Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Partitioning Shapes

Partitioning involves dividing an object, set of objects or number of objects into parts. Students will be given a shape and they will divide the shape into equal parts.  According to Johnathan Fisher “Knowing that shapes, sets and quantities can be partitioned into equal-sized parts, and understanding the importance of equal-sized partitions, is fundamental to recognizing the part-whole relationship between the numerator and denominator in fractions”.  He also states, “One important aspect of partitioning is the reasoning that students employ to indicate how they know the partitions are equal sized, and how they could justify this to somebody else by drawing, showing or explaining. Students may need to fold or cut and overlay the pieces to illustrate their point”. Another aspect is that of encouraging students to find multiple ways of partitioning the same shape into the same number of equal parts.⁹

The area model represents a fraction as a part of a rectangle that is divided into equal parts. The area model will provide students with a visual. Models can help students clarify ideas that are often confused in a purely symbolic mode. Area models of fraction representation focus on the area of the whole unit being divided into equal portions.

My approach to teaching this unit is using the gradual release model. The gradual release of responsibility model is a traditional approach to teaching that is centered around the idea that a teacher will model a skill or strategy for a few minutes, briefly allow students to practice with some support, and then release the students to begin practicing on their own. The gradual release of responsibility model of instruction requires that the teacher shift from assuming “all the responsibility for preforming a task … to a situation in which the students assume all of the responsibility”. This approach has five stages that lead to effective learning. ¹⁰

The stages are as follows:

1. The first is the no awareness stage. This is the “I do” stage or focused instruction. This is when the teacher models how to solve or work through something aloud so students can see an example of how to think about the new information.
2. Next, is the Awareness stage where the student becomes aware of the focused instruction after the teacher has modeled.
3. Performing with more help is the “We do.” This is the Guided practice where the teacher and student work together.
4. Performing with less help is the “You do with a peer.” Students work with peers to solve problems.
5. Perform with no help is the “You do” by yourself. Students work independently to solve problems.

During this model I will start with the “I Do” portion of the model. I will provide an anchor chart for each lesson. All the students should focus and pay attention as the teacher models. Students should not have anything in their hands. All eyes should be on me, the teacher. During this time students are only watching and listening to me model and demonstrate several problems. During the “We Do”, students will work with me and guide me through a problem. Students will demonstrate that they understand and have grasped the concept of showing fractions using paper strips. Next, students will turn and talk to work on a problem with a partner. They will show a given fraction using number strips. They will then explain their thinking. This will allow all students to participate and learn from one another. The last part of the gradual release calls for the students to work independently. This is the “You Do” portion of the gradual release model.  Once students are working independently, I plan to circulate the room to make sure all students are engaged in their learning.  I will be checking for understanding and to see who needs intervention as I circulate the room. I will start pulling a small group of students who need extra support.  This unit will help struggling students and will help prepare them for more geometry concepts as well as algebraic thinking.