Author:
ERIN WOLFHOPE
Subject:
Mathematics
Material Type:
Lesson, Textbook
Level:
Upper Primary, Middle School
Tags:
License:
Creative Commons Attribution Share Alike
Language:
English
Media Formats:
Text/HTML

Education Standards

Introduction to Fractions

Introduction to Fractions

Overview

This workbook was created through the efforts of three instructors at Scottsdale Community College in Scottsdale, Arizona. Any individual may download and utilize a digital copy of this workbook for free.

Introduction

and composite numbers. We will then begin working with fractions and the concept of “parts of a whole”. The table below shows the specific objectives that are the achievement goal for this lesson. Read through them carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the end of the lesson to see if you can perform each objective.

  • Find all factors of a number. 
  • Write the prime factorization for a composite number. 
  • Find the LCM (Least Common Multiple) of two numbers. 
  • Recognize a fraction as part of a whole. 
  • Identify fractions represented by shading.
  • Write mixed numbers as improper fractions.
  • Write improper fractions as mixed numbers.
  • Find equivalent fractions.
  • Write fractions in simplest form.
  • Compare fractions.
  • Graph fractions on a number line.
  • Simplify fractions that have a “1” in the numerator or denominator.
  • Simplify fractions that have a “0” in the numerator or denominator.
  • Solve applications with fractions.

Key Terms

The key terms listed below will help you keep track of important mathematical words and phrases that are part of this lesson. Look for these words and circle or highlight them along with their definition or explanation as you work through the MiniLesson.

  • Factors
  • Prime Number
  • Composite Number
  • Prime Factorization
  • Exponential Form
  • Factored Form
  • Least Common Multiple (LCM)
  • Fraction
• Numerator
• Denominator
• Proper Fraction
• Improper Fraction • Mixed Number
• Quotient
• Remainder

Mini Lesson

Factors

The factors of a number divide the number evenly (with remainder zero).

Example 1: Find all factors of 24.

Prime Factorization

A prime number is a whole number that has only itself and 1 as factors.

(Example: 2, 3, 5, 7, 13, 29, etc...)

A composite number is a whole number that is not prime (i.e. has factors other than itself and 1). Every composite number can be written as a product of prime factors. This product is called the prime factorization.

Example 2: Find the prime factorization of each of the following. Write the final result in exponential form and factored form.

72

 

600

Least Common Multiple

Example 3: The LCM of two numbers is the smallest number for which both numbers are factors. For example, the LCM of 2 and 4 is 4. The LCM of 3 and 5 is 15. Find the LCM of 8 and 10:

Multiples of 8 are: ____________________________________________________

Multiples of 10 are: ____________________________________________________

Some common multiples of 8 and 10 are: ______________________________________

The LEAST COMMON MULTIPLE of 8 and 10 is _________________

You Try

Example 4. List the factors of 18.

 

Example 5. Find the prime factorization of 270.

 

Fractions

Suppose I buy a candy bar to split with two of my friends. What number could we use to discuss how much of the bar each of us would get? Well, if we have 1 bar and it is split

into 3 equal pieces, then we would say that each person gets \(\frac{1}{3}\) of the bar. The number \(\frac{1}{3}\) is called a fraction because we use it to represent part (one part) of a whole (3 pieces).

The fraction \(\frac{1}{3}\) can be represented by the shaded part in each of the following diagrams. Notice that in each diagram, the whole is a different shape or set of shapes but the use of the fraction \(\frac{1}{3}\) still applies.

Example 6: Identify the fraction represented by the shaded part of each figure.

You Try

Example 7: Draw two different figures or sets of figures that are 3/4 shaded.

Vocabulary of fractions:

• The top number in a fraction is called the numerator.

• The bottom number in a fraction is called the denominator.

• Fractions for which the top number is smaller than the bottom are called proper fractions.

• Fractions whose numerator is larger than the denominator are called improper fractions and can be written as what are called mixed numbers.

Example 8: Identify the fraction represented by the shaded part of each figure.

 

Example 9: Express as an improper fraction.

\(2\frac{1}{4} \)

\(12\frac{1}{3}\)

You Try

Example 10. Write the steps to convert a mixed number to an improper fraction.

 

 

Example 11: Express as a mixed number.

\(\frac{42}{5}\)

\(\frac{53}{9}\)

\(\frac{84}{7}\)

You Try

Example 12. Write the steps to convert an improper fraction to a mixed number.

Example 13. Express \(\frac{57}{11}\) as a mixed number.

Example 14. Express \(8\frac{1}{5}\) as an improper fraction.

Equivalent Fractions

Each rectangle below has the same amount of shaded area. The simplest way to represent the shaded areas as a fraction is as \(\frac{1}{4}\). All of the listed fractions are equivalent to \(\frac{1}{4} \).

\(\frac{1}{4}\)

\(\frac{2}{8}\)

\(\frac{3}{12}\)

 

Example 15: Which of the given fractions are equivalent to \(\frac{2}{7}\)?

\(\frac{4}{14}\)

\(\frac{6}{18}\)

\(\frac{10}{35}\)

\(\frac{14}{28}\)

Example 16: Find for fractions equivalent to \(\frac{1}{5}\).

 

Fractions in Simplest Form

Fractions are in simplest form if they are completely reduced. To completely reduce a fraction, remove all common factors other than 1 from the numerator and denominator. Leave fraction answers always in simplest form.

Example 17:  Write the following fractions in simplest form.

\(\frac{4}{16}\)

\(\frac{28}{54}\)

\(\frac{360}{495}\)

You Try

Example 18: Find two fractions equivalent to \(\frac{3}{8}\).

Example 19: Write \(\frac{40}{72}\)in simplest form.

Comparing Fractions

To compare fractions, create equivalent fractions with the same denominator then compare the numerators.

Example 20: Which is larger, \(\frac{4}{5}\) or \(\frac{6}{7}\)?

Graphing Fractions on a Number Line

Example 21: Divide the given line into units of length \(\frac{1}{6}\), label each tick mark, then plot and label the following numbers: \(\frac{0}{6},\frac{2}{6},\frac{5}{6},\frac{6}{6},\frac{9}{6}\). Provide alternate forms if possible.

One and Zero

Example 22: Complete the following table.

NumberComputationSimplified ResultGeneral Rule

\(\frac{4}{1}\)

 

 

 

\(\frac{4}{4}\)

 

 

 

\(\frac{0}{4}\)

 

 

 

\(\frac{4}{0}\)

   

 

Application of Fractions

Answers

You Try Problems:

Lesson 2 – Introduction to Fractions

4: 1, 2, 3, 6, 8, 18
5: 270 = 2 x 33 x 5
7: Answers vary
13: 5 2 11

14: 41 5

18: Some possible answers:
19: 95
24: 14 of the senators voted against the bill.

 

Example 23: There are 14 men and 12 women in Professor Bohart’s MAT082 class. What fraction of the students in the class are women?

GIVEN:

GOAL:

MATH WORK:

CHECK:

FINAL RESULT AS A COMPLETE SENTENCE:

You Try

Example 24. The local PTA group approved a fall carnival by a vote of 15 to 5. What fraction ofthe PTA group voted against the bill? Remember to reduce the final result.

GIVEN:

GOAL:

MATH WORK:

CHECK:

FINAL RESULT AS A COMPLETE SENTENCE: