Math 131

Spring 04

Models for Multiplication and Division

(Especially for Fractions and Decimals)

 

Multiplication:

 

1.  Repeated Addition

 

            This is the first model children learn for multiplication of natural numbers.  So, for example, we think of  as seven kids with six cookies each, and note that there are  cookies total.

 

            This model easily extends to multiplication of a natural number and a rational number.  For example, we can think of  as representing a situation where 7 kids each eat  of a pizza, so the total number of pizzas eaten is .

 

 

2.  Array Model

 

            With natural numbers, we can represent  with an array of dots or squares:

 

 

 

 

 

 

 

 

Note that with this model, it is easy to see that , which is an illustration of the commutative property; it is much harder to use repeated addition to see that multiplication is commutative.

            In this model, the unit is one square or one dot.  To represent rational number multiplication, we must subdivide our unit.  The large square below represents 1 square unit, and the diagram shows .  The shaded region is the product, which consists of eight small rectangles.  Each small rectangle represents  of the square unit; so the product is .

 

 

 

 

 

 

 

                   

 

 

 

 

 

                                                

 

We can also use an array model to represent products where one or both of the multiplicands is greater than one; we just have to be careful to remember that our unit is a  square; each unit is highlighted in bold below.  The model below shows .  Note that there are 56 grey small rectangles, each of which represents  of a unit.  Also note that by rearranging the grey rectangles, it’s fairly easy to see that 9  units are shaded in.

 

 

 

 

                                                 

 

 

 

 

 

 

 

 

 

                                                             

 

 

3.  Fraction of a Fraction

 

            With simple products, such as , it’s often helpful to view the product as “half of ,” which is clearly .  Using an array model, the horizontal lines represent , while half of the rectangles representing  are shaded further to mark the product.

 

 

 

 

 

 

 

 

 

 

 

 

Division:

 

1.  Portion Model

 

            When asked for a situation to represent the division problem , most students suggest a problem like, “Three kids want to share twelve cookies equally, how many will each child get?” 

            The portion model can get confusing when dealing with fractions.  Suppose we want to model the problem  with a similar problem about cookies.  Do we say that “  of a kid wants to share  of a cookie equally,” which doesn’t exactly make sense?

            We can make the situation a little better by rewording our original problem in a way that makes it easier to mimic: “We have 12 cookies, which represent 3 portions of cookies.  How many cookies are in one portion?”

            Thus, for the other problem, we have  of a cookie, which represents  of a portion, and we want to find out how many cookies are in one portion.  Since  of a cookie represents  of a portion, a whole portion must be  cookies.