Making Sense of Multiplying Fractions

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SWBAT reason about the product of two numbers using number sense.

Big Idea

Students participate in a fish bowl activity where they listen to classmates' cooperative discussion about reasoning when multiplying fractions.


10 minutes

This lesson is designed around the concept of multiplying fractions less than 1 compared with multiplying fractions greater than 1.  Students have been working with this concept throughout the year (starting with multiplying decimals and then with multiplying fractions).  This lesson comes toward the end of the unit on multiplying fractions.  It is a fundamental concept for this grade level.  In order to make it as meaningful as possible for the students, I have designed this lesson to be unique from the structure of a typical math lesson.

To launch this lesson, I start by sharing my CCSS book of standards.  

"Did you know that every teacher in the country has book that tells us exactly what you need to learn when you are in our class?  It doesn't tell us how to teach you (we have our brains and the text book for that) but it lists out just what you need to learn."

I show them the 5th grade pages and the introduction.  I point out that the list of standards in the introduction matches their standards based report card.  Then I show the standards explained in detail.  

Today, to start the lesson, I am going to ask you to think as teachers.  You will take a look at one of the standards in our list of things that you need to know, and together, we will figure out what that standard means, and make up some examples.

Guided Practice

15 minutes

To begin this lesson, I post standard 5.NF.5 on the board.  I have broken it down into smaller parts to make it more accessible to the students.  Then, we work through interpreting it together. 

Without actually multiplying.....

"Students will be able to explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number.

And explain why multiplying a given number by a fraction less than 1  results in a product smaller than the given number."

In order to make my approach more clear, I list out my prompting/probing questions.

• Who would like to read the first part of this standard?

• Wow, that seems complicated.  Almost like it is written in another language when I hear it the first time.  Can we look at it more closely and try to make sense of it?  What are some words that we know and understand? (Multiply, product, fraction greater than 1).  As students share these words, I circle them or underline them in the standard)

• You said you know the word product.  What does that tell me about the problems we will be working with today? (They are multiplication problems). I write a X on the board.  

• You said you know about fractions greater than 1.  Who can tell me more about this? (They are mixed numbers and improper fractions).  

• Lets make an example to help us make sense of this standard.  Who can tell me an example of a fraction greater than 1? I write the fraction on the board to the left of the x sign.

• Now it says a given number. Hummm. Any given number.... ok thats easy.  Who can give me any given number they choose for this example?  I write this number to the right of the x sign.

• Then, I revisit the standard.  It says that I have to think about the product (answer) with out actually multiplying.  So, with out finding the actual answer, lets talk about what we can expect to get for an answer if we multiply the given number by a fraction that is greater than 1.  

See "breaking down the standard" resource for a visual from the class discussion.

This same pattern continues with the second part of the standard - what if you multiply a given number by a fraction that is less than one? 

Independent Practice

20 minutes

For the independent practice portion of this lesson, students are provided with a handout from the text book.  Students work in groups of 5 or 6 to practice reasoning about multiplication of fractions to determine if a given number will be made larger or smaller based on the multiplier fraction's size.

Students will "practice without pencils" a routine we have established throughout the year.  In a practice without pencils lesson, the focus of the activity is the discussion, not recording thinking.  Students may choose to use pencils to jot down notes.

Before students begin working, I explain the end result, for our group share today; the fish bowl.  I set the expectation beforehand so students can prepare themselves for the upcoming activity.  The independent practice provides students time to work in groups to prepare for the closing fish bowl activity.

A few video clips are attached. These clips show students practicing for the fish bowl activity.  They are provided with a handout that has over 20 problems in which students must reason to the solution.

Fish Bowl

20 minutes

One group at a time serves as the "fish".  They sit around the horseshoe table while the class circles around them.  Class members watch and listen to the discussion that the "fish" have about each problem.

In order to engage more students, I select members from the crowd to write the problems for the fish to solve. 

The purpose of this activity is to provide students with an authentic opportunity to explain their thinking and listen to others (MP 3).

This is the first time I have organized a fish bowl activity for this class.  For this reason, the majority of the discussion will consist of a student sharing his/her thoughts and reasoning and then the other members of the group agreeing or disagreeing.  The fish bowl approach is a great tool that can also be used in situations where there is a mathematical argument to be worked out. To introduce this method of sharing,  I choose to use a situation that would have minimal debate.

Ticket Out

5 minutes

It is one of my personal goals to take time to summarize each day.  I use a variety of techniques to help the students reflect on their experience.  Often times, I can use these reflections multiple ways.  


The ticket out serves multiple purposes:

  • this is a reflection

  • this is open ended feedback to teacher

  • this takes an emotional temperature

  • and can be used for purposes of continuos assessment.