# Decomposing and Composing Fractions

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## Objective

SWBAT break apart fractions to their unit fractions, then compose them back.

#### Big Idea

Fractions can be broken apart into unit fractions.

## Whole Class Discussion

15 minutes

In today's lesson, the students learn to compose and decompose fractions by breaking fractions apart into sums of fractions with the same denominator.  This aligns with 4.NF.B.3b because the students decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. The students justify decompositions by using a visual fraction model.

Before the students begin the lesson, I ask the question, "When might you need to add fractions in your everyday lives?"  I give the students a few minutes to think about this.  Student responses:  If I have a piece of cake and want another piece; when my mom gives me 2 pieces of pizza, and then gives me some more.  I let the students know that they are correct.  I share with them that we can take fractions and break them apart, or we can compose fractions by adding them together.  In the Decomposing and Composing Fractions.pptx powerpoint on the Smart board, I refer to the example.  With the fraction 4/10, we can break the fraction apart in different ways.  First, we can add 1/10 + 1/10 + 1/10 + 1/10= 4/10.  Another equation for 4/10 is 2/10 + 1/10 + 1/10 = 4/10.  Also, we could add 2/10 + 2/10 = 4/10.  In this example, the fraction 4/10 has been decomposed three different ways.

In the next example, I let the students know that we can compose fractions.  To compose fractions, you are connecting fractions together by adding.

Examples:

2/5 + 1/5 +1/5 = 4/5

1/3 + 1/3 + 1/3= 3/3

In this example, we composed the fractions 4/5 and 3/3 by adding fractions with like denominators.  Notice that in the equation 1/3 + 1/3 + 1/3, the numerators are all the number 1.  This is called a unit fraction.  All fractions can be broken apart into unit fractions.  This is shown with a visual model.  (The visual model is to give the students a clear picture of the concept.)

I tell the students that this skill is important because of some of the reasons they stated earlier. (It is important that the students know what the skill is important to their lives.)   If mom wants to make sure that 4 people get pizza, she can decompose the fraction 8/8 into 2/8 + 2/8 + 2/8 + 2/8.  "If mom wanted to bake a pie for dinner for the family, but she didn't know how many slices of pie they each wanted, what could she do?"  Student responses:  She can bake the pie and let them get how much they want; she can ask them how many pieces they want.  Let's just say that mom asked each person for the number of slices of pie they wanted, and she got the following response:  dad - 2 slices, daughter 2- slices, son - 2 slices, and mom wanted 1 slice.  Mom now knows how many slices of pie she needs.  Mom can cut the pie into 7 slices.  She knows this because 2/7 + 2/7  + 2/7  + 1/7= 7/7, which is 1 whole pie.

You will now have an opportunity to practice this skill independently.

## Skill Building/Exploration

20 minutes

For this activity, the students work independently on composing and decomposing fractions.  I give each student a Decomposing and Composing Fractions.docx activity sheet and fraction strips.  The students must decompose and compose fractions, then justify their answers  by using the the fraction strips for a visual model (MP5).

For the identified fractions on the activity sheet, the students must look at the denominator to determine how many pieces are in the whole.  The students must look at the numerator, then break apart that fraction into sums of fractions in more than one way, then record it as an equation.  In the second part of the activity, the students can compose any fraction they choose by using the unit fraction.  The students use the fraction strips to model their equations.

The students are guided to the conceptual understanding through questioning by me.   As they work, I monitor and assess their progression of understanding through questioning.

1.  What is the sum of your equation?

2. How can you break the fraction apart in more than one way?

3.  What is a unit fraction?

4.  How can you use a model to represent the fraction?

Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing: http://www.softschools.com/math/games/fractions_practice.jsp

My Findings:

I found that the students did well on this activity.  With all of the work that we have done on fractions so far, I feel that it really helped them with this concept.   They knew the meaning of the numerator and denominator, therefore, they could break the fraction apart in different ways.  The common mistake that needed correcting from this lesson was on the unit fractions.  The students should have added with unit fractions, but some of them did not.  By asking the students, "What is a unit fraction?", this helped guide them to correct their mistake.  (I feel that some of them made the mistake because they did not read the directions carefully.)  This lesson was important to me because in the past I have assumed that the students could decompose fractions.  I felt this way because of their introduction to fractions in third grade.  I learned last year as my students were taking a test that was designed to prepare them for the PARCC assessment, that some of them could not break a fraction apart in different ways.  Therefore, this year, I wanted to make sure we practiced the skill.

## Closure

15 minutes

To close the lesson, we review the answers to the problems.  This gives those students who still do not understand another opportunity to learn it.  I like to use my document camera to show the students' work during this time.  Some students do not understand what is being said, but understand clearly when the work is put up for them to see.

I feel that closing each of my lessons by having students share their work is very important to the success of the lesson.  Students need to see good work samples (Student Work) (Students Work - Composing Fractions.jpg), as well as work that may have incorrect information.  More than one student may have had the same misconception.  During the closing of the lesson, all misconceptions that were spotted during the activity will be addressed whole class.

I collect all papers from the students.  All struggling students identified as I monitored during their independent activity will receive further instruction in small group.