During the warm up of this lesson, students practice ordering fractions on a number line. This is a skill they have practiced, and I present to them related unit fractions including halves, fourths, and eighths. Another option would be thirds and sixths. This practice of ordering fractions on a number line helps students to develop a conceptual understanding of the quantities represented.
To begin the mini-lesson I take a piece of paper and I cut it into two pieces, but I purposely do not fold or have it marked in half. I cut it close to half, but not so differently it is visually clear to determine. I keep the pieces separated so that they can not be measured side by side. I repeat this with a piece of paper that has been marked with equal thirds and cut on the lines. I ask the students, "Do these three pieces show equal thirds?" The students respond quickly "Yes." My next question, as the basis of this lesson, is "How can you prove that?"
After a few responses from the students, I model for them the language I will be asking them to use during their presentations and group work with their shapes. This includes, "These pieces are equal thirds because I can compare each rectangle and see that they are all the same size in length and width."
The shapes the students will be using vary and include triangles, hexagons, rectangles, squares and circles. Some of these are divided into equal units, and others are divided into non-equal units. I want the students to be able understand and explain when a shape is divided into pieces it does not always represent equal unit fractions.
Each group of students are given different shapes and units to discuss within their group and decide if it contains unit fractions. During the group work, the students are not cutting apart their shapes and individual units because I want them to construct their argument for their position. The Common Core Mathematical Practice of explaining your reasoning and critiquing the reasoning of others (MP3) is important because students will need to communicate their thinking during the presentation portion of the lesson. I require each student in the pair or group to be able to provide their explanation, because each one of them will be contributing at some point during the presentation.
Each group presents their shapes and provides explanations about the fractions represented and if they are unit fractions. Students listening to the presentations are encouraged to ask questions about the process used to determine if the shapes show equal unit fractions.
After the groups present, students cut apart their shapes and make the final determination for equal unit fractions.
Another option is to use the worksheet as an assessment on this day or on a different day.