After the students worked with their partners, Ms. Cariello encouraged them to talk about their interpretations of the problem in a whole-class discussion. The students then completed the rest of the problem set. Following this lesson, Ms. Cariello was still concerned about the students' confusion and wanted to furnish another context for them to think about these ideas. She thought it important that the students have opportunities to recognize the similarities among the different representations of the fractions they had considered, as well as to discern the differences among them. On the next day, Ms. Cariello introduced these tasks by challenging the whole class to imagine that one of the rods was named a particular number and then to figure out the name of each of the other rods, whose lengths are proportional to the one named rod.
Flexibility is needed during the lesson.
The two boys read the problem individually, commented to each other that it was difficult to understand, and then sat quietly, each rereading and thinking alone about possible ways to begin. After checking to see that the two trains were the same length, he pushed aside the red rods one at a time, quietly counting the number that he had used. Brian turned to Ryan and asserted that he now knew what he was doing. Ryan, I'm trying to figure . . . I don't get it.