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Mathematics

Lesson 1: Introduction to Fractions and Mixed Numbers

Are You Hungry?

If 16 people try to share an 8 piece pizza, how many pieces does each person get? (Assume everyone wants the same amount of pizza)

The Need for Another Type of Number

Definition: A fraction (or rational number) is a number in the form a/b where a and b are integers and b =/0. A fraction is one way to show parts of a whole.

Remember: the fraction bar mean division

a (top number) is called the numerator
b (bottom number) is called the denominator

Sets of Numbers

Name of SetExampleSymbol
Natural Numbers{1,2,3,...}N
Whole Numbers{0,1,2,...}W
Integers{...,-3,-2,-1,0,1,2,3,...}Z
Rational NumbersAny number that can be written as a fractionQ

A Closer Look at Fractions

box divided into 4 parts, one part shadded1 part shaded
4 equal parts
one-fourth
box divided into 6 parts, five parts shadded5 part shaded
6 equal parts
five-sixths
3 boxs divided into 3 parts, two boxes completely shaded, one box with 1 part shaded7 part shaded
3 equal parts
seven-thirds

Showing Understanding

How can we show 1/2, 2/3, 1/4, 5/6 ?

  1. How do the fractions compare to 1?
  2. What does the top number represent?
    The bottom number?

Examples

Draw a picture with shaded parts to show each fraction:

Given the numbers 245 and 542, answer the following:

  1. 2/3
  2. 5/5
  3. 7/4

Definition: If the numerator < the denominator, the fraction is a proper fraction.
If the numerator > the denominator, the fraction is an improper fraction.

Examples

Given the fractions:

  1. 2/3
  2. 5/5
  3. 7/4
  1. State the numerator and denominator of each fraction.
  2. Which fractions are proper fractions?
  3. Which fractions are improper fractions?
  4. How does each fraction compare to one?

FYI

  • Proper fractions are less than one.
  • Improper fractions are either equal to one or greater than one.

Examples

Blast From the Past!

Simpify:

  1. 3/1
  2. -3/-3
  3. 0/3
  4. 3/0

Examples

Graph each on separate number lines:

  1. 3/5
  2. -2/3
  3. 1/2

Challenge Question

If I work 5 whole days and then one third of a day, how many days did I work in total?

Challenge Question

If I work 5 whole days and then one third of a day, how many days did I work in total?

What do we have to do to get our answer? 5 + 1/3

5 + 1/3 = 5 1/3 ← 5 1/3 is a mixed number

How can we illustrate this?

How can we graph this?

Key Information

  • An improper fraction is greater than 1; this allows us to write it as a mixed number.
  • A mixed number has two parts:
    whole number + proper fraction
    A mixed number is the sum of a whole number and a proper fraction.

Seeing the Connection

5 + 1/3 = 51/3 = 16/3

51/3 = 3 • 5 + 1/3 = 16/3

Mixed Numbers → Improper Fractions

51/3 = 3 • 5 + 1/3 = 16/3

  1. Multiply the denominator of the fraction by the whole number.
  2. Add this product to the numerator of the fraction.
  3. Use the answer from (2) as the numerator of the fraction, keep the same denominator.

Examples

Write each mixed number as an equivalent improper fraction:

  1. 72/3
  2. 34/5
  3. 51/9

Note:

Positive Mixed Numbers

51/3 = 5 + 1/3

Negative Mixed Numbers

-51/3 = -5 + (-1/3)

-51/3 = -5 - 1/3

The rule to express negative mixed numbers as improper fractions is the same as the rule for expressing positive mixed numbers as improper fractions.

Example

Express the following mixed number as an improper fraction in lowest terms.

-51/3

-3 • 5 + 1/3 = -16/3

Example

Draw the following mixed number using shaded regions.

13/5

Improper Fractions → Mixed Numbers

13/5 = 5 13 = 2 Remainder 3 = 23/5

Examples

  1. 17/5
  2. 23/4
  3. 62/3

Group the Equivalent Fractions Together:

1/2   -1/2   -1/2   1/-2   -1/-2

Reviewing Concepts

Evaluate Each:

  1. |7/8|
  2. |-2/3|

Assessment: Introduction to Fractions and Mixed Numbers