What happens when something unexpected occurs? It is a question that I reflect on quite often, not just in teaching mathematics but in all types of situations. My son, who is autistic, is an extreme follower of rules. More than anyone I know, established routines govern his life and provide him comfort. However, when habits are broken, learning opportunities happen.

A prime example of a routine for him is the path we take to school or the store. If we take a different route, it will typically lead to frustration. At this moment, I have to make a choice as a parent:

- Offer him a new routine or rule to make sense of the change.
- Allow him to struggle with strategies he has used in other situations to make sense of what is going on.

The quick fix is to offer a new rule such as “this is the path we will now take on Tuesdays” instead of struggling with “it is ok that we take this route this one time, we have to be flexible, use some of your strategies.” The problem with the quick fix is that while it works now, what happens next Tuesday? Will I remember to go the new way?

I tell that personal anecdote as it is similar to some of the thoughts I have been having about the Algebra 1 curriculum that is currently under construction at Desmos. The unexpected happens for students in Algebra when they first encounter quadratic equations. Students spend most of middle school mathematics learning about linear patterns through proportional reasoning and constant rates of change. However, when teaching quadratic equations, there is a potential tendency to jump to various rules and define new situations for students (some of which do not work in every case). Here are just a few:

- F.O.I.L. (First, Outside, Inside, Last)
- Factoring Rules with Boxes or X’s
- Standard Form
- Factored Form
- Vertex Form

Not saying that defined structures, rules or formulas are not an important part of Algebra. The current unit Desmos has released addresses the last three and a slew of vocabulary words related to quadratics. The build-up of conceptual understanding through patterns and visual representations is where the unit shines. It allows students to use familiar strategies instead of rules to make sense of quadratics.

For mathematicians, analyzing patterns and exploring visual representations are two transferable strategies when encountering a new or unexpected situation.

#### Analyzing Patterns

Devoting time to exploring the pattern of quadratic functions can be a helpful step in transitioning from linear to quadratic thinking. Students come to Algebra from years of pattern explorations in earlier grades and this should feel somewhat familiar to them, but with a new twist.

At the beginning of the unit students are asked to build a pattern and then extend it to step ten with some Desmos interactive magic.

#### Exploring Visual Representations

After exploring the pattern of quadratics, students have space to play with the graph. Understanding the visual representation of a quadratic in the form of a graph is pivotal to developing connections to various types of quadratic equations.

Students are encouraged to manipulate the graph of quadratic functions using the vertex, concavity, and the concept of symmetry to create some visual art.

There are a variety of approaches to teaching any mathematics content, but are we overloading students with a set of rules that may not work next Tuesday? Consider anchoring students in analyzing patterns and exploring visual representations as transferable mathematical strategies to lean on in any situation.

Side note: If you have not made some parabola art yet, highly recommended, it is a lot of fun.