All Fractions Are Not Created Equal

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Students will be able to identify unit fractions and recognize that the comparisons are valid only when the two fractions refer to the same size whole.

Big Idea

If you were really hungry, would you want 1/4 of a large pizza, or a quarter of a small pizza? This is the type of question students will be able to answer after this experience.

Whole Group Lesson

20 minutes

Before gathering the students, I will ask them to talk at their tables about the fractions 1/2 and 2/4. They are to determine if those two fractions are equivalent and why they think so, or not.  The purpose of this discussion is to focus students on comparing fractions in preparation to thinking about the size of the whole.

As the students gather in the community area, I show them two pizza boxes (donated by the local pizzaria) which are closed. Inside one box is a large pizza. Inside the other box is a small pizza. Both are cut into fourths.

I tell the children that they are to imagine themselves as very, very hungry.  I then ask them to turn and tell their partners which pizza box they would want to choose.  If they chose one box, they will get ¼. In the other box they will also receive ¼.

The idea here is to create conversation about how all fraction names do not mean that the size of the fraction (or amount) is the same. I am hoping that they will ask about what size pizza is in the box.  If they do, I will engage them in the conversation of why they are asking, probing to get at why it matters.

If the conversation doesn't move there on its own, I ask one student to chose two students to come up and help open the boxes. They open their boxes at the same time and we then have a visual to support the discussion of choices as a whole class. 

Next, I place a purple rod on the document camera, telling students that this rod represents ONE. In order to model using the rods as different WHOLES, I ask the students to find what rods could be used to make 1/2 of the purple. (With my fraction rods, that choice would be red, because two of them equal the whole and the denominator requires two pieces.)  

I have the students trace their WHOLE rod and then discover the unit fractions, recording in their reflection journals.  This is a critical step, as it helps students move from the concrete to the visualization phase.

To solidify thinking, and allow me to get a sense of who's grasping this, we do this with different unit fractions and different wholes for a few more examples.

The unit fractions to be addressed are 1/4, 1/6, 1/5 and then we move to finding other fractions, such as 4/5 or 2/3.

Once I'm satisfied that students understand what they are thinking about, and representing, they ready to work on their own.

Active Engagement

20 minutes

Prior to today's lesson, the students had an opportunity to explore fraction rods, discovering the patterns of how different rod lengths (distinguished by different, consistent colors) can be aligned to create lengths equal each other.  

Today, students use the rods to compare fractions based on various wholes - which means I must establish with the students that although prior to this lesson we all agreed that one particular rod represented whole, today we will use a variety of representations of wholes.  If you chose to use the rods, be sure to give the children plenty of exploration time prior to the lesson, as they will need to be accurate with their use of the tool. If it is very confusing for students to use the same tools, but in slightly different ways, you could use fraction strips in the prior lesson, and rods in this lesson.

The students are sent off with their partners to work through defining fractions based on the size of the given whole using a table derived from Exploring Fractions, a Model Curriculum Unit created for the Massachusetts Department of Elementary and Secondary Education.  

As you work with students, confer with them in a way that shapes their communication to be more precise mathematically (MP6).  Many times students may be able to "do" the work, but struggle to show/explain their thinking. We know that rigor goes beyond "knowing and even applying. It is also found in the expectation of clear, specific, and precise mathematical communication.

In this clip, my student knows what he is doing, but his challenge is finding a way to express it.  I prompt him to build his model and talk me through it.  At first, he just keeps talking!

Many may not finish in this session, which is okay, as the task of explaining thinking is the key. You may choose to be done with this lesson after a day, or finish up the next day. 

Review and Closing

5 minutes

As a review, I have the students share what they have done on the board and have students discuss any mistakes or critiques. 

I then repeat that all fractions depend on the size of their whole and remind them of the pizza problem.  I also inform them that the next lesson will be more work with the same concept, but we may use different tools. 

Telling them this is important, as it reminds them that fractions are not always just part of a pizza or a candy bar, but could be anywhere and everywhere. 

My homework assignment is for them to look around their home for as many examples off 1/4 as possible. We will open the next day's lesson sharing these findings.