In this lesson
What a decimal really is
A decimal is a fraction with a power-of-ten on the bottom.
Multiplying and dividing decimals is multiplying and dividing fractions. The rules don't change, only the way the answer is written.
So . Top × top, bottom × bottom. A tenth times a tenth is a
hundredth, and 12 of them is 0.12.
See it on a 10 × 10 grid
The unit square holds 100 small cells. Shade a columns from the
left and b rows from the bottom. The cells that are shaded both
ways are the product, drawn at literal scale.
Try it
Click any cell, or drag the sliders.
12 shaded cells out of 100
Each row is one tenth. Each column is one tenth. The whole grid is one — so 100 small cells equal 1.
Try 0.4 × 0.3. The dark cells form a 4-by-3 block, 12 cells out
of 100, and 0.12 reads off the readout. Try 0.5 × 0.4: half the
columns, four-tenths of the rows, 20 cells out of 100, 0.20.
The grid is the same picture as the fraction-multiplication grid from the last lesson, just rescaled. Tenths instead of thirds and quarters.
When the result grows or shrinks
A decimal between 0 and 1 acts like a proper fraction. Multiplying by it shrinks the result; dividing by it grows it. Same rule as fraction multiplication and division, just written with decimal points.
For example, 0.5 × 0.4 = 0.2 is smaller than each factor, while
12 ÷ 0.5 = 24 is much larger than the dividend. Both are
consequences of the size rule.
Dividing decimals: the equivalent-expressions trick
Division is the harder direction. There's a clean trick from the curriculum: multiplying both the dividend and the divisor by the same factor doesn't change the quotient. Use that to turn a decimal-÷-decimal into an integer-÷-integer.
So 0.6 ÷ 0.2 is the same as 6 ÷ 2, which is 3. Multiply both
sides by 10. Same answer, easier arithmetic.
Try it
Multiply both numbers by 10. The quotient stays the same.
Scaling the dividend and divisor by the same factor doesn't change the quotient — it just rewrites the same division in a tidier form. A stacked ladder of division expressions. The student picks a starting decimal-by-decimal expression and taps a 'multiply both by 10' button; the expression rewrites as the dividend and divisor each gain a factor of ten, and the ladder grows by a row. The quotient is identical on every row.
quotient: 3
The trick also handles the case where the dividend is whole and
the divisor is a decimal less than 1. 12 ÷ 0.5 asks "how many
halves fit into 12?" The answer is 24. Multiply both sides
by 10 to clear the decimal:
The same identity in three places: 36 ÷ 4 = 9, 3.6 ÷ 0.4 = 9,
and 0.36 ÷ 0.04 = 9. Three views, one division.
Worksheet
Try these to put the idea into practice. The goal is fluency. Work one approach per question, then keep whichever fits your head best.
Practice · Not graded
MA.7.NUM.4Practice the idea
01 / 05
What is 0.4 × 0.3?
Multiple choice: what is zero-point-four times zero-point-three? Four cards: zero-point-twelve, zero-point-seven, one-point-two, zero-point-one-two.Decimal products
0.5 × 0.4 = 0.20
0.3 × 0.3 = 0.09
0.6 × 0.2 = 0.12
Show common mistakes
Student says
“0.4 × 0.3 = 1.2. I just multiplied 4 × 3.”
What it reveals
Forgot the place-value piece. Multiplying tenths by tenths gives hundredths, not tenths.
Targeted response
As fractions: 4/10 × 3/10 = 12/100 = 0.12. Two decimal places in, two decimal places out. The DecimalAreaShader shows it: a 4-by-3 block of small cells covers 12/100 of the unit square.
Student says
“0.6 ÷ 0.2 = 0.3. Divided 6 by 2, kept the decimal.”
What it reveals
Tried to divide the digits without rescaling. Division of decimals isn't the same as division of the digits with a decimal point appended.
Targeted response
Multiply both sides by 10 first: 0.6 ÷ 0.2 = 6 ÷ 2 = 3. The dividend and divisor scale together, so the quotient stays the same, and now the arithmetic is whole-number easy.
Student says
“12 ÷ 0.5 = 6. Division shrinks.”
What it reveals
Whole-number bias. Expected every quotient to be smaller than the dividend, but dividing by a decimal less than 1 grows it.
Targeted response
0.5 = 1/2. Dividing by 1/2 asks 'how many halves fit into 12?' Twenty-four. The size rule: dividing by a value less than 1 grows the result.
Going further
Decimal multiplication shows up throughout the next lesson. Order of operations applies the same precedence rules to integers, fractions, and decimals; BEDMAS resolves the order whether the numbers are mixed or all of one kind.
In Grade 8, scientific notation reads decimal multiplication
backwards: 3.2 × 10⁻⁴ is just 3.2 ÷ 10000 = 0.00032. And in
proportional reasoning (the very next outcome, MA.7.NUM.5),
percentages are decimal multiplication: 35% of a number is
just 0.35 times the number.