## IM Commentary

The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task engages students in Mathematical Practice Standard 5, Use appropriate tools strategically. The students are directed to create a number line marking 1/6, 2/6, 3/6, 4/6, and 5/6. Then they carefully cut out a strip of paper of 5/6 length and fold it in half and then fold it in half again. The paper folding along with the connection to the number line helps students visualize that when you divide a fraction in half, it is the same as multiplying ½ by 5/6. The same thinking corresponds to folding the strip once again in half. Folding the paper in half twice is really the same as folding it into fourths or multiplying 1/4 by 5/6.

As students become proficient in MP.5, they will be able to consider a tool’s usefulness and consider its strengths and limitations, as well as know how to use it appropriately. The solution pathway that a student selects needs to make sense to him/her to be able to explain and justify (MP.3). Students will come to realize that certain methods/tools are more efficient and they will abandon less useful tools in favor of more appropriate strategies/tools. Some tools might make more sense for a student to use depending on their level of understanding vs. their need for efficiency. While some tools might make more sense for beginning to understand a concept, others might make more sense once students are searching to make calculations in the most efficient way.