Multiplying Mixed Numbers
Prerequisite Skills: Understanding of and ability to convert fractions and mixed numbers,
multiplying fractions by fractions, like and unlike denominators, distributive property, use of the
generic rectangle, and prime factorization
Objective: Students will learn and use three methods for multiplying mixed numbers:
Decomposition with Distributive Property, Converting to Improper Fractions, and Using the
Generic Rectangle
Lesson:
Today we are going to apply what we have learned about multiplying with whole numbers and
multiplying with fractions, to multiplying mixed numbers.
We discovered the other day, that when we multiply fractions, the product is smaller because it is a part of
a part. Are there any predictions or thoughts on what might happen if we multiply two mixed numbers
together?” “Who can tell me what a mixed number is again?” (lead to the conclusion that a mixed
number has some wholes and some parts of wholes)
Let’s look at this expression: 2
3
1
•1
4
3
If you were to estimate a product for this multiplication expression, what would your product be?
If we just look at the whole numbers, we have 2 1 which = 2. But we also have some fractional
parts in our mixed numbers, so we could estimate that our product will be a little more than 2
wholes.
In our estimation, we were looking at the wholes and parts of wholes separately, which leads to
our first method for multiplying mixed numbers, DECOMPOSITION and the DISTRIBUTIVE
PROPERTY. Who can refresh our memories about what decomposition means? And can we
have a volunteer to explain the distributive property?
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DECOMPOSITION and the DISTRIBUTIVE PROPERTY
2
3
1
•1
4
3
Let’s decompose these mixed numbers so they are a little easier to work with
3⎞ ⎛
1⎞
⎛
= ⎜ 2 + ⎟ • ⎜ 1+ ⎟
⎝
4⎠ ⎝
3⎠
And now we can go ahead and use the distributive property
1⎞ 3 ⎛
1⎞
⎛
= 2 ⎜ 1+ ⎟ + ⎜ 1+ ⎟
⎝
3⎠ 4 ⎝
3⎠
Let’s continue to simplify this expression
=2 +
2
3
3
+
+
3
4
12
Now we’ll need to make our denominators common, so we can easily add them
=2 +
=2 +
2⎛ 4⎞
3 ⎛ 3⎞
3
⎜⎝ ⎟⎠ + ⎜⎝ ⎟⎠ +
3 4
4 3
12
8
9
3
+
+
12 12
12
=2 +
8+9+3
12
=2 +
20
12
We know that there are some wholes in here because the numerator is larger than the denominator
=2 +
12
8
+
12 12
We know that the +
12
8
makes a whole and also that +
is not in simplest form,
12
12
so let’s prime factor it
11
= ( 2 + 1) +
=3
2i2i2
2i2i3
2
3
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YOU TRY:
1
3
3
• 3
8
4
IMPROPER FRACTIONS
We can also use what we know about mixed numbers and improper fractions, to simplify this expression
2
3
1
•1
4
3
How can we rewrite these wholes so that they will be easy to add to the fractions
4 4 3
3 1
= ⎛⎜ + + ⎞⎟ • ⎛⎜ + ⎞⎟
⎝4
4
4⎠
⎝3
3⎠
⎛ 4 + 4 + 3 ⎞ ⎛ 3 + 1⎞
=⎜
⎟⎠ • ⎜⎝
⎟
⎝
4
3 ⎠
=
11 4
•
4
3
Now that they are improper fractions, we can simply multiply across (at this point, as a variation, you
could also go directly to prime factoring the numerators and the denominators)
=
11 4
•
4
3
=
44
12
Again we see some wholes in this fraction because the numerator is larger than the denominator, and in
the Grade 5, we usually turn it back into a mixed number. We have a couple of ways we know to do that,
I’m going to use the division method this time.
= 44 ÷ 12
=3
8
12
We know that
8
is not in simplest form because the numerator and the denominator share more factors
12
in common than 1, so we need to prime factor it to get it in simplest form.
11
=3 +
=3
2i2i2
2i2i3
2
3
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YOU TRY:
1
3
3
• 3
8
4
GENERIC RECTANGLE
Since many of you like using the generic rectangle as a tool for multiplication, I’m going to show you
how the generic rectangle works with multiplying mixed numbers. Remember that the generic rectangle
is basically the distributive property, just organized in a rectangle.
2
3
1
•1
4
3
2
1
3
4
+
3
4
2
+
1
3
2
3
3
12
Now we need to add up our partial products
=2+
3 2
3
+ +
4 3 12
We can use the work we did in the previous methods to add these products
=2+
20
12
=2 +
12
8
+
12 12
11
= ( 2 + 1) +
=3
2i2i2
2i2i3
2
3
YOU TRY:
1
3
3
• 3
8
4
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