Unit 3, Lesson 7
Finding an Algorithm for Dividing Fractions
Accelerated 6
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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.
Complete the diagram to show how many s are in 6.
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.
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:
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:
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.
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”
Partner A: Find the value of each expression by completing the diagram.
How many s are in ?
How many s are in ?
Partner B:
Elena said, “If I want to divide 4 by , I can multiply 4 by 5 and then divide it by 2 or multiply it by .”
Find the value of each expression using the strategy Elena described.
Discuss with your partner:
Complete this sentence based on what you noticed:
To divide a number by a fraction , we can multiply by and then divide the product by .
Select all the equations that represent the sentence you completed.
To answer the question “How many s are in 4?” or “What is ?”, we can reason that there are 3 thirds in 1, so there are thirds in 4.
In other words, dividing 4 by has the same result as multiplying 4 by 3.
Fraction bar diagram. 12 equal parts. Each part labeled “the fraction 1 over 3.” One part labeled “1 group.” Total labeled “4” and “unknown quantity groups.”
In general, dividing a number by a unit fraction is the same as multiplying the number by .
How can we reason about ?
We already know that there are or 12 groups of s in 4. To find how many s are in 4, we need to put together every 2 of the s into a group. Doing this results in half as many groups, which is 6 groups. In other words,
Fraction bar diagram. 12 equal parts. Each part labeled “the fraction 1 over 3.” Two parts labeled “1 group.” Total labeled “4” and “unknown quantity groups.”
or
In general, dividing a number by a fraction is the same as multiplying the number by , which is the reciprocal of the fraction. Reciprocals are numbers that when multiplied equal 1.
Two numbers that multiply to equal 1 are reciprocals.