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* *Reflection: Standards Alignment
Same Numerator, Different Denominators - Section 2: Active Engagement

Before teaching with the Common Core State Standards in mind, I would not have done a lesson like this. I always taught students to compare fractions to a half and gave them "tricks" as to how to decide if if was greater than, less than, or equal to 1/2, without ever asking them why. I did not even consider it, thinking it was too much and, frankly, it wasn't in the TE. Now I know, aligning with the standards, offering rich tasks, and trusting the students and myself, brings about much more learning than a page of problems.

Listening to my students explain their reasoning, and asking them to challenge themselves to problem solve with journal prompts and performance tasks has made me a better teacher. The winner's in this are the students. Never before have I had the depth of thinking in my classroom. Never before have a just gotten quiet and traveled the road the students put before me.

To my surprise and delight, we have traveled further than I have ever taken them before…because they are at the head!

Journal-you will love the way the students lead you through the standards.

*New For Me*

*Standards Alignment: New For Me*

# Same Numerator, Different Denominators

Lesson 1 of 5

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

#### Mini Lesson

*10 min*

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.

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#### Active Engagement

*20 min*

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!

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#### Closing

*10 min*

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.

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- UNIT 1: Developing Mathematical Practices
- UNIT 2: Understanding Multiplication
- UNIT 3: Using Multiplication to Find Area
- UNIT 4: Understanding Division
- UNIT 5: Introduction To Fractions
- UNIT 6: Unit Fractions
- UNIT 7: Fractions: More Than A Whole
- UNIT 8: Comparing Fractions
- UNIT 9: Place Value
- UNIT 10: Fluency to Automoticity
- UNIT 11: Going Batty Over Measurement and Geometry
- UNIT 12: Review Activities