Proper fractionsThe 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.

What do we mean by the term unit fraction?

Unit FractionsUnit fractions are fractions whose numerator

is 1:

1 1 1 1 1

2 7 24 100 8

Operations with fractions• Addition

• Subtraction

• Multiplication

• Division

Adding and subtracting fractions

1/2 + 1/3

Mixed numbers• Meaning of

325

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

312

423

Multiplying fractions

• Repeated addition model

• Area model

Multiplication of fractions

• Fraction as operator

• The multiplication algorithm is best explained by the area model.

Use an area model to multiply1/2 by 5/7

Multiply 2 1/3 by 1 5/6

Dividing fractions• Division of fractions is most easily

understood as repeated subtraction.

212

12 ÷ ½

11 divided by 1 1/2

Multiplicative Inverses• We know that division is the inverse of

multiplication.

52110

5210

Multiplicative inverses• The multiplicative

inverse of a is 1/a

• The multiplicative inverse of a/b is b/a

11

aa

1ab

ba

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

Worksheet: Dividing Fractions

Problems

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.