Division with Mixed Numbers

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SWBAT divide improper fractions and mixed numbers using the standard algorithm.

Big Idea

The meaning of division is the same for whole numbers and fractions; you are taking a total and creating equal groups.

Think About It

7 minutes

Students work in pairs to solve the Think About It problem.  Based on the work we've done in the previous lessons in this unit, students should be able to complete parts a and b of this problem, but part c is likely to be difficult.  That's okay!  The purpose of this Think About It is to have students access what they already know about dividing fractions by fractions.

For part c, some students might be able to reason that 2 1/4 is half of 4 1/2, and therefore 4 1/2  ÷ 2 1/4 must be 2.  Many students won't make that connection this yearly in the school year.

I let students know that in this lesson, they'll learn how to divide with mixed numbers. 

Intro to New Material

15 minutes

In this lesson, students will be asked to solve problems involving dividing fractions, whole numbers, mixed numbers, and improper fractions.  Following the standard algorithm won't be difficult for students, but I also want them to anticipate the size of the quotient, relative to the dividend and divisor.  Students are going to round and use compatible numbers to estimate the quotient of the problems in this lesson.

To start the Intro to New Material section, I guide students through the problems from the Think About It section.  First, I ask students to compare part A to part B.  I want them to articulate that part B involves improper fractions.  Then, I ask students to turn and talk with their partners about how thinking about improper fractions might help lead to a solution path for part C.  I ask someone to share out what they discussed.  I'm looking to hear something similar to "We can convert the mixed numbers into improper fractions so that we can multiply the dividend by the reciprocal of the divisor, just like we do with fractions."

Before I have students complete the division, I ask students what a good estimate of the quotient would be.  I want them to state that we can use compatible numbers and say that 4 ½ is about 4 and 2 ¼ is about 2, and 4/2 = 2, so our answer should be about 2.

I provide students with these Steps for Division with Mixed Numbers.  You can use them as scaffolded notes and have students fill in blanks throughout.  I don't choose to do that for this lesson, because students know how to convert mixed numbers into improper fractions and they know how to divide with fractions.  I'd rather students use class time solving problems in this lesson.  I do project the steps during independent work time.  I also have a number of hard copies available for students who need the extra support.  

I use questioning to guide students through completing the other examples in this section with me.  


Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  While students are working, I circulate around the room and check in with each group.  I am looking for: 

  • Are students organizing their work?
  • Are students estimating using compatible numbers to assess their answers for reasonableness? 
  • Are students correctly converting the mixed numbers into improper fractions?
  • Are students correctly writing a multiplication number sentence to solve the problem, using the reciprocal of the divisor?
  • Are students simplifying their answers, if needed?

I'm asking:

  • How did you solve this problem?
  • How do you know your answer makes sense?
  • What's a reciprocal?

After 10-12 minutes of partner work time, students complete the Check For Understanding problem independently.  I circulate as they work, and quickly scan for strong work samples.  I display student work on the document camera, and ask the class to give feedback on the work.


Independent Practice

20 minutes

Students work on the Independent Practice problem set.  While students work, I circulate around the room, checking students' work. 

Common mistakes for students who don't have a firm grasp on this content tend to center around the reciprocals.  Some students will try to use the reciprocals of both the dividend and divisor, while others will use the reciprocal of the dividend.  As a first level of intervention here, I'd want to be sure that students had the visual anchor.  I'd also make sure that they are using estimation before they begin, and that they're comparing their estimate to the actual quotient they come up with.  I don't teach them a memory device (I know some teachers use 'Keep, Change, Flip.'  I've encountered students in higher grades who know the Keep, Change, Flip catch phrase and aren't clear on what they're keeping, what's changing, and what's getting flipped.


Closing and Exit Ticket

8 minutes

After 20 minutes of independent work time, I bring the class back together.  I have students explain to their partners the mental math they used for problem 8/9.  After students have 2 minutes to talk, I then ask for students to share with the class what they did.  I stick with this conversation until we exhaust all strategies.  This problem incorporates mental math, and isn't requiring students to practice using the standard algorithm.  It's important that students are able to reason about problems in a variety of ways.

I then project a work sample that I create.  In my sample, I use the reciprocal of the dividend, rather than the divisor.  My work space is very neat and organized.  I ask students to first take time to look at and analyze my work.  After 1 minute of silent think time, I ask students to give me feedback on what I've produced.  

Students then complete the Exit Ticket independently to end the lesson.