Fraction Division Using the Standard Algorithm

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Objective

SWBAT divide fractions using the standard algorithm.

Big Idea

The traditional algorithm for division with fractions is to invert the divisor and multiply.

Think About It

7 minutes

Students work independently on the Think About It problem.  Students will likely access visual strategies for finding the values in Sets B and C.  Once they've evaluated all of the expressions, students write about any observations that they make.

After 3-4 minutes of work time, I have students share their observations with their partners.  I then have 3-4 students share out the observations they've made with the class.  I want students to talk about the relationship between reciprocals (although they won't have the word for this) - that 1/3 and 3/1 are 'flipped' versions of one another.  I want them to articulate that there's a relationship between multiplying by 3 and dividing by 1/3.  

Intro to New Material

12 minutes

In this lesson, students will move to the more abstract with fraction division and apply the standard algorithm when finding the quotient.  It's important that this lesson comes after students have had the chance to master the visual strategies - it can be difficult to reason about the quotient using only the standard algorithm.  The conceptual understanding is an important foundation for students to have before this lesson.

To start the Intro to New Material section,  I have students fill in the guided notes at the bottom of the Think About It section.  The key ideas:

  • To divide with fractions, we can multiply the dividend by the reciprocal of the divisor.
  • Two numbers whose products are one are reciprocals of each other.

 

The remainder of this section is very procedural, giving students practice with the standard algorithm.  Students follow these Steps for Dividing Fractions to find the quotients.

 

 

Partner Practice

17 minutes

Students work in pairs on the Partner Practice problem set.  As they work, I circulate around the room and check in with each group.  I am looking to see that students are correctly finding the quotients, using the standard algorithm.  Work should be neat and organized, and easy to follow.

As students work, I am asking: 

  • How did you solve this problem?
  • What is the relationship between (the divisor) and (it's reciprocal)?  (for example, 'what's the relationship between 1/4 and 4/1?)
  • What could you do to check your work? 

 

An exemplar response for the 'explain' question with Problem 3 would be:  To divide 2/7 by 1/4, I found the reciprocal of 1/4, which is 4/1.  I multiplied 2/7 by 4/1 to determine the quotient is 8/7.  I simplified to 1 1/7.

After students have 8 minutes of work time, I display two incorrect solutions for problem 1.  I prepare these ahead of time, to address the most common student errors that will occur.  In the first work sample, I use the reciprocal of the dividend, instead of the divisor.  In the second work sample, I use the reciprocals of both the divided and divisor.  I ask students to explain to me what mistake I've made (I'll often ham it up here and say that my fictitious friend BoBo did this work, and students love telling me what BoBo has done wrong).

 

 

 

Independent Practice

20 minutes

Students work on the Independent Practice problem set.  

Students may have difficulty finding the reciprocal in Problem 4.  If I see a student stuck here, I'll ask them about how they found the reciprocals in Problems 1-3.  Often, just this is enough for them to speculate that they'll need to use 1/6.  If not, I'll ask other guiding questions, like what the denominator is on any whole number.

Students can draw area models with these problems, if they'd like to (although most find the standard algorithm to be an efficient breath of fresh air!).  If I have any fast finishers, I may ask them to go back and draw a visual representation for Problem 9.

Closing and Exit Ticket

8 minutes

After independent work time, I'll bring the class back together for a quick debrief.  I'll have students clap out their answer for Problem 9, as a way to check student thinking.  I'll then ask students to jot down what another appropriate expression could be for this problem (I'm looking for 5/8 x 4/1).  I'll cold call on a student to share and justify his/her response.

Students then work on the Exit Ticket independently to close the lesson.