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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
▪ 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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