ED 411/518
Fall 2006
Modeling Multiplication
(in response to Class #10 assignment)
This document reviews and explains the area and repeated addition models for multiplication. The
examples were chosen to be related (using 3, 2, 0.3, 0.2 as factors) so that you can easily compare cases
to see what is similar and different.
In all of the examples below, a square with an area of 1 square unit and side lengths of 1 (linear) unit
will be considered the whole or the unit.
Repeated Addition Interpretation
The repeated addition interpretation of multiplication defines a x b as: b added together a times.
Another way to think about this is that a x b means: the sum of a groups of b.
3 x 2:
Using a repeated addition interpretation, the problem 3 x 2 is defined as 2 + 2 + 2, or as the sum of 3
groups of 2. That is, the 3 refers to the number of groups, and the 2 indicates the size of each group.
This problem can be modeled as follows:
Counting the number of unit squares in 3 groups of 2 shows that 3 x 2 = 6.
Mathematics Methods Planning Group
University of Michigan
ED 411/518
Fall 2006
3 x 0.2:
In the problem 3 x 0.2, the 3 refers to the number of groups, and the 0.2 indicates the size of each
group. That is, 3 x 0.2 means 0.2 + 0.2 + 0.2, or 3 groups of 0.2.
0.2 can be represented by cutting a unit square into tenths and shading two of the tenths. Thus, the gray
region below represents two groups of size 0.1, or a total of one group of size 0.2:
3 x 0.2 can be represented as adding 0.2 together 3 times, in other words, as the sum of 3 groups of
0.2:
Another way to represent 3 x 0.2 is to show the 3 groups of 0.2 on the same whole:
Both representations show that 3 x 0.2 = 0.6 because, with 3 groups of 0.2, six-tenths of a whole (0.6) is
shaded.
Mathematics Methods Planning Group
University of Michigan
ED 411/518
Fall 2006
0.3 x 0.2:
Just as before, in a repeated addition interpretation, the first factor indicates the number of groups and the
second factor indicates the size of a group. Therefore, 0.3 x 0.2 is interpreted as 0.3 of a group of 0.2.
To model this, first represent the number in one group of size 0.2:
Now we want to see 0.3 groups of 0.2. The 0.3 indicates the number of groups of 0.2. Note that because
0.3 is between 0 and 1, this means that there will not be a whole group of 0.2, only part of a group.
To find 0.3 of a group of 0.2, first cut 0.2 into tenths, or ten equal parts:
Then, to show 0.3 of a group of 0.2, mark off three of the tenths:
Thus, 0.3 x 0.2, or three-tenths of a group of 0.2, is shown by the diagonal marks above. To find the
value of this product, consider how much is represented by the diagonally-marked section. To figure that
out, it helps to partition the whole into smaller squares of equal size.
Since each small square is 0.01 of the whole and six of the small squares are marked, the diagonally-
marked section represents 0.06, so 0.3 x 0.2 = 0.06.
Mathematics Methods Planning Group
University of Michigan
ED 411/518
Fall 2006
Below are two other examples of a repeated addition interpretation from the Class #10 assignment:
2 x 0.8:
Using repeated addition, 2 x 0.8 is interpreted as 0.8 + 0.8 or 2 groups of 0.8:
The shaded amount fills one whole plus another six-tenths. Therefore, 2 x 0.8 = 1.6.
0.2 x 0.8:
Using repeated addition, 0.2 x 0.8 is interpreted as 0.2 of a group of 0.8. To model this, start with a
group of 0.8:
To find 0.2 of a group of 0.8, cut the 0.8 into tenths, and then mark off two of the tenths:
To find the value of this product, consider what portion the diagonally marked section is of the entire
whole by partitioning the whole into the smaller squares:
Since each small square is 0.01 of the whole and 16 of the small squares are marked, 0.2 x 0.8 = 0.16.
Mathematics Methods Planning Group
University of Michigan
ED 411/518
Fall 2006
Area Interpretation
The area interpretation of multiplication defines a x b as: the area of a rectangle of width a and length b.
2 x 0.8:
To model 2 x 0.8, draw a rectangle with side lengths of 2 (linear) units and 0.8 (linear) units:
The product is the area inside the rectangle (shown in gray above). Since each small square is 0.01 of
the whole, and there are 160 small squares inside the 2 by 0.8 rectangle, this is more than one whole
(which is 100 small squares). Therefore, the shaded area, represented as 2 x 0.8, equals 1.6 or one
whole and 0.6 of a whole.
0.2 x 0.8:
To model 0.2 x 0.8, draw a rectangle with side lengths of 0.2 (linear) units and 0.8 (linear) units:
The product is the area inside the rectangle (shown in gray above). Since each small square is 0.01 of
the whole, and there are 16 small squares inside the 0.2 by 0.8 rectangle, 0.2 x 0.8 = 0.16. 16 squares
that are each 0.01 in area equal sixteen hundredths, or 0.16. 1
1
It is important to notice that both interpretations give rise to the same numerical values for the products, even though there are
different units attached to the factors and the products in the two interpretations. One nice feature of the area interpretation is
that it makes visible the commutativity of multiplication – a x b = b x a – something that is not at all evident in the repeated
addition interpretation; why are 7 baskets of 5 apples the same number of apples as 5 baskets of 7 apples? We often show this
using rectangular arrays, which, when using square tiles, are a kind of representation midway between repeated addition and the
area interpretation.
Mathematics Methods Planning Group
University of Michigan
ED 411/518
Fall 2006
Mathematics Methods Planning Group
University of Michigan