# Decomposing and Composing Mixed Fractions

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## Objective

Students will be able to decompose fractions and represent them as compositions of fractions in a variety of ways.

#### Big Idea

Given a set fraction amount students will be able to decompose into a sum of fractions in more than one way.

## Anticipatory Set

20 minutes

I invite students to the carpet to demonstrate how to justify breaking apart (decomposing) fractions using visual fraction models. First, I remind students of the concept of turning mixed numbers into improper fractions.  I tell them that sometimes fractions need to be emphasized by using a visual fraction model.  This will allow students to actually see how fractions are decomposed.

Example:

I use a similar model to check students' understanding.  After students understand the process of decomposing fractions, I move them towards the same concept, however, I demonstrate how to covert an improper fraction to a mixed number.  I explain that decomposing the fraction into a sum of a whole number and a number less than 1 can shed some light on how to solve the problem.  I want to see if students can apply the same concept to mixed numbers, so I ask them to think of different ways to write mixed numbers.

Hopefully students can use their prior knowledge of whole numbers to understand fractions.

Example:

knowing that 1 = 3/3, they see:

5/3=3/3+2/3=1+2/3=1  2/3

I repeat this strategy until students can apply the concept on their own by explain how and why mixed numbers can be decomposed

Mathematical Practices:

MP.1. Make sense of problems and persevere in solving them.

MP.2. Reason abstractly and quantitatively.

MP.3. Construct viable arguments and critique the reasoning of others.

MP.4. Model with mathematics.

## Guided Student Practice

20 minutes

Material:  printable fraction strips.pdf

I ask students to move to their assigned groups.  I restate the purpose of this lesson by discussing what students have already learned.  I ask students to tell me what they have learned about fractions so far.  Students say they have learned how to add fractions with like denominators, and how to rewrite a fraction as a sum in a variety of ways.

I check to see if students fully understand what they are to do.  I ask students, "what are some different ways you can write 8 as a sum?"

(8=4+4; 8=2+6; 8=1+1+3+3) It is my goal to connect them to the given task.

I pose a problem for each group to solve.

Beth had 5/8 of a pound of chocolate candy to put into three jars.  How much could she put into each bowl if the amount of chocolate candy did not have to be the same in each bowl?  Discuss your problem among your group, and write your answer in an addition sentence that looks like 5/8= ______+_______+________

I tell students that they will have about 10 minutes or so to solve their problem together.  As students are working I move into facilitator mode because I want to see how students are thinking.  I want to know how they are solving their problem, and I want to know if they can think of additional ways to solve this equation as well.  I stop to ask some students how to use the fraction wall to decompose a fraction into a sum of addends.  For instance,  can you look at 4/8 and see that it can be 2/8+2/8, or 1/8, 1/8, 1/8, 1/8?

As students are responding to my questions I use what they say to identify any misconceptions and to note any mastery of this given skill.

## Independent Closing

20 minutes

I transition students back to their seat so they can add fractions with like denominators and write their answers in simplest form.  Because some students generally forget that you do not add the denominators if that are alike, I offer some students the option of using models to represent the problems they are solving.  As students are working, I take note of the different ways students use math models to solve their equation.  Because it is essential for students to be able to rewrite a fraction as a sum in a variety of ways, I ask students to provide 2 or 3 different ways and explain how they solved their problem.

Students are given manila paper to illustrate how to use addition to show a mixed number in a variety of ways.  Teacher's Model

Graph paper is also a great way to support struggling learners. They can use the lines to help them illustrate equal parts of a whole.