Equivalent Fractions

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Objective

SWBAT find equivalent fractions with multiplication and division, then show a visual representation by using models.

Big Idea

Equivalent fractions are fractions that represent the same amount. They can be created by multiplying or dividing, as well as represented with models.

Opener

15 minutes

In today's lesson, the students learn to find equivalent fractions. This aligns with 4.NF.A1 because the students will explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.

To begin the lesson, I give each student a piece of white paper and two crayons.  These materials are used to help the students discover the meaning of equivalent fractions.  I also have a piece of paper and two crayons.  First, I direct the students to fold the piece of paper in half.  I fold my paper to give the students a visual of which direction to fold the paper.  I want all of the papers the same so that the students will clearly see the equivalent fractions.  I explain to students that fractions are equal pieces, so we must line the paper up evenly to have the same size pieces.  The students open the paper.  I ask, "How many pieces do  you see?"  The students respond that there are two pieces.  I let them know that this is the denonimator, which is the bottom number of the fraction.  (The students should be familiar with the terms denominator and numerator from 3rd grade.  I just review it to refresh their memories.)  Next, I direct the students to color the top half of the paper with one of their crayons.   I color the top part of my paper along with the students.  I feel that is is important for teachers to model for the students.  I ask, "How many pieces are shaded?"  The students respond one.  "What fraction of the paper is shaded?"  The students respond one-half.  I instruct the students to write the fraction 1/2 at the bottom of the paper.  Upon completing this, the students fold the paper into fourths.  On the back of the paper, the students shade two of the pieces with the other crayon.  (Make sure the students are shading the top section of the paper - this should be on the opposite side of the paper were they shaded the one half.)  I ask, "What fraction is shaded?"  The students respond 2/4.  

Our lesson for today is equivalent fractions.  I direct the students to look at the two fractions.  I ask, "What do you notice about the 1/2 and 2/4?"  I give the students a few minutes to think about this.  "Share your thoughts with your neighbor."  I walk around to listen in on what the students are saying to their neighbors.  Some responses I hear:  I notice that both of them look like 1/2, and I notice that they both look like 2/4 on both sides. I share these responses with the whole class and ask "who is right?"  The majority knew that both were correct.  If we are talking about equivalent fractions, somebody tell me, based upon what we just did, "What is an equivalent fraction?"  The student that I call on shares with the class that equivalent fractions are two fractions that are even.  I ask the class is "even" the correct word to use.  Another student corrects this by saying "equal" is the correct word. 

I let the students know that today we will find equivalent fractions using models and objects.

 

Whole Class Discussion

15 minutes

After the opening activity, I bring the class back together as a whole to discuss equivalent fractions.  I let the students know that today, we learn how to create equivalent fractions.  This aligns with 4.NF.A1 because the students explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models.  The students use multiplication and division to find equivalent fractions.  The fraction strips are used to give a visual of the equivalent fractions.  

Before discussing Equivalent Fractions.pptx, I wanted to clear up a misconception that I heard while walking around the room to listen to the students discuss the two shaded fractions.  (I addressed this misconception with the two students immediately, but I wanted to bring it before the whole class because I want to make sure that there are no other students with the same misconception.)  

When discussing the shaded fractions, I heard two students say that one side shaded was the whole (the side with the shaded 1/2) and the other side is half of the whole (the side with the shaded 2/4).  I wanted to clear that up before we got any further into the lesson.  My question to those two students was, "How is that a whole when you shaded it from part of something?" I wanted them to understand the concept of what a fraction actually means.  "It is a part of a whole."  

 I explain to the class that the sheet of paper is the "whole."  A fraction is a part of a whole.  If you have a pizza, the pizza is the whole.  The slices would be a fraction of the whole pizza.  

When you are dealing with fractions, if you are trying to figure out if they are equivalent, they must have the same whole.  (I hold up a piece of 8 1/2 x 11 copy paper and a piece of 9 x 12 construction paper.)  I can't use these two pieces of paper to shade 1/2 and 2/4 because the construction paper is larger than the copy paper.  That is not the same whole.  The 1/2 on the construction paper is larger than the 1/2 on the copy paper.  (I demonstrate by folding the construction paper and comparing it with 1/2 of the copy paper.)  This gives the students a conceptual understanding that we must use the same whole when finding equivalent fractions.  

On the Smart board, I show the students fraction strips of 1/2 and 2/4.  The students see from the fraction strips that these two fractions are equivalent.    (This is another visual for the students to see that 1/2 and 2/4 are equivalent.)  Beneath those two fractions, I have 6 fraction boxes.  I ask, "How many would we have to shade to equal 1/2 and 2/4?"  I call on a student and she says that we shade three boxes.  I shade the boxes for the students to see.  "Are they equivalent?"  They all say yes.  "What is this fraction?"  The majority of the students yell out 3/6.  However, I heard someone say 6/3.  "The bottom number, which is the denominator, is always the total number of pieces," I explain again to help this student understand the denominator.

