# Fractions II

**Questions**

• Are whole numbers fractions ?

• If Annie got 1/8 of a pie and Manny got 1/13 of a pie, who ate more pie?

▪ If Annie got 2/3 of a pie and Manny got 3/4 of a pie, who ate more pie?

▪ How can we compare fractions ? Are there some that are easier to compare?

**Comparing fractions**

▪ Convert fractions into equivalent ones that are easier to compare.

▪ OR

▪ Use your fraction sense .

But, I could also think about how far each of these is from 1!

**Problem**

▪ Arrange these fractions from smallest to largest without converting to equivalent fractions , decimals, drawing pictures . Use your fraction sense and reasoning tools:

▪ Is it possible to put a fraction between any two fractions on a number line?

▪ Why or why not?

▪ We say that the set of fractions is dense.

▪ Can you come up with a way of adding two fractions with equal denominators ? Give an example and show on a model why your method works .

▪ What about adding fractions with different denominators ?

**Definition**

▪ Let be any two fractions. Then

**Misconceptions**

▪ Some students initially view the addition of fractions as adding the numerators and denominators as follows :

Using this example discuss why such method for addition is unreasonable.

**Exercises:**

**Properties?**

▪ What properties of whole number addition do you think fraction addition has?

▪ Closure?

▪ Commutativity?

▪ Associativity?

▪ Additive identity?

**Definition**

▪ Let be any two fractions. Then

**Mental math and properties of addition**

**Multiplication: a whole number times a fraction **

▪ Use the repeated addition approach:

**Multiplication: a fraction times a whole number**

▪ Repeated addition doesn’t make sense for

▪ We can think of taking one third of 6.

**Multiplication: a fraction times a fraction**

▪ Use the new approach!

is one fifth of

Picture

Take one fifth:

**Definition**

▪ For any two fractions we define

**Question**

▪ Is Why or why not?

**Properties of multiplication**

▪ All the properties of whole number multiplication and

▪ Multiplicative inverse property:

▪ For every nonzero fraction there is a unique fraction such that

**Problem**

▪ During one evening Kathleen devoted 2/5 of her time to mathematics, 3/20 of her time to Spanish, 1/3 of her time to biology and the remaining 35 minutes to English. How much time did she spend studying her Spanish?

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