Is 1/3 Always Equal to 1/3?: A Journal Exercise

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Students will be able to explain why the same fraction may not equal the same proportion.

Big Idea

Applying knowledge already learned is an important step in becoming a fluent math student. This journaling exercise stretches children to use concepts previously introduced.

Mini Lesson

10 minutes

Begin the class by asking the students to discuss with their partners what 1/3 means.  Listen for them to suggest that it is one part of 3 equal parts, or that an object is divided into 3 equal sections. This shouldn't be much of an issue at this point. 

Next, prompt them to discuss if 1/3 always equals 1/3.  As they discuss, listen in for some confusion, which is to be expected.  At this point, it may still be difficult for some of your students to understand that the fraction size depends on the size of the whole.

Bring the class together and listen to their reasoning.  Then, place 4 red Cuisenaire rods in a line on the overhead, as well as 4 purple rods in a line underneath the red.  

Ask the students to name the parts in the red line, as well as in the purple line. They should be able to name them as fourths. 

Now ask student partnerships to make a line of 4 green rods and another line of 4 brown rods.  Prompt them to discuss the names of those unit fractions with their partners.  Then as a class ask how a red, purple, green, and brown rod could all be called 1/4.  Are they the same?  Do they equal each other?  

Some of my students were still a bit confused and wondered how they could all be named the same if they represented something different.  I responded by reminding them that we have 3 Lauren's in our class.  Then I asked, are they the same girl?  No, but they have a name that is the same, but represents something different. With fractions, like people, we need names to represent the equal parts, but the wholes are completely different. 

Active Engagement

25 minutes

In order to gauge students on their understanding of this, after two lessons, I give them a prompt:

Marcus looks at the number lines below and says they can’t both be correct because 1/3 is in different spots on each line. Explain why you agree or disagree.

(This prompt is in the resources of this section formatted to be printed on mailing labels so students can just stick it in their Journal. Because it is a label format, there are no number lines provided to make the comparison, so I draw the two number lines on the board.)  

The number lines presented to students are of the same length, but show the fraction 1/3 in different locations. 

The students are expected to create a model of the question using the number lines, and respond to the question's statement that the 1/3 represented on two different number lines could not be correct if they were not in the same location.

Many students still struggled, but I let them.  I purposefully created this lesson to assess the understanding of the students thus far.  You will see in the samples, that some students understood and could communicate with detail, their understanding. Others had a response, but could not communicate their thinking. 

Wrap Up

10 minutes

To wrap up the lesson, I ask all of the students to come to the community area with their journals. I then ask the children to share their journal responses in groups of three.  I alert them to the fact that they will later be sharing out their partner's ideas, and not their own, so they have to listen carefully and ask remember to questions if they don't understand what their partner is saying. 

This student was sharing her thinking and including the thinking of her partner.  She stumbles on the explanation of her partner, so her partner chimes in!

As the groups shared, we added information on the board from ideas heard and learned. You can see in the photo in this section that the students went back to our lesson about the slices of pizza in a small pizza not being the same as the slices in a large pizza.

That suggestion really helped some students.  For the last 5 minutes of the class, students are asked to return to their desks and add to, or revise their journals with new understanding.