Multiplication as Repeated Addition
Fred has three trays of cookies.
Each tray has 4 cookies on it. How
many cookies are there altogether?
4 + 4 + 4 = 12
3 x 4 = 12
Multiplication as an Array or Area Model
Jan’s patio is 3 square blocks wide, and
4 square blocks long. How many blocks make up her patio?
4 3
4
3
Dave’s wall measures 12’ x 8’. If he wants to paint that wall, he needs enough paint to
cover how many square feet?
Multiplication as a Cartesian Product
Jonas has 3 hats: Grey, Orange, Black. He also has 4 scarves: White, Red, Purple, &
Taupe. How many hat-and-scarf possibilities are there?
W W W
R R R
G P O P B P
T T T
{(G,W), (G,R), (G,P), (G,T),
(O,W), (O,R), (O,P), (O,T),
(B,W), (B,R), (B,P), (B,T) } ….. 3 x 4 Total
Properties of Multiplication
Closure
If, for any x & y Î A, (x * y) ÎA ,
Then A is Closed under addition
Ex1 { 0, 1, 2, 3 … } CLOSED
Ex2 { 1, 3, 5, 7, … } CLOSED
Ex3 { 2, 7, 12, 17…} NOT CLOSED
Properties of Multiplication for Whole #s
Commutative
For two whole #s a & b, a * b = b * a
i) (7*3) * 1 = 1 * (7*3)
ii) (8*2) * 5 = 5 * (8*2)
iii) (8*2) * 5 = 5 * (2*8)
Properties of Multiplication for Whole #s
Associative
For three whole #s a & b & c,
(a * b) * c = a * (b * c)
(7*3) * 1 = 7 * (3*1)
Properties of Multiplication for Whole #s
Multiplicative Identity
1 is the Multiplicative Identity: A unique
number such that, for any whole # x,
x * 1 = x = 1 * x
Distributive Property of Multiplication…
…Over Addition:
a (b + c) = ab + ac
21(12) = 21(10 + 2)
= 21*10 + 21*2
= 210 + 42
…Over Subtraction:
a (b - c) = ab - ac
23(18) = 23(20 - 2)
= 23*20 - 23*2
= 460 - 46
YOU TRY: 6 x 47 and 7 x 19
Division as Sharing
(Partitive Model)
Sam has 12 cookies that he wants to give
all away to 3 friends. He wants each friend
to have the same number of cookies. How
many cookies will each friend get?
Division as Repeated Subtraction
(Measurement Model)
Sam has 12 cookies that he wants to give
all away to as many friends as possible.
He decides to make gift baskets with 3
cookies in each basket. How many friends will each receive cookies?