## Reflection: Developing a Conceptual Understanding Three Ways to Solve a Problem - Section 3: Guided Problem Solving: Three Ways to Solve a Problem

The transition from our detailed study of guess and check, to graphing systems, and then onto algebraic substitution yields all sorts of opportunities for students to make sense of why algebra and the tools that compose it exist.  Upon teaching this sequence again, I am excited to remember how students engage in some big ideas over the course of the unit.

One ideal outcome for this unit, beyond straightforward mastery of the canonical systems trifecta - graphing, substitution, elimination - is that students really understand mathematics as a process of choosing the right tool for the right job.  That this is even a question is lost on many of my students when we start working together.  There are processes to learn, my students think, whether there's good reason to learn them or not.  If my students saw substitution in such a way, then I'd be missing an important opportunity.  The point of this unit is guide students to the point where, by the time they learn about substitution, they are amazed by it, and grateful that such an elegant tool exists.

That's where today's "Three Ways to Solve a Problem" finds us: we review guess and check and graphing more time, before seeing that - WOW! - this whole substitution thing makes so much sense, and actually feels easier than not using algebra!

With that in mind, we must also investigate the converse: that if substitution is not the easiest way to solve a problem, then we shouldn't force ourselves to use it.  Isn't it actually harmful to do so?  To force such practice when it's unnecessary may prevent students from really getting it.  The active mind with which we choose an appropriate process is what I hope all of my students will develop during their 9th grade year.  This unit is essential to that goal.

At the Intersection of Formal and Not
Developing a Conceptual Understanding: At the Intersection of Formal and Not

# Three Ways to Solve a Problem

Unit 9: Systems of Equations
Lesson 9 of 20

## Big Idea: Over the course of today’s lesson, students will beg for an algebraic solution, understand on their own why it’s great that algebra was invented, and see the beauty of an elegant solution.

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43 minutes

### James Dunseith

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