Lesson 10Dividing by Unit and Non-Unit Fractions
Learning Goal
Let’s look for patterns when we divide by a fraction.
Learning Targets
I can divide a number by a non-unit fraction
by reasoning with the numerator and denominator, which are whole numbers. I can divide a number by a unit fraction
by reasoning with the denominator, which is a whole number.
Lesson Terms
- reciprocal
Warm Up: Dividing by a Whole Number
Problem 1
Work with a partner. One person solves the problems labeled “Partner A” and the other person solves those labeled “Partner B.” Write an equation for each question. If you get stuck, consider drawing a diagram.
Partner A:
How many 3s are in 12?
Division equation:
How many 4s are in 12?
Division equation:
How many 6s are in 12?
Division equation:
Partner B:
What is 12 groups of
? Multiplication equation:
What is 12 groups of
? Multiplication equation:
What is 12 groups of
? Multiplication equation:
What do you notice about the diagrams and equations? Discuss with your partner.
Complete this sentence based on what you noticed:
Dividing by a whole number
produces the same result as multiplying by .
Activity 1: Dividing by Unit Fractions
To find the value of
Problem 1
For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression.
Value of the expression:
Value of the expression:
Value of the expression:
Print Version
For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression.
Value of the expression:
Value of the expression:
Value of the expression:
Problem 2
Examine the expressions and answers more closely. Look for a pattern. How could you find how many halves, thirds, fourths, or sixths were in 6 without counting all of them? Explain your reasoning.
Problem 3
Use the pattern you noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.
Problem 4
Find the value of each expression.
Activity 2: Dividing by Non-unit Fractions
Problem 1
To find the value of
Complete the diagram to show how many
s are in 6. Elena says, “To find
, I can just take the value of and then either multiply it by or divide it by 2.” Do you agree with her? Explain your reasoning.
Problem 2
For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression. Think about how you could find that value without counting all the pieces in your diagram.
Value of the expression:
Value of the expression:
Value of the expression:
Problem 3
Elena examined her diagrams and noticed that she always took the same two steps to show division by a fraction on a tape diagram. She said:
“My first step was to divide each 1 whole into as many parts as the number in the denominator. So if the expression is
My second step was to put a certain number of those parts into one group, and that number is the numerator of the divisor. So if the fraction is
Which expression represents how many
Problem 4
Use the pattern Elena noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.
Are you ready for more?
Problem 1
Find the missing value.
Lesson Summary
To answer the question “How many
In other words, dividing 4 by
In general, dividing a number by a unit fraction
How can we reason about
We already know that there are
or
In general, dividing a number by