisor × quotient + remainder = dividend; for multiplication swap the factors and multiply again.
Estimate first, then write the column: Round 123 × 45 to 120 × 45 = 5400; a column answer of 5535 is the same size, so it is plausible. A wildly different answer usually means a shifted row or a missing partial product.

Common mistakes

When borrowing twice in a row, subtract 1 from the column that lent before using it.
Never drop a 0 in the quotient: where the digit is too small to divide, write 0 in that place.
Add every partial product at the end; stopping after the first one is the most common mistake.
The remainder must be smaller than the divisor, otherwise the quotient digit was too small.

Page keywordscolumn additioncolumn subtractionlong multiplicationlong divisioncolumn method worksheetcarrying and borrowingvertical arithmetichow to do column additioncolumn subtraction with borrowinglong multiplication stepslong division with remainderhow to line up column arithmeticcarry and borrow worksheetblank column sum worksheetone box per digit templatevertical arithmetic generatorwhy partial products shiftdivision with zero in the quotientcheck division with multiplication

Related tools

Related chapters