Name: ______________________
Multiplication of Decimals
Multiplication of a decimal by a whole number can be represented by the repeated addition
model. For example, 3  0.14 means add 0.14 three times, regroup, and simplify, as shown below:
3  0.14 =
Estimate: roughly 0.45
Actual: 0.42
Regrouping
There are three separate cases of multiplication, each with its own representation. The first case has
already been described: multiplication of a whole number by a decimal.
Try these two multiplication problems, modeling what is shown in the example above:
2  0.43 =
Estimate:
Actual Answer:
Regrouping
4  0.19 =
Estimate:
Actual Answer:
Regrouping
©Dr Brian Beaudrie
pg. 1
How about this one?
5  0.26 =
Estimate:
Actual Answer:
+
Multiplication of a whole number by a decimal is an important step in helping a child conceptualize
multiplication of two decimals.
The second case is the opposite of the first case: multiplication of a decimal by a whole number. For
example, suppose you going to find the solution to 0.28  2. Of course, you could use the
__________________ law, which allows you to rewrite the equation as 2  0.28; but since most 3rd
graders don’t understand that law, let’s try a different approach:
Since we are multiplying a decimal by 2 (in this case) begin with two decimal squares, and shade
each with 0.28. Then, combine them into one decimal square (with regrouping, if necessary).
0.28  2 = 0.56
joining
regroup
Go ahead and try it: (you can combine the joining and regrouping steps):
0.42  2 =
Estimate:
Actual Answer:
join and regroup
The third case involves multiplying a decimal by a decimal. For this activity, we will only do tenths
multiplied by tenths. The method we employ might look a bit familiar… we did something similar to it
when we did fractions.
©Dr Brian Beaudrie
pg. 2
Suppose we wanted to multiply 0.3 by 0.4. Looking at our decimal square we would do the following:
We would divide the tenths
along the horizontal axis into
tenths as well, as many tenths as
we needed. From there, we
would shade in up to 0.3 and
over to 0.4. From that, we
would then count the shaded
boxes to obtain our answer.
So, 0.3  0.4 = .12
Try out this idea on a few examples:
0.5  0.7 =
0.3  0.8 =
Questions:
1) How could you represent thousandths using decimal squares?
2) Why do we only do examples that show tenths multiplied by tenths? Why would showing an
example having tenths multiplied by hundredths be difficult?
3) What are some of the limitations of using the decimal squares?
©Dr Brian Beaudrie
pg. 3
Division of Decimals
With division of decimals, there are also three separate cases to consider. The first case, division of a
decimal by a whole number, is done using the partitive model of division.
Example: The problem 0.54  3 is essentially asking to
separate 0.54 into three sets of equal size. If we were
to estimate a range for our answer, we know that
our answer will be more than 0.1 but less than 0.2
because we can split the columns (tenths) into three
equal groups once, but not twice. Therefore, to use
this method, we will divide up the columns (tenths)
first (one for each group), then divide the remaining
squares (hundredths) into 3 equal groups, as shown
to the right. We can then see that each color uses
exactly eighteen squares, so we know: 0.54  3 = .18
Use the method (estimating first) described above to do the following examples.
a) 0.56  4 =
Estimate:
Actual:
b) 0.72  3 =
Estimate:
Actual:
For the second case, dividing a whole number by a decimal, the repeated subtraction
(measurement) method of division is the model to employ. In this method, you count the number of
sets equivalent to the divisor (the second number) that are in the dividend (the first number). For
example, in the problem: 3  0.6 = , you are being asked “how many sets of 0.6 can you take away
from three whole units?” So, starting with three whole units, you will make groups of size 0.6, as shown
below:
So, we know that 3  0.6 = 5, since we ended up with five different colors.
©Dr Brian Beaudrie
pg. 4
Use the repeated subtraction model to find the correct answer to the following problem:
2  0.5 =
Estimate:
Actual:
The third case concerns dividing one
decimal by another decimal. For this,
you will also use the repeated
subtraction model of division. For
example, with 0.76  0.19, you need to
find out how many sets of 0.19 exist in
0.76. So, you would “pull out” groups
of 19 little squares at a time, as shown
on the left. So, you will have:
0.76  0.19 = 4.
What happens, though, when you
don’t have a whole number solution?
Well, consider 0.23  0.04. The decimal
square to the right shows what
happens…you end up with six groups,
each size 0.04, with three extra groups
left over. If I look at 0.03/0.04, I can see
that it is exactly 0.75 of another group
of size 0.04. Therefore, I have:
0.23  0.04 = 6.75
Use your decimal squares and the method described above to find the following quotients. Estimate
first where appropriate.
a) 0.72  0.24 =
Estimate:
©Dr Brian Beaudrie
Actual:
b) 0.55  0.2 =
Estimate:
Actual:
pg. 5