Multiplication and Division
of Fractions and Decimals
Session 3
January 15, 2013
Huntersville Elementary
A number with 3 digits after the decimal
point is rounded to 4.8 when rounded to
the nearest tenth.
What is the smallest this number could be?
What is the largest this number could be?

Pizzas, pies, and candy bars
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Number lines/Sentence strips
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Fraction bars/Centimeter grid paper
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Clocks
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Percent strips- percent and decimal equivalents
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Arrays (4 x 6 and 5 x 12)

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Benchmarks (containers)
Decomposing fractions 7/8 = 3/6 + 1/4 + 1/8
(Fraction Track Game)
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Pattern blocks (unit fractions)

Investigations’ Bar diagrams

Paper folding and Open arrays

Equations
Fraction Conjectures
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To multiply unit fractions, multiply the
denominator times the denominator and
To multiply any two fractions, multiply the
numerator times the numerator, and the
denominator times the denominator
When you multiply a whole number times a
fraction less than 1, the answer is smaller than
the whole number
When you multiply a whole number times a
fraction greater than 1, the answer is greater
than the whole number


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When you multiply two fractions that are both less
than 1, the product is a fraction smaller than either
of the factors
When you divide a whole number by a fraction less
than 1, the answer is larger than the whole number
When you divide a fraction by a whole number, the
answer is smaller than the whole number and the
fraction
 10 ÷ ½ = 10 x 2
 10 x ½ = 10 ÷ 2
The cafeteria made lunches for the fifth graders
going on a field trip. They were in four different
groups so the number of sandwiches differed. The
sandwiches were all the same size.
Group
Group
Group
Group
One had 4 students sharing 3 subs
Two had 5 students sharing 4 subs
Three had 8 students sharing 7 subs
Four had 5 students sharing 3 subs
Did each student get a “fair share?”
If not, which group ate the least? Most? How
do you know?
Next trip we want to guarantee that each
student will receive 2/3 of a sub
Using large paper, create a chart for the
cafeteria to help them know how many subs
to make for up to 15 students
 What patterns do you notice?
 What strategy could cafeteria workers use for
any number of students?
 If you knew there were 8 subs made, how
could you figure out how many students
could each get 2/3 sub?
Model this situation using numbers and symbols.
What does 6 ÷ ½ mean?
How does this problem relate to
multiplication?
 How
is multiplication and
division of whole numbers
connected to multiplication
and division of fractions?
 What
is meant by “there are
two wholes when dividing
fractions?” Give an example..
Rounding Decimals
Using the number line below, label the point
1.68
1.7
1.6
How did you know where to place
the number?
Rounding Decimals 1.68
Underline the rounding place
 Circle the digit to the right
 If the circled digit is 5 or greater, increase the
underlined digit
 If the circled digit is less than 5, leave the
underlined digit as it is
 Drop the digits to the right of the underlined
digit
4.923
Round to the nearest hundredth
What about 2.97 rounded to the nearest tenth?

Expanded Notation
Write the following number using expanded
notation: 689,738
Use exponents when possible
What number is 5,000 less than this?
What number is 200 more?
Note the suggestions in the snap-ins!
Decimal Subtraction Problems
Mercedes had 1.86 grams of gold. She used
0.73 grams of it in a piece of jewelry. How
much gold does she have left?

Use of Hundredths Grids
 Subtract in parts
 Add up from 0.73 to 1, and then from 1 to 1 and 86
hundredths
How can equivalent decimals
help you to subtract these two
numbers?

Use 4 Hundredths grids to show the following
4 x 0.01
4 x 0.1
How can we use what we know about multiplying
fractions to help us solve the two problems above?
4x1=
4 x 10 =
4 x 100 =
4 x 0.1 =
4 x 0.01 =
4 x 0.001 = ?
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25
25
25
25
25
x
x
x
x
x
0.01
0.1
1
10
100
=
=
=
=
=
0.25
2.5
25
250
2,500
2x7=
2 x 0.7 =
Use a number line from 0 to 2 to show the
answer to the second problem
Use the same number line to show the answer
to the following: 2 x 0.07
(notice the use of running context p CC110111)
32 x 0.8 =
2.56
25.6
256
Problem: 185 x 0.4 =
If the answer to 185 x 4 is 740, how can we
use reasoning to determine the answer to the
above problem?
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Multiply the numbers like they are whole numbers
and then think about the size of the factors and
place the decimal point so the product is the right
size.
Multiply a whole number by a decimal, and the
answer has the same number of decimal places as
the decimal number being multiplied. Or, if each
of the numbers has one decimal place, then the
answer has two decimal places.
Therefore solve:
42 x 36 = 1,512
4.2 x 3.6 = ?????
0.2 x 0.4 =
Think about one of the conjectures we made about
multiplying 2 fractions less than 1
How can this conjecture and what you know about
the relationship between decimals and fractions
help you solve this problem?
2÷1=?
2 ÷ 0.1 = ?
2 ÷ 0.01 = ?
What does 2 ÷ 1 mean?
Will the answer to each of these be greater
or less than 2? How do you know?
How can we use 2 hundreds grids to represent
these situations?
25
25
25
25
25
x
x
x
x
x
100 = 2,500
10 = 250
1 = 25
0.1 = 2.5
0.01 = 0.25
25
25
25
25
25
÷
÷
÷
÷
÷
100 = ____
10 = ____
1 = 25
0.1 = 250
0.01 = 2,500
What patterns do you notice?
How can you use your understanding of fractions to
figure out the answers to the
two unsolved problems?
Bonus: use the pattern in the division problems to figure out the
answer to 25 ÷ 0.001
6.8 x 2.3 ≈ 1.4
74 x 8.1 ≈
5.6
166 x 0.08 ≈ 1.66
14
140
56
560
16.6 166
What is the closest estimate for each?
Is the closest estimate greater or less than the
actual answer?
How do you know?
18 ÷ 6 =
18 ÷ 0.6 =
How could you write these problems as
missing factors problems?
Draw a number line from 0-20
Solve each of these using the number line.


Problems can be solved by thinking about
both numbers as whole numbers and then
reasoning about where the decimal belongs
Problems can be solved by thinking about the
division problem as a missing factor problem
to solve or to check the reasonableness of the
answer
97.5÷ 6.5 =
21.52 ÷ 0.8 =
Resources
DPI Fraction Unit
Test-item bank
CMS Wiki
Unpacking Document
There are easy rules for multiplying and
dividing fractions and decimals. So…..
Why don’t we
just teach them
the rule?
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Thinking about multiplying and dividing
fractions, why do you think it has been
said that “fractions are the pathway from
arithmetic to algebra?
Is multiplying or dividing fractions more
difficult? Explain your reasoning.
Is multiplying or dividing decimals more
difficult? Explain your reasoning