Are They Really The Same?

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Objective

SWBAT explain why fractions are equivalent.

Big Idea

Students often wonder how two things so different can end up being the same, in this lesson students will be exploring the relevance and how to determine if fractions are equivalent through a variety of ways.

Opener

20 minutes

Fraction Pad

I ask students, “Where would you most likely use fractions in the real world?”  Students' responses vary.  I ask, “What do you think fractions tell us?”  Some students respond by saying how many.  I write the fraction ½ on the board while listening to the remainder of the responses.  I carefully guide students' responses to ensure they understand that the bottom number (denominator) tells how many parts something is broken into, and the top number (numerator) tells how many of those parts we are working with. I repeat this step a couple of more times using fraction models to represent the fractions to give students a visual picture of how it is actually broken down into parts of a whole. I repeat this using 3/4, 2/3, and 7/8  just to ensure students understand parts of a whole , and how to interpret the different parts of a fraction.

Probing questions: 

1.  How do you know what the size of a fraction is?  

2.  How do I know my answer makes sense?  

3.  How can I use models to compare fractions?  

4.  How does understanding and using equivalence help us solve problems? 

5.  How can I use language to explain my thinking about the size of fractions?

Students could respond to some of the questions, however, they will need further practice to grasp the entire concept. These questions will help guide students to understand that the value of the numerator and denominator helps you compare and reason about fractions.

 Mathematical Practices:

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

MP.2. Reason abstractly and quantitatively.

MP.4. Model with mathematics.

MP. 8. Look for and express regularity in repeated reasoning.


Group Performance Task

20 minutes

Candy Bar Fractions.docx

Materials: Hershey's candy bar (have students each bring $0.50, so you can purchase them in advance)

I want my students to work with fractions a bit on their own to help them gain a deeper understanding of them so they can communicate their reasoning using appropriate mathematical language. 

 To give students a reasonable idea about what they will be doing.  I write the fraction 3/12 on the board.  I ask them what the number 12, the denominator, means.  ( It tells how many equal parts the whole is broken into.) I pass out candy bars to each student.  I draw an illustration of a Hershey chocolate bar on the board. I section it off into 12 parts.  I tell students to carefully separate each piece of the candy bar and show me what 3/12 looks like.  I also have students demonstrate 3/12 of the candy bar by shading in the correct amount of squares on their template of the candy bar.  I repeat this process by asking them to show me 6/12.  Next, I ask students to show me 1/2.  I want students to visually see what happens when we met equivalent fractions (It's the same amount; the whole is just divided into different parts.)  I ask students to draw 2 candy bars on their paper and section them into 12 equal parts.  Then, I ask them to shade in the correct amount of squares so that 1 candy bar illustrates 6/12 and the other illustrates 1/2. 

I focus their attention to the essential questions to ensure students are focused on the main idea of this lesson. 

Discussions may be with a partner or whole class.  Students should be able to construct viable arguments and critiquing the reasoning of others MP3

I say, "I think you guys are ready to explore fractions on your own!"  Each group is given 4 fractions to represent on large crate paper.  ( 2/3; 5/10; 7/12; 4/7) I point out that they should make sure that the parts are represented equally.

 

 

Create and Discover

20 minutes

 

Materials: Student Model.docx Pair work

I want my students to be able to view equivalent fractions from many different angles.  In this hands- on activity I ask students to move with their assigned partner to explore a bit more.  It is my goal to gear students toward understanding that equivalent fractions are an important concept when ordering and comparing fractions.  My students tend to assume that wholes are the same size.  To get them moving in the right direction, I draw two fractions on the board. For struggling students I use fraction manipulatives. I ask students, “Are the wholes the same size?” Some students pause for a bit to explore using their fraction strips.  Some students use pattern blocks to determine if the fractions are equivalent. It is essential for students to determine if the fractions are equivalent before moving to a procedure to find equivalent fractions.

I tell students that they will have about 15 minutes to discuss how to determine if the fractions are equivalent.  I want students to take turns explaining how they determined their answer.  Students share responses about the relationship between the pictures.  I give students a quick example so that they can fully understand what they will be doing in this activity.  (Each of the larger pieces is broken up into 3 little pieces) Some students question their ability to explain their answer correctly. I tell them that when we share our responses with the rest of the class, we will have an opportunity to refine and connect the appropriate mathematical language. 

As students are working I circle the room reinforcing how to determine if fractions are equivalent.  I use students’ responses to determine if more time should be given for them to explore equivalent fractions.

Student Sample

Journal It!

10 minutes

Materials:  note taking paper.pdf

Some of my students have a hard time reflecting on their learning. To help students deepen their understanding of the important concepts, I help them connect to real-life applications of the math they are learning. I ask students, “Why do you have a hard time explaining your answers?” Several students say it is easy to verbally explain it, but they don't know how to put it into words.  I place students in pairs and tell them to explain it to their partner and their partner will write the explanation.  *Students need to understand that explaining your answers helps you remember how to solve the problem because you are thinking about and writing down the steps.

 Because I want my students to have more time practicing on writing an explanation for problem solving, I give them about 8 minutes to write in their math journal. I tell them they are going to be writing about their problem-solving discussion they had earlier in the lesson.  I want to see if they can recall how they solved the problem logically. As students are working, I circle the room to check for understanding.  

 

1. How do you know what the size of a fraction is?  

2.  How do I know my answer makes sense?

  3.  How can I use models to compare fractions?  

4.  How does equivalence help us solve problems? 

I use students' responses to check the level of understanding and determine if additional practice is needed.