To find the value of , Elena thought, “How many s are in 6?” and then she drew this tape diagram. It shows 6 ones, with each one partitioned into 2 equal pieces.

Fraction bar diagram. 12 equal parts. 1 part shaded and labeled “one half” and “1 person.” Total labeled “6” and “unknown quantity groups.”

For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression.

1. 

   Value of the expression: 

2. 

   Value of the expression: 

3. 

   Value of the expression: 

Look for a pattern in the expressions and their values. Talk to your partner about how to find how many halves, thirds, fourths, or sixths were in 6 wholes, without counting all the parts.

Use the pattern you noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.

Find the value of each expression.

Fraction bar diagram. 18 equal parts. Every third part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 3. Total labeled 6.

1. Complete the diagram to show how many s are in 6.

2. Elena says, “To find , I can take the value of and then either multiply it by or divide it by 2.”

   Discuss with your partner why Elena’s method works.

Use the diagram and Elena’s method to find the value of each expression. Think about how to find that value without counting all the pieces in the diagram.

1. 

   Fraction bar diagram. 24 equal parts. Every fourth part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 4. Total labeled 6.

   Value of the expression: 

2. ​​​​​​

   

   Fraction bar diagram. 18 equal parts. Every third part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 3. Total labeled 6.Fraction bar diagram. 18 equal parts. Every third part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 3. Total labeled 6.

   Value of the expression: 

3. 

   Fraction bar diagram. 36 equal parts. Every sixth part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 6. Total labeled 6.

   Value of the expression: 

Elena noticed that she always took the same two steps to show division by a fraction on a tape diagram. She said:

“First, I would partition each 1 whole into as many parts as the number in the denominator. For , that number is 4, so the diagram would have 4 times as many parts.

Next, I would put a certain number of those parts into one group. For , I would put 3 of the s into each group and see how many groups there are.”

Which expression represents the result of taking these two steps to find ?

Be prepared to explain your reasoning.