Improper Fractions and Mixed Numbers

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Objective

Students will be able to identify and write mixed numbers as improper fraction and improper fractions as mixed numbers.

Big Idea

Given a set of mixed fractions students will be able to identify and write mixed numbers as improper fractions.

Anticipatory

10 minutes

It is important to involve as many as students as possible to peak their interests for learning about fractions, since fractions are somewhat difficult for students to understand. I wanted to think of a way to inform my students of this particular lesson in a Kid- friendly sort of way.  I think about what my students need to know in order to master this lesson. 

Introduction:

Now that students have been working with the foundation of fractions, we are ready to move towards a deeper understanding of the use of fractions.  To connect students to prior knowledge, I tell them we have already learned about writing equivalent fractions.  Today we will learn how to change mixed numbers to improper fractions.  

To see what they already know about the stated process I ask students to tell me what are some ways they know how to write numbers that are equivalent to the given fractions?  Students respond by saying equivalent fractions, decimals, and percents. However, some students do not respond with any of these possible answers. I explain that it is ok, I just want to know where you guys are so far in their learning.  

 

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.5. Use appropriate tools strategically.

MP.7. Look for and make use of structure. 

Fraction Pad

Modeling

20 minutes

Because the complexity of this lesson is a higher. I begin by calling students in a whole group setting.  I write mixed numbers and improper fractions on the board. I further explain that these vocabulary words are an essential part in this lesson.  I ask students to explain what the vocabulary words mean.  If they are unable to do so.  I give them brief  visual and explanation of what they mean... 

Vocabulary

An Improper fraction has a top number larger than (or equal to) the bottom number, A Mixed Fraction is a whole number and a proper fraction combined. 

 After that, I pose this problem to prove the structure students need in order to identify the differences between mixed and improper fractions. 

Problem:

Pam took 1 1/3 of the pizza home with her. How can you write 1 1/3 as a fraction? 

I allow students to work in pairs to discuss and solve the given problem.  As they are working, I ask why and how this could be done.  I take notes of student’s response to see if I need to provide additional teaching in order for them to be successful.  I give them about five minutes or so to solve the given problem.  I do not want them to do anything other than use what they know so far to solve... If possible…..  Taking the necessary steps during teaching fractions is key to students achievements.

 After that, to clear up any misconception I demonstrate how to solve the given problem.  First, I ask students what two parts are combined in the number 1 1/3.  Students are unable to answer. I say (1 and 1/3) are combined.   A number that has a whole number part and a fractional part is called a mixed number.  A fraction in which the numerator is equal to or greater than the denominator is called an improper fraction. 

 Since this is the second time I have stated the correct meaning of improper and mixed fractions, I ask students to explain how they can use models to show the problem.   I ask students what fraction is equivalent to 1 1/3?  What type of fraction is 4/3?  Again, I am reintroducing them to the terms and focusing their attention to the intended purpose of this lesson. 

 Next, I go a bit further by explaining step by step how to:  I write 1 1/3 on a piece of paper.  I multiply the whole number part by the denominator part.  As I write I ask students to solve.  So, I ask what did you get by multiplying the given numbers. (3)  Then, I add the numerator of the fraction to 3.  I ask what you get. (4)Then I write this as the numerator and keep the same denominator.  I ask what fraction I write. 4/3

I explain that this is another way to change a mixed number to an improper fraction. 

I repeat this problem-solving technique until a level of understanding is reached.

Guided Practice

20 minutes

Because it is important for students to see and hear the correct way to problem solve I ask them to gather closer to me on the carpet, so we can discuss mixed fraction and improper fractions together. 

Home Connection:

My students think that every fraction we talk about is related to ½ or is ½.  I definitely wanted to know why they suggested this notion.  One response was, “Whenever my mom gives me a piece of her candy bar, she always gives me half.”

When we teach fractions we always tell students that a fraction is a piece/part of a whole, however, when the student asked her mother for a piece she gave her half  of the candy bar. Therefore, the student always relates a piece/fraction to a half, which is similar  to the fraction 1/2. As teachers we never think about how students learn at home.  Having students share what they know help me to understand exactly where to connect to their current level of knowledge.

 For about five minutes or so, I allow students to create equal parts/piece by using a halving strategy-diving an object in half, dividing the resulting halves in half, and so on, until there are enough pieces to share with their friends.  Hopefully, students should be able to visually and mentally see that mixed numbers are just pieces of a whole. If it does not it is ok at this point because I want to build knowledge.

Practice

 I refocus student’s attention to the intended purpose of this lesson by  having students restate the big idea with me.  (Students will be able to write improper fractions as mixed numbers and mixed numbers as improper fractions.)  

Because students generally have a hard time understanding this process, I model another way to write an improper fraction as a mixed number/.  I write 10/3 on the board.  Then, I find the number of wholes in 10/3.  (3) Then I write the left over parts as a proper fraction.   (To do this divide 3/10) 1/3 The mixed number is the sum of the wholes and the parts.  (3 plus 1/3 equals 3 1/3)

I allow students to move into their assigned groups to work on changing improper fractions as mixed numbers and mixed numbers as improper  fractions.   To do this I give them a guided practice sheet to practice this process.  As students are working together solving the given problems. I ask students to identify and explain how to write the whole number in the mixed fraction.  Then, I ask them to add the whole number fractions and the proper fraction to get an improper fraction.  As students continue to work I choose to keep repeating the same questions throughout the given time.  I do this to provide the support until students are able to repeat the process on their own.  

If some students are having difficulty working on this process.  I take note of their misconceptions and apply it towards re-teaching later own.  Some students need more exposure to working with fractions than others.  

Independent Closing

20 minutes

Because some students were experiencing difficulty understanding how to write improper fractions as mixed numbers and mixed numbers as improper fractions. 

  I have them to return to their seats. I ask them to illustrate and use fraction strips to test for reasonableness as they work on changing mixed fractions into  improper fractions.  I encourage them to use models as needed to support them in their learning.

While students are working on the work. I ask several students to tell me the difference between improper and mixed fractions.  I always make sure that students understand the objective of a given lesson.

Students response:

 First, fractional amounts greater than one can be represented using a whole number and a fraction.  Whole number amounts can be represented as fractions.  When the numerator and denominator are equal, the fraction equals 1. I have them to write an improper fraction as a mix number and explain the process by writing it in their math journal.  Then I have them to write a mix number as an improper fraction with an written explanation. Having students explain their answers help them to build on their problem solving skills.

After students are finished working on their given assignment, I ask student volunteers to share what they have learned, I encourage students to illustrate and explain their answers on the board. I used their responses to generate an anecdotal record of how I can improve and re-teach this lesson.