What a decimal really is
A decimal is a fraction with a power-of-ten on the bottom.
That single fact is the whole lesson. 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
Drag the sliders. The dark cells are the product.
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.
Three approaches, one answer
The Alberta curriculum names three approaches to decimal multiplication: expressing each decimal as a fraction, using the standard algorithm, and drawing an area model. They all land in the same place.
Three approaches, one answer
as fractions
410×310
= 12100
= 0.12
standard algorithm
| 0.4 | |
| × | 0.3 |
| 0.12 |
one place + one place = two places
area model
12100 = 0.12
Same number, three ways of seeing it.
The standard algorithm is the fastest at the till. The fraction form is the easiest to explain. The area model is the easiest to picture. Three views, one number.
Multiplying makes things bigger... unless
Just like with fractions, the everyday rule "multiplying makes things bigger" doesn't survive contact with decimals between 0 and 1.
0.5 of 0.4 is less than half of 0.4 — it's smaller than each
factor. Same reason as fractions: a decimal less than 1 is a number
less than 1, and multiplying by a number less than 1 shrinks.
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 same idea works the other way too. 36 ÷ 4 = 9, and so does
3.6 ÷ 0.4, and so does 0.36 ÷ 0.04. They're three views of the
same division.
Where it shows up in real life
Alberta's sales tax is 5% — the only tax on a regular receipt, since
Alberta has no PST. Five percent is 0.05 written as a decimal, and
the tax line on a $48.00 purchase is just 0.05 × 48.00.
GST math at the till
Receipt — Alberta
No PST in Alberta — the GST line is the only tax.
The tip-jar math is the same shape. A 15% tip is 0.15 × bill. A
30%-off coupon is 0.30 × sticker price taken off. Every cash
register in the province runs decimal multiplication a few thousand
times a day.
Worksheet
These aren't graded. The goal is fluency — try one approach per question, then pick the one that fits your head best.
Question 1 of 4
Try it
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
Going further
Decimal multiplication shows up everywhere in the next lesson. Order of operations applies the same precedence rules to integers, fractions, AND decimals — once a problem mixes them, BEDMAS is what keeps everything tidy.
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.