Multiplication and Division for Fractions

1
Multiplication and Division for Fractions

• Sec. 5.3

2
Review of Def. of Multiplication

• What does 3 ? 4 mean?
• 3 groups of 4 i.e. 4 4 4 (repeated addn.)
• Rectangular array

What does 3 ? ¼ mean? 3 groups of ¼ i.e. ¼
¼ ¼
3
Review of Def. of Multiplication

• What does ¼ ? 3 mean?
• Take 3 as one whole, break it up into 4 equal
  parts, and take one of them.

How about ?
Note that 3?4 was three groups of four.
Similarly, this means one-third of a group of
four-fifths.
4
Understanding Multiplication of Fractions
One-third group of four-fifths.

• 3 ? 4

Three groups of four
1 whole
5
Definition of Multiplication

• Let be any fractions. Then

6
Another Example

• Draw a picture to solve and then simplify

Recall what one whole equals
Cut them all up the same.
7
General Example
b rows of d columns b ? d pieces in whole a rows
of c a ? c pieces in consideration
8
Properties for Fraction Multiplication

• Closure for Fraction Multiplication
• Commutative for Fraction Multiplication
• Associative for Fraction Multiplication
• Multiplicative Identity for Fraction Multiplic.
• Distributive for Fraction Multiplication
• Multiplicative Inverse for Fractions (new)

9
Division

• We can use the quotative model and a picture to
  understand the division problem 15 ? 5
• Think How many groups of 5 can I make?

Similarly, solve the division problem 7 ? 2 ?
3 groups with 1 left over. That 1 is ½ of a group
of 2. Hence 3½ groups (of 2).
10
Division of Fractions

• A similar idea motivates division of fractions.
• Draw a picture to show what 4 ? ½ means

Since there are 8 groups of ½ (of one whole) in 4
wholes, 4 ? ½ 8.
Hence, one-half of one whole
1 whole
11
Division of Fractions

• Draw a picture to find the value of (and show
  what it means to do)

3
Its like using a measuring stick to measure the
division!
1
3
2
12
Division of Fractions

• Draw a picture to find the value of (and show
  what it means to do)

But we dont have enough for another group. We
have only one piece which is one-half of a group
of two-sixths.
1
2
13
Division of Fractions with Common Denominators

• Let be fractions with b ? 0, c ? 0,
  then .
• When the denominators are the same, we only have
  to compare numerators. Hence we are trying to
  see how many groups of c can be made out of a
  pieces.

14
Division of Fractions

• How would you handle ?
• Find equivalent fractions that have common
  denominators (which we know how to do).

1
2
15
Algorithm for Division of Fractions aka Invert
and Multiply Theorem

• Let be fractions with
  then .
• Proof

Fund. Thm. of Fractions
Defn. Division of Fractions with Common Denom.
Multiplication of Fractions
16
Invert and Multiply and Missing Factor
Approach
Example of Invert and Multiply Method
Example of Missing Factor Approach
But we know
Hence
17
Exponents

• 53 5?5?5
• am a?a?a? ? ?a means a multiplied m times
• (a2)?(a3) (a?a)?(a?a?a) a?a ? a?a?a a5
• am ? an amn
• These properties hold even if a is a fraction.
• Ex

m factors
18
More Exponent Properties
Examples
In general, if a is an integer with a ? 0 and
n is a natural number, then
19
Extending Exponents to Fractions

• In the last two slides, weve seen properties of
  exponents and how they also apply to fractions.
  Lets keep going

20
Extending Exponents to Fractions
A fraction to a power
Changing from a negative exponent to a positive
exponent on a fraction
21
Some Concluding Problems

• Put in simplest form using positive exponents
• 272 ? 93
• 363?? 124
• ( 5)2