Transcript Fractions, Part 2 - University of Arizona Math

Proper fractions
The value of the numerator is less
than the value of the denominator.
Proper in this case does not mean
correct or best.
Improper fractions
The value of the numerator is greater than
or equal to the value of the denominator.
Unit Fractions
Unit fractions are fractions whose numerator
is 1:
1
2
1
7
1
24
1
100
1
8
Mixed numbers
• Meaning of
2
5
3
Writing mixed numbers as
improper fractions
• The algorithm that is taught in schools
obscures the meaning. This is true for
many algorithms, which are “efficient”
ways of carrying out operations.
Write mixed number as improper
fraction and vice versa
2 13
23
4
Operations with fractions
• Addition
• Subtraction
• Multiplication
• Division
Adding and subtracting fractions
1/2 + 1/3
Multiplying fractions
• Repeated addition model
• Area model
Multiplication of fractions
• Fraction as operator
• The multiplication algorithm is best
explained by the area model.
2/3 of 2 1/2
Mixed number times mixed
number
Dividing fractions
• Division of fractions is most easily
understood as repeated subtraction.
12  2
11 divided by 1 1/2
Multiplicative Inverses
• We know that division is the inverse of
multiplication.
10  2  5
1
10   5
2
Multiplicative inverses
• The multiplicative
inverse of a is 1/a
• The multiplicative
inverse of a/b is b/a
1
a 1
a
a b
 1
b a
Dividing fractions
Because division is the inverse operation of
multiplication, dividing a number by a
fraction is equivalent to multiplying the
number by the multiplicative inverse,
called the reciprocal, of the fraction.
Exploration 5.12
• “Drawn to scale”
• Part 1 Use reasoning not algorithms to
answer #1
• Part 2 Write justifications for the following:
– #1: 3, 6, 8, 13, 16
– #2: 1, 2, 7, 8, 9, 13, 15, 16
Extra Practice
• 1. You have from 10:00 - 11:30 to do a project. At 11,
what fraction of time remains? At 11:20, what fraction of
time remains?
• Use a diagram to explain how you know. Are there
certain diagrams that are more effective? Discuss this
with your group.
Extra Practice
• 2. Is 10/13 closer to 1/2 or 1?
• Use a diagram to explain how you know.
Are there certain diagrams that are more
effective? Discuss this with your group.