This task can also be used as an instructional task where students work in groups and explain to partners and/or the class how their diagrams represent \(5\times4\) and later \(5\times\frac\). While working on the task, the students can also look for patterns in the diagrams that they use to represent products that include a fraction as a factor. The task is designed so that students who look for and make use of structure (MP 7) can move forward even if they have not seen or have forgotten how to extend the meaning of multiplication to this case. Some students may be very comfortable with this and can adapt their diagrams right away, while others may not know what to do at this point. The third question assesses whether students can extend multiplication of whole numbers to the case where one number is not a whole number. The area representation will be especially useful for future work, especially when multiplying a fraction by a fraction. The teacher can sequence subsequent lessons to help students who only use a set representation to draw, e.g., a number line, and then see how that relates to the abstraction used in writing equations. They may use a set representation, length representations such as a number line or a bar diagram, or a rectangle where the side lengths correspond to the factors and the area corresponds to the product. Understanding whether students have access to one or more of these representations is useful for planning future lessons. After knowing how students understand multiplication and the different diagrams they use, the teacher can better integrate their prior knowledge into future lessons about fraction multiplication.įor the first two questions, the goal is to see what students know about the meaning of multiplication based on how they represent multiplication with contexts and diagrams versus simply having strategies to solve multiplication problems. This task can be used at the beginning of a fourth grade unit on multiplication with fractions to evaluate what students know about multiplication.
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