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The second way we discussed modeling math was through discussing the different interpretative models (i.e., understanding dividing fractions). For the sharing brownie problem, division was interpreted as a partitive model where it was used to see how many pieces of brownie each of the four children received. As students’ examined this task, their interpretive model of the problem was evident. Many of the students started by drawing the five brownies and were able to divide them equally “fair share” among the four children. However, they struggled with labeling the pieces and writing the fraction.

Other students were able to count the 5/4 but were unable to see that this was the same as one whole and V. As students shared their solutions with the class, they noted that the groups had different models of “wholes” resulting in a rich class discussion about how fractions are part of a whole and that “whole” can be different. During the lesson, students who had no exposure to mixed numbers were able to explain their understanding of 1 V using words such as one whole and V more, a whole and a quarter, or one big piece and then V of the piece. This notion of understanding that dividing can be represented as a partitive model is important. Just as it is important that division can also be represented with a measurement or quotitive model, which repeatly measures off a portion, for example, when interpreting 1 У2 divided by V which can be interpreted as how many V cup servings can I get from 1 У cups.

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