Comparing Fractions

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Objective

SWBAT compare fractions by using the benchmark fraction 1/2, then justify their answer with fraction strips.

Big Idea

Using benchmark fractions and drawing visual models are two strategies to compare fractions.

Whole Class Discussion

15 minutes

In today's lesson, the students learn to compare fractions by using the benchmark fraction 1/2, then justify their answer with fraction strips.   This aligns with 4.NF.A2 because the students are comparing two fractions with different numerators and denominators by a benchmark fraction, recording their findings using >, <, or =, then justifying their answer by using fraction strips.   

The Comparing Fractions.pptx power point is on the Smart board as we begin class.  I remind the students that we have been working on fractions for a few lessons now.  I ask, "What can you tell me about fractions?"  I give the students a few seconds to think about the question.  Some student responses: 1) You can draw out a fraction to find the mixed number, 2) a numerator is how many they are talking about, and the denominator is how many pieces in all, and 3) when you draw it out, if the numerator is bigger  than the denominator, you will have more than one whole.  I remind the students to draw congruent figures with equal-sized pieces when they draw their models.  

 Real-World Scenario:

Tom says that 4/6 is greater than 3/9.  Peter disagrees with Tom.  Peter thinks that 3/9 is greater than 4/6.  Who is correct?

I give the students a few seconds to think about this question.  One student responds that Tom is correct.  She says that if you look at the  4 and the 3, the 4 is bigger than the 3.  Another student disagreed with this student.  She said that 3/9 is bigger because I learned in the 3rd grade that the smallest fraction is the biggest.  (I knew that the student was referring to the denominator being larger when they are cut in less pieces, but I needed her to explain more to see if she would take the numerator into consideration in her answer.)  She goes on to say, "When you look at fraction strips, the bigger the fraction, the smaller the strip. I explain to the class that yes, the more pieces you divide something into, the smaller the pieces.  I give the example of cutting a cake into 6 pieces or into 9 pieces, the slices will be bigger on the cake with 6 pieces.  However, we have to look at the numerator to see how many pieces we are talking about.  I told them that if the numerator for both fractions was a 1, then yes, this would work.  If the numerators are not common, then this does not work.  I explain to the students that we are comparing fractions with unlike denominators.  If the denominators are not the same (common), you can not compare by using the numerator.  If the fractions have common denominators (same), like 4/8 and 6/8, you can compare by using the numerator.  Therefore, 4/8 < 6/8.

I tell the students, in this problem, we are comparing fractions with unlike denominators.  There are several strategies that can be used to compare the fractions.  We could find a common denominator by finding a number that is a multiple for both the numerator and the denominator.  We could also use a benchmark fraction, as we did with numberlines.  The strategy that we are using today is using the benchmark fraction 1/2 to compare fractions.  If you compare two fractions, and 1 is more than 1/2 and the other is less than 1/2, then the fraction that is more than 1/2 is greater than the fraction that is less than 1/2. 

If you compare two fractions and they both are less than 1/2 or more than 1/2, draw a model of both fractions to compare them.

Let's practice.

Tom says that 4/6 is greater than 3/9.  Peter disagrees with Tom.  Peter thinks that 3/9 is greater than 4/6.  Who is correct?

What is the denominator in 4/6? 6.  If we are going to use the benchmark fraction of 1/2, I need to divide my 6 in half.  What is half of 6?  3.  Therefore, 3/6 is equal to 1/2.  If we have 4 out of 6, is it less than half, equal to half, or more than half?  The students agree that 4/6 is more than half because it is bigger than 3.  Next, we need to look at the fraction 3/9.  Nine is an odd number, so to take half of it, you will have a remainder.  Let's think about 8 first.  What is half of 8?  The students respond that it is 4.  If 4 is half of 8, then 4.5 is half of 9.  This means that my numerator for the fraction 3/9 would have to be 4.5 to equal 1/2 of the fraction.  What is the numerator?  3.  Is 3 bigger than 4.5?  The students know that 3 is not bigger than 4.5.  So I ask, "Is 3/9 less than 1/2, equal to 1/2, or more than 1/2?  Less than 1/2 is the response from the majority of the students.  

If 4/6 is more than 1/2 and 3/9 is less than 1/2, it is easy to compare the numbers.  This means that 4/6 > 3/9.  Tom was correct.

 Let's try another one.

Carrie ate 2/8 of her large pizza.  Sarah ate 3/6 of her large pizza.  Who ate more pizza?

Again, I ask questions of the students to help lead to the answer.

What is half of 8?

Is 2 less than, equal to, or more than half of 8?

What is half of 6?

Is 3 less than, equal to, or more than half of 6?

 After we go through the questions, the students come to the conclusion that

 

2/8 ___less than 1/2___

 

3/6 ___equal to 1/2____

Therefore, 2/8  < 3/6


 


 

Skill Building/Exploration

20 minutes

Before the students work in collaborative groups, I want them to take a few minutes to try and solve the problem on their own.  By giving the students independent work time before working within groups, allows the students to read the problem, plan a strategy to solve, then begin solving the problem.  When it is time for group collaboration, the students will have something to talk about within the groups.  

After the independent work time, the students are put into pairs.  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 give each pair  a Comparing Fractions Activity Sheet.docx and fraction strips. The students must work together to compare fractions with different denominators by using the benchmark fraction 1/2, recording their answer with >, <. or =, then justifying their answers using fraction strips (MP5). If they can't determine the correct comparison by using the benchmark fraction 1/2, then the students draw a model of each fraction to make the comparison.  After they have their answer, the students must use the fraction strips to show a visual comparison of the fractions to justify their answer.   By the end of the lesson, each student should know how to compare fractions based on a benchmark.  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. Is the fraction less than 1/2, equal to 1/2, or greater than 1/2?

2. How can you use a model to help you compare the fractions?

3.  Does the fraction strips validate your answer?

4.  Explain 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.mathplayground.com/fractions_compare.html

 

My Findings:

 As I walked around while the students worked as pairs, I had to help several students with using the benchmark fraction 1/2.  There are some students that can easily grasp a concept, but there are others that struggle.  I used questioning to help guide these struggling students.  Some students could not quite figure out the 1/2 without seeing models.  Therefore, in some cases, we had to draw a model of the denominator and circle half of them.  If the denominator was an odd number (as in our example in the whole class discussion), the students had to have an understanding of counting by halves.  I don't think that the idea of counting by halves connected with some of the students.  (Sometimes, students learn things and can't apply them in other situations.)  

Using a benchmark is not an easy concept for most students, therefore, I expected the students to struggle.  Some of the students wanted to take the easy way out and just use the fraction strips, but I encouraged them to think about the concept.  Look at the denominator and determine half of it.  Check to see how the numerator compares to that number.  I know that the students will get better at using benchmarks.  The more that they are exposed to a skill, the easier it will get.  We will continue to use benchmarks when applicable to the lesson. 

Closure

15 minutes

To close the lesson, I have one or two pairs 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 (Student Work 1) (Student Work 2), 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.  

 Each student completes an exit ticket to complete individually.  On their paper, they compare the fractions 2/3 and 4/6.  Group activities are great, but I need to know how well each student is doing on their own. The exit tickets are collected at the end of class.  This gives me further data on how the students are comprehending individually.   All struggling students identified from the data on the exit tickets will receive further instruction in small group.