Math Background
Multiplying by a Fraction
We develop fraction meanings by relating them to the equal groups meaning of
multiplication.
Markers come in sets of 6.
Alta has 3 sets.
6 taken 3 times ⴝ
3ⴛ6ⴝ
markers
3 sets of 6
markers
Isabel has ᎏ31ᎏ of a set of 6 markers.
6 taken ᎏ31ᎏ times ⴝ
markers
ᎏ1ᎏ ⴛ 6 ⴝ
markers
3
1
_ set of 6
3
ᎏ1ᎏ of 6 ⴝ ᎏ1ᎏ ⴛ 6 ⴝ 6 ⴜ 3 ⴝ ᎏ6ᎏ ⴝ 2
3
3
3
1
Because ᎏ3ᎏ is 1 of 3 equal parts, finding ᎏ31ᎏ ⴛ 6 requires dividing 6 into 3 equal parts.
Students review multiplication comparison problems, where the ᎏ31ᎏ times as many
1ᎏ
language helps to establish the meaning of multiplying by a unit fraction, ᎏd
, as
1ᎏ
meaning dividing by d (the reciprocal of ᎏd
).
Students solve more difficult fractional multiplication problems that do not divide
evenly, by multiplying each 1 whole or each unit fraction that makes the
multiplied quantity.
ᎏ1ᎏ ⴛ 2 ⴝ ᎏ1ᎏ ⴙ ᎏ1ᎏ ⴝ ᎏ2ᎏ
7
7
7
7
0
1
2
0
1
2
ᎏ3ᎏ ⴛ 2 ⴝ ᎏ3ᎏ ⴙ ᎏ3ᎏ ⴝ ᎏ6ᎏ
7
7
7
7
nᎏ
n ⴛᎏ
w
1ᎏ
ᎏⴝᎏ
They also see that finding ᎏd
ⴛ w ⴝ n ⴛ ᎏd
ⴛ w ⴝ n ⴛ ᎏw
.
d
d
3
1
3 ⴛᎏ
2
.
In the second example above, with w ⴝ 2: ᎏ7ᎏ ⴛ 2 ⴝ 3 ⴛ ᎏ7ᎏ ⴛ 2 ⴝ 3 ⴛ ᎏ72ᎏ ⴝ ᎏ
7
In all of this work, students are asked to observe that multiplying by a fraction
makes a smaller number because you are taking only a part of the whole number.
This is true for proper fractions.
Multiplication and Division with Fractions
Teaching Unit 5 (Continued)
General Pattern
Multiplying a Fraction Times a Fraction
Students work on their MathBoards solving real-world problems and then
find the general pattern for multiplying fractions across repeated
examples.
In Lesson 4, for example, you will tell the class that a board is ᎏ54ᎏ yd long.
Ask how long ᎏ32ᎏ of the board would be. On the front of the MathBoards,
students will write ᎏ32ᎏ ⴛ ᎏ54ᎏ ⴝ to the right of the fraction bar for fifths.
You will begin by labeling the bar and circling four of the fifths.
0
–
5
1
–
5
2
–
5
3
–
5
4
–
5
5
–
5
2
–ⴛ4
–
3 5
Fifths
ᎏaᎏ ⴛ ᎏcᎏ
b
d
The class will discuss how to find the denominator:
• How can you show ᎏ32ᎏ of the circled ᎏ54ᎏ? It is difficult to take ᎏ32ᎏ as a chunk,
but you can take ᎏ32ᎏ of each fifth.
• If you 3-split each fifth, how many parts will you have? 15
• What denominator will our answer have? 15
Then students will label the new divisions (15ths) below the bar and circle
ᎏ2ᎏ of each of the four circled fifths to find the numerator.
3
1
–
5
0
–
5
0
—
15
1
—
15
2
—
15
3
—
15
2
–
5
4
—
15
5
—
15
6
—
15
3
–
5
7
—
15
8
—
15
9
—
15
4
–
5
10
—
15
11
—
15
12
—
15
5
–
5
13
—
15
14
—
15
15
—
15
Discussion of what the fraction bar shows will lead to the product.
• How many fifteenths are circled? How do you know? 8; there are 4
groups of 2 circled.
2ⴛ
4
ᎏ
• What is the answer to ᎏ32ᎏ ⴛ ᎏ54ᎏ? ᎏ
ⴝ ᎏ18ᎏ
3ⴛ5
5
• When you take a fraction of a fraction, you are dividing the whole into
smaller parts. Why do you get a larger denominator? A larger
denominator means that the whole is divided into more but smaller
parts.
aᎏ
< 1 (that is, a < b), the product will be less than ᎏdcᎏ because you
• When ᎏb
will only take part of (a ⴢ c of) the new unit fractions b ⴢ d.
UNIT 5 OVERVIEW
Find the denominator:
b-split each dth
ᎏaᎏ ⴛ ᎏcᎏ ⴝ ᎏxᎏ
b
d
bⴢd
b ⴛ d is the new
denominator.
Find the numerator:
c groups of a
a ⴢᎏ
c
ᎏaᎏ ⴛ ᎏcᎏ ⴝ ᎏ
b
d
bⴢd
a ⴛ c is the new
numerator.
Find the product:
multiply tops
ᎏᎏᎏ
multiply bottoms