Many people have learned that to divide a fraction, we “invert and multiply.” This week, your student will learn why this works by studying a series of division statements and diagrams such as these:

• can be viewed as “How many s are in 2?”

  

  Because there are 3 thirds in 1, there are , or 6, thirds in 2. So dividing 2 by has the same outcome as multiplying 2 by 3.

• can be viewed as “How many s are in 2?”

  

  We already know that there are , or 6, thirds in 2. To find how many s are in 2, we need to combine every 2 of the thirds into a group. Doing this results in half as many groups. So , which equals 3.

• can be viewed as “How many s are in 2?”

  

  Again, we know that there are thirds in 2. To find how many s are in 2, we need to combine every 4 of the thirds into a group. Doing this results in one fourth as many groups. So , which equals .

Notice that each division problem above can be answered by multiplying 2 by the denominator of the divisor and then dividing it by the numerator. So can be solved with , which can also be written as . In other words, dividing 2 by has the same outcome as multiplying 2 by . The fraction in the divisor is “inverted” and then multiplied.

Here is a task to try with your student:

1. Find each quotient. Show your reasoning.
2. Which has a greater value: or ? Explain or show your reasoning.

Solution:

1. 1. 21. Sample reasoning:
   2. 7. Sample reasoning:
   3. . Sample reasoning: . The fraction is two times , so there are half as many s in 3 as there are s.
   4. . Sample reasoning:
2. They have the same value. Both equal 10. and .