Intuition The one core ideaMultiplication is a fast way to add equal groups , and every big multiplication is just breaking numbers into place-value chunks, multiplying the chunks, and adding . Master three things — counting equal groups, place value, and "splitting then adding" (the distributive law) — and every table, long-multiplication column, and area-model rectangle becomes obvious.
This page builds, from absolute zero, every symbol and idea the parent note Multiplication topic leans on. Nothing here assumes you've seen the notation before. We go in order — each brick sits on the one below it.
Before we can multiply, we need the things we count with .
The numbers 0 , 1 , 2 , 3 , 4 , …
Definition The equals sign= = the same amount ." It is a balance, not an instruction. 3 + 1 = 4 the very same amount as "four" — two names for one quantity.
Look at the picture: both pans hold four dots. The bar = =
Definition Addition, written+ a + b a b +
Intuition Why we care hereThe parent note says multiplication is repeated addition . So addition is the engine , and multiplication is the shortcut for when the piles are all the same size . If you can't picture + ×
Adding two piles is easy. Adding the same pile over and over (3 + 3 + 3 + 3 Addition — carrying and place value for the carrying details we reuse later.
Several piles that each hold the same number of things. "Four groups of three" = four piles, three things in each.
Intuition Read the pictureFour boxes, each with three orange dots. Counting them one dot at a time is tedious. Counting by groups — "three, six, nine, twelve" — is faster. That "counting by equal jumps" is multiplication.
Definition The multiplication sign× a × b a b a b a × b = a copies b + b + ⋯ + b 4 × 3 = 3 + 3 + 3 + 3 = 12 ×
Definition Factor and productIn 4 × 3 = 12 4 3 factors ; the answer (12 product . Factor = ingredient; product = result.
× × Algebra — Expanding Brackets , the letter x × x
To see why 4 × 3 = 3 × 4 grid (a rectangle of dots).
Intuition Same dots, two countsCount the rows: 3 + 3 + 3 + 3 = 12 4 + 4 + 4 = 12 Same dots , so the totals must match. That's why a × b = b × a commutativity .
⇒ ⇒ therefore " or "which forces." When we write "same dots ⇒ a × b = b × a the picture on the left proves the statement on the right .
This is the single most important brick for long multiplication.
A digit's worth depends on its column . In 47 4 tens column so it means 40 7 ones column so it means 7 47 = 40 + 7 = 4 × 10 + 7 × 1.
Intuition Read the pictureThe number 47 4 7 left multiplies its worth by 10 placeholder zero the parent note adds in the second row of long multiplication.
Full details live in Place Value & Number Systems — but the box above is all the parent note needs.
The engine of every multiplication method.
( ) first, as one lump . ( b + c ) b c
Intuition Why the topic can't live without it47 = 40 + 7 47 × 6 = ( 40 + 7 ) × 6 = 40 × 6 + 7 × 6 never multiply the scary number whole — you split it by place value and add friendly pieces. Long multiplication and the area model are both just this law used repeatedly. Deep-dived in Distributive Law .
Common mistake Don't stop halfwaya × ( b + c ) = a × b + c a every term inside the brackets. Picture a rectangle of height a b c both sub-rectangles are a
Definition Squaring — the special casea × a a × a a squared ," written a 2 2 square because a × a a Squares & Square Roots .
÷ 12 ÷ 3 3 12 × 4 × 3 = 12 12 ÷ 3 = 4 is knowing division; see Division — inverse of multiplication .
Addition plus combine piles
Equal groups same size piles
Commutativity a times b = b times a
Times 10 shift left add a zero
Distributive law split then add
Tables 1 to 20 learn half
Read it upward: whole numbers feed addition, addition feeds equal groups, equal groups feed multiplication; place value plus the distributive law feed long multiplication and the area model.
Cover the right side and see if you can answer each before revealing.
What does the whole number 0 , 1 , 2 , 3 , … All counting amounts including zero; no fractions or negatives.
What does = The two sides are the same amount — a balanced beam, not an instruction.
Write a × b b a a b
Name the parts of 4 × 3 = 12 4 3 12
Why does a × b = b × a One dot grid can be counted by rows or by columns — same dots, same total.
What does ⇒ "Therefore / which forces" — the left proves the right.
Break 47 47 = 40 + 7 = 4 × 10 + 7 × 1
Why does × 10 Every digit shifts one column left, so its worth grows tenfold.
State the distributive law in words. a ( b + c ) a b a c a ( b + c ) = ab + a c
Why is a × ( b + c ) = ab + c a every term inside the bracket, so it's ab + a c
What does a 2 a × a a
What question does 12 ÷ 3 How many groups of 3 12
Parent topic — Multiplication Addition — carrying and place value Place Value & Number Systems Distributive Law Division — inverse of multiplication Squares & Square Roots Algebra — Expanding Brackets 
