Readings for Session 9 – Models for Multiplication of Whole Numbers
How does the operation
of multiplication used in the following problem relate to sets?
The Moorhead Youth
Hockey League has eight teams in an age group with fifteen
players on each team. How many kids play hockey in this age
group?
The league consists of a set of
eight teams and each team is a set of fifteen players. We note
that each player is on only one team. We consider all the
pairings of a player with a specific team to be a set. The
cardinality of this set of player-team pairings is 120. The
league with eight teams of fifteen players on a team has a total
of 120 kids playing hockey.
Team 1
Team 2
Team 3
Team 4
Team 5
Team 6
Team 7
Team 8
Also, note how the solution is the cardinality of the union of the eight disjoint sets (teams). In other words, we may consider the problem to be an addition problem where
8
× 15 = 15 + 15 +
15 + 15 + 15 + 15 + 15 + 15 = 120.
This relationship
motivates the following definition for multiplication of whole
numbers.
Repeated
Addition Definition for Multiplication of Whole Numbers.
Given a whole number a
≠
0 of equal sets, each containing
b elements, we define
0
∙
b
= 0 and
The numbers
a and
b are called
factors and
ab is the
product.
Notations used for multiplication:
ab = a
∙
b = a
×
b = a
∗
b = (a)(b).
When we add these addends, the sum is 210, so 5
×
42 = 210.
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