Same Numerator, Different Denominators

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Objective

Students will be able to compare and explain fractions with the same numerator by reasoning about their size.

Big Idea

Understanding that fractions are related to their whole and that comparing fractions means one must compare the same size whole is critical to student's basic understanding of the concept.

Mini Lesson

10 minutes

To open the lesson, students gather in the community area with a white board and marker. I remind them that we have been looking at models of fractions and noticing equivalencies. Today we are going to begin to explore how to compare fractions that might look "equal", but are not.

I ask the students to discuss with their shoulder partner whether they think fractions with the same numerator, but different denominators, can be equal.  Many say yes.

Then, I have them draw two rectangles as close to the same size as possible. They are to divide one rectangle into fourths, and the other one into fifths. When that is done, I ask them to shade in 3/4 and 3/5 and label the rectangles with their fraction they are showing.

Now students discuss with their partner whether these fractions are the same -- because the numerators are the same. Now, they say no, these fractions are not the same.  Together we determine how to write the comparison: 3/4 > 3/5.

We walk through several similar examples, changing the numerators while keeping same as well as different denominators. Each time, students compare and indicate the sizes using the greater than, less than, or equal symbols.

Active Engagement

20 minutes

Journaling about fractions has been a focus for my class this week.  Journaling gives me a way to "get into their heads" and understand the breadth of my students' ability to communicate their mathematical reasoning, as well as the depth of their understanding and application of what they have learned. Today is no different.

I give the students a prompt which reads:

"Write and model two fractions with the same numerator. Which one is greater?  Explain."

I allow the students to roll dice to create the numerators and denominators, or they can create their own two fractions.

The students write and expressed themselves differently. In this clip, my student is explaining to me how he can tell his comparison is correct. He explained the size of the denominators, rather than discussing the numerator.

This student explains how he knows his comparison.  I did not prompt him to explore the idea that 2/4 equals 1/2, as the prompt asks him to consider fractions with the same numerator.  Causing him to see it differently at this point may have confused him.  I may have considered it with other students.  These are the types of teaching decisions we make every second of every lesson!

Closing

10 minutes

To close the lesson, students share their entries with each other, revise, and turn them in to me for scoring against our rubric.

I then ask them to look around the room and find examples of fractions with 3 as a numerator. As they do this (as a community), we compare the regions (wholes) and review that when you compare fractions, you must compare wholes of the same size.