If we look at 1/2, 2/4, and 3/6, we should be able to notice something.  The students notice the pattern with the fractions.  One student says that the top number is counting up 1, 2, and 3.  Another student notices that the denominator is counting up by 2's.  "If this pattern continues, what would be the next equivalent fraction?"  I give the students a minute to think about this question.  One of my students who tries to hide behind other students when working in groups, raised his hand to answer the question.  "The top number is 4 and the bottom number is 8."  I probe further by asking, "What is the name of that fraction?"  He says 4 over 8.  Another student added, 'It is four eighths."  I explain to the students that we can also find equivalent fractions by using multiplication or division.    If my fraction is 1/2, I can find an equivalent fraction by multiplying the top number and the bottom number by the same number. You can use any number other than 1.  "Who can tell me why we can't use the number 1 to multiply or divide?"  I was proud that the students remembered their property of one, which says that any number multiplied by 1 equals that same number.  If I pick the number 2, then I multiply 1 x 2 = 2 and  2 x 2 = 4.  This shows you that 1/2 is equivalent to 2/4.  When you do this, you must make sure you use the same number for the numerator and denominator.  (The students will see the connection between multiplying to find equivalent fractions and get a visual of this when they use fraction strips for their hands-on activity.) 

I reinforce to the students that in equivalent fractions, we must refer to the same whole.  If you have 1/2 of something, I have 2/4, and another person has 3/6, we all have the same amount if we are talking about the same whole.  "You can't have a large pizza and I have a small pizza, and we say we have the same amount if I have 2/4 and you have 1/2.  That is not going to be true because you are going to have more than me if you have a large pizza and I have a small pizza.  My 2/4 will be smaller than your 1/2. It has to be the same size and the same shape. 

Group or Partner Activity

20 minutes

I give the students practice on this skill by letting them work together.  I find that collaborative learning is vital to the success of students.  Students learn from each other by justifying their answers and critiquing the reasoning of others.

For this activity, I put the students in pairs.  I give each pair an activity sheet and fraction strips. The students must work together to find equivalent fractions using fraction strips (MP5). The students must use the fraction strips to show the equivalent fractions.  Also, the students use multiplication or division to find equivalent fractions.  By the end of the lesson, each student should know how to multiply or divide to find equivalent fractions.  The fraction strips should be used as a resource to check their answers.  For example, if they say 1/2 and 2/4 are equivalent, they should show this by placing the fraction strips to show that they are equivalent.  

The students are guided to the conceptual understanding through questioning by their classmates, as well as by me.  The students communicate with each other and must agree upon the answer to the problem.  Because the students must agree upon the answer, this will take discussion, critiquing, and justifying of answers by both students.  As the pairs discuss the problem, they must be precise in their communication within their groups using the appropriate math terminology for this skill.  As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.  

As they work, I monitor and assess their progression of understanding through questioning. 

1. Can you multiply the numerator and denominator by different numbers?

2. Do the fraction strips show that the fractions are equivalent?

3.  What number should you not use when multiplying or dividing to find equivalent fractions?

4.  Explain how you came up with your answer.

As I walked around the classroom, I heard the students communicate with each other about the assignment.  I hear the classroom chatter and constant discussion among the students.  Before Common Core, I thought that a quiet class working out of the book was the ideal class.  Now, I am amazed at some of the conversation going on in the classroom between the students. 

 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/fractions/equivalent_fractions/games/

 

My Findings:

Each pair was able to find some equivalent fractions using multiplication or division.  I noticed that finding equivalent fractions with multiplication was easier for the students than division.  Some students (who still do not know their multiplication facts) were having a hard time finding a number that could divide evenly into the numerator and denominator.  The fraction strips helped the students determine if the fraction they found was actually equivalent or not.

Closure

15 minutes

To close the lesson, I have one or two students share their answers.  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 by 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, 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 group activity will be addressed whole class.  

To assess how well the students learned the skill, I give them an Exit Ticket to complete before leaving my classroom.  On the Smartboard, I write the fraction 2/10.

The students use their own paper to find an equivalent fraction for 2/10.  I collect the papers as the students exit my classroom to go to their next class.

Results:

Out of 26 students in my homeroom class, 17 students were able to give me an equivalent fraction.  The remaining 9 students struggled with the concept.  I will work with the 9 struggling students in small group the next day while the other students are completing their "Do Now" assignment at the beginning of class.