## IM Commentary

This task is meant to be used in an instructional setting. Part (b) of this task is intentionally left open-ended to encourage students to develop the habit of looking for patterns that might hint at some underlying structure as described in Standard for Mathematical Practice 7, Look for and make use of structure.

This kind of work is also related to making conjectures and determining whether those conjectures are true or not as described in Standard for Mathematical Practice 1, Make sense of problems and persevere in solving them. Some of the things that students might notice about the numbers in the table don't really go anywhere; in fact, that is the nature of making conjectures: they don't always turn out to be true.

Part (c) is meant to engage students in what should naturally follow after noticing a pattern, namely, investigating whether it always holds and if so, explaining why. So this task also has students engage in Standard for Mathematical Practice 3, Construct viable arguments and critique the reasoning of others.

A variant on this task would remove the scaffolding from part (c). In either case, having small objects, like chips or coins, might help students develop their explanations.

Submitted by Sherri Martinie and Melisa Hancock of Kansas State University, Manhattan, KS to the first Illustrative Mathematics Task writing contest.

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. Possible secondary practice connections may be discussed but not in the same degree of detail.

Even though this task illustrates several mathematical practices, it is being used to highlight Mathematical Practice Standard 7. As students follow the given rule, they should start to see that all of the entries are odd (MP.8). In order to understand why they are all odd, students need to use the structure of even and odd numbers, which is the point at which MP.7 comes into play. The task then asks them to explain what they see, so that they will engage in MP.3.