What it does
A product like 23 × 47 is the area of a rectangle 23 wide and 47 long. Cutting that rectangle into pieces whose areas are easy to work out is the whole secret of mental multiplication. It works because multiplication distributes over addition: (20 + 3) × (40 + 7) = 20 × 40 + 20 × 7 + 3 × 40 + 3 × 7.
This page draws the rectangle and cuts it two ways:
- From a round number: write both numbers as a round base plus or minus a little, for example 103 = 100 + 3 and 97 = 100 − 3. The picture is a big square of the base, with strips added for numbers above it and strips taken away (hatched) for numbers below it, plus a small corner. When the two numbers are the same distance either side of the base, the strips cancel: 103 × 97 = 100² − 3² = 9991.
- Split by place value: hundreds, tens and ones make a grid of easy products, which you add up.
It fits Class 8 Mathematics, the chapter on the distributive property and multiplication tricks (we distribute, yet things multiply), and leads straight into the algebraic identities. The same pictures with letters instead of numbers are the algebraic identity tiles; quick percentages in your head are practised on the percentage bar.
How to use it
- Type two whole numbers (1 to 9999), or tap a quick example.
- Choose From a round number or Split by place value.
- With a round number, the page picks a handy base for you (a hundred near both numbers, otherwise a ten near their middle). Type your own base to try another, or press Pick for me to go back.
- The picture is not to scale: small pieces are drawn bigger so their labels fit. Read the steps under it, and the shortcut line when the numbers sit evenly around a round number.
A worked example
103 × 97 with base 100:
- 103 = 100 + 3 and 97 = 100 − 3.
- Big square: 100 × 100 = 10 000.
- Strip for +3: 100 × 3 = +300. Strip for −3: 100 × (−3) = −300.
- Corner: 3 × (−3) = −9.
- 10 000 + 300 − 300 − 9 = 9991.
98 × 96 with base 100: 10 000 − 200 − 400 + 8 = 9408. Both strips are taken away, and the 2 × 4 corner was taken away twice, so it is added back once.
23 × 47 by place value: 800 + 140 + 120 + 21 = 1081.
Check yourself
Four fresh questions each time: a pair either side of a round number, two numbers just below 100, two just above 100, and a two-digit product. Work them out in your head, type the answer and press Check. Nothing is saved.
FAQ
Which base should I choose? One that is easy to square and close to both numbers. 100 for numbers in the 90s and 100s, 50 for numbers near 50, 1000 for numbers near 1000.
Why is the corner added back when both numbers are below the base? Each hatched strip removes the corner, so it is removed twice but should only go once. (−x) × (−y) = +xy says the same thing.
Is this from the NCERT book? It covers the same topic with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- The distributive property, (a + b)(c + d) as an area, and (B + x)(B + y) = B² + B(x + y) + xy are standard school algebra. Wording, pictures and examples are fibub's own.
- Checked against ordinary multiplication for every pair of numbers up to 400, with bases 10, 50, 100 and the automatic base.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.