Fraction by Fraction Division

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Objective

Students will be able to model division of fractions by another fraction. Students will be able to divide a fraction by a fraction using the standard algorithm.

Big Idea

How do I divide that?! Dividing fractions by fractions.

Curriculum Reinforcer

5 minutes

Today's Curriculum Reinforcer reviews multiplication with decimals and rewriting fractions with a common denominator. The first task reads as follows: 

The local health club is advertising a special for new members: no initiation fee to join and only $34.50 per month for the first year. If Andy joins the health club for one year, how much will he spend on membership?

The remaining problems review an important skill that we will use as we learn how to find the quotient of two fractions.

Change the following sets of fractions so that they have common denominators.

        1.)  2/3 and 3/4              2.)  1/2 and 5/6              3.)   7/8 and 4/5

 

Engagement

5 minutes

To start off today's new learning, I will review with my students the meaning of dividing a fraction by a fraction. To begin this work, I ask my students to come up with a bulleted list of things that they think a person should know about dividing fractions. I tell them that we will use this list as we review the procedures for dividing one fraction by another.

In our work we will cover some of the following themes:

  1. Dividing fractions is similar to multiplying fraction
  2. To divide fractions you multiply by the reciprocal.
  3. Dividing fractions is not that different from dividing whole numbers when you really look at what is happening when you divide.
  4. Dividing fractions can be shown using visual models

I do not expect that my students will come up with these items in their bulleted list, but the brainstorming activity primes their thinking and gives me the opportunity to build on their initial ideas.

In this lesson, I will be sure to teach students the standard algorithm for dividing fractions while connecting the standard algorithm to the different modeling methods of solving fraction by fraction division for the purpose of promoting deeper understanding and retention of the subject matter. I will also refresh my students' memories when it comes to the vocabulary words; reciprocal and simplify.

Instruction & Teacher Modeling

10 minutes

When teaching fraction by fraction division, it is important to stress the fact that students are figuring out how many fractions of one particular size can fit into another fraction of a particular size. I reiterate this point over and over with my students. I demonstrate this in my Fraction By Fraction Division video, as well as sharing some of the other ways that I teach this skill to my students. 

During this section of the lesson, I model some problems for my students. 

Example #1: 2/3 divided by 1/6 means, how many 1/6 size fraction pieces can I fit into or can you get out of 2/3  size fraction piece.

I generally show my students four methods to solve this problem:

1)   Visually, using fraction models - So my students can see what is happening when I divide a fraction by a fraction.

2)   Using repeated subtraction - So my students can connect what they know about whole number division to fraction division.

3)   Using like denominators - So that my students can see that if I cut everything up into the same size pieces, it is very easy to determine how many pieces fit. Also, this will further connect the concept to whole number division because once the denominators are the same, all you have to do is divide the numerators.

4)   Using the standard algorithm for dividing fractions - I ensure to always go back to the standard algorithm. While the other methods help students to see what is going on, we still must connect those methods back to the standard algorithm to promote fluency.

Once I have modeled Example 1 using multiple methods, I will complete another example that will show how a smaller fraction is divided by a fraction that is larger using the same four methods.

Example #2: 3/4 divided by 1/2  means, how many  size fraction pieces can I fit into or can you get out of 3/4 size fraction piece.


Try It Out

10 minutes

After modeling some problem solving with division of fractions, I will give my students a Guided Practice Worksheet for individual practice. I plan to give my students seven minutes to complete this work.

As my students are working, I will travel the room answering questions and asking probing questions to enhance understanding. After the allotted time for this assignment, I will provide students with the answers to the problems. I will ask students to self-grade their work. Then, I will use their performance to arrange groups for the following activity:

  1. Those students that performed well on the Guided Practice will work independently on the exploration worksheet.
  2. Those that did not perform well on the Guided Practice will work in a teacher led group on the same exploration worksheet along with correcting the Guided Practice worksheet.


Independent Exploration

20 minutes

As discussed above, during today's student exploration, I will group students based on their performance during the Try it Out.  I plan for this section of the lesson to take about 15 minutes.

Here is the Fraction Problem Solving worksheet for today's Exploration.

Many students will work independently on this worksheet. But, students who are confused will work together in a teacher led group. I want to be able to monitor progress and address misconceptions directly.  


 

Closing Summary

20 minutes

To wrap up this lesson, we will do some partner work. I will have my students partner up using their Four Corner Partners. The students will be given five minutes to discuss the problems presented on the Exploration Assignment. They will compare their answers and solutions. They will talk about their chosen method of solution. They will debate about, critique, and discuss each other's work. This partner work will include the students who worked individually, and, the students from my working group. I want these students to have the opportunity to share what they have learned and to hear explanations from a classmate. 

Finally, my students will complete a Ticket Out the Door. For today's exit assessment I will ask my students to complete this problem on an index card and give it to me on their way out the door:

What is 4/5 divided by 2/3?