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# 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

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 .

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?

Subtraction

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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