Complex Fractions

Getting Started

The student does not have an effective strategy for rewriting complex fractions as equivalent simple fractions.

The student:

• Rewrites the complex fraction as a multiplication replacing the main fraction bar with a multiplication symbol.
  
  
  
   
• Rewrites the complex fraction as a division and then divides the larger numerator by the smaller numerator and the larger denominator by the smaller denominator.
  
  
  
   
• Rewrites the complex fraction as a multiplication of the reciprocals of both the numerator and denominator.
  
  
  
   
• Rewrites the fractions within the complex fraction in decimal form (either correctly or incorrectly) and then attempts to use long division to divide.
  
  
  

Can you explain your strategy for rewriting these complex fractions?

Should 18 divided by be greater than or less than 18?

Should divided by 15 be greater than or less than ?

Review the concept of division of fractions using visual models. For example, guide the student to interpret a problem such as ÷ as determining how many halves are in . Illustrate the problem with a visual diagram such as an area or number line model. Then review strategies for dividing fractions. Be sure the student understands that a fraction bar can be interpreted as division so that can be rewritten as ÷ . Guide the student to then use his or her understanding of fraction division to rewrite complex fractions as simple fractions.

Moving Forward

The student makes an error working with fractions that is unrelated to rewriting complex fractions as simple fractions.

The student demonstrates that he or she can correctly rewrite complex fractions as equivalent multiplications. However, the student makes another type of error when working with fractions. For example, the student:

• Rewrites 18 as a fraction in a nonequivalent form such as .
  
  
  
   
• Rewrites the reciprocal of 15 as .
  
  
  
   
• Reduces both numerators by a common factor.
  
  
  
   
• Attempts to divide both the numerator and denominator by a common factor when the complex fraction is written as a division.
  
  
  

Can you explain your strategy for rewriting these complex fractions?

Is 18 really equivalent to ? How can you rewrite 18 as a fraction in an equivalent form?

How can you rewrite 15 as a fraction in an equivalent form? What is the reciprocal of this fraction?

Can you explain your strategy for writing fractions in lowest terms?

Review writing whole numbers as fractions in equivalent forms and finding reciprocals. Make clear that a number such as 15 is equivalent to . Then guide the student to use this representation to determine the reciprocal of 15.

Review the process of reducing products of fractions before multiplying (e.g., rewriting  as ). Be sure the student understands the rationale for dividing both a numerator and a denominator by a common factor. Show the student that, for example, is not equivalent to (which was obtained by dividing both numerators in the original expression by three) by finding each product and showing that they are not equal. Also, address attempting to divide both the numerator and denominator by a common factor in a division problem.

Provide more experience with multiplying and dividing fractions and whole numbers in the context of simplifying complex fractions.

Almost There

The student makes a computational or other minor error.

The student correctly rewrites each complex fraction as an equivalent multiplication but:

• Makes a multiplication error.
  
  
     
   
• Makes an error when reducing fractions before multiplying.
  
  
  
   
• Rewrites  as a mixed number as if it were .
  
  
  
   
• Writes a final answer as its reciprocal.
  
  
  
• Does not write final answers in lowest terms.

I think you made an error in this problem. Can you find and correct it?

Is equivalent to 6? What is 6 as an improper fraction?

Can you check to see if you wrote your answers in lowest terms?

If needed, assist the student in locating his or her error and ask the student to make corrections. Provide additional opportunities to rewrite complex fractions as equivalent simple fractions in lowest terms.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly rewrites each complex fraction as an equivalent multiplication, completes each multiplication, and writes each final answer in lowest terms.

1. 27
2. 
3. 

Are there other equivalent forms in which these answers could be written?

Do you know any other strategies for writing complex fractions as simple fractions?

Introduce the student to another approach to rewriting fractions as equivalent simple fractions. For example, choose one of the complex fractions on the Complex Fractions worksheet and ask the student to rewrite the fractions in the numerator and denominator with a common denominator (e.g., rewrite as ). Then guide the student to find a strategy for rewriting the fraction as a simple fraction (e.g., = ÷ = = =27). Ask the student to consider why it might be advantageous to first rewrite the fractions in the numerator and denominator with a common denominator and when this step might be omitted.