## Reflection: Developing a Conceptual Understanding A Graph is a Set of Points - Section 1: Opener / Mini Lesson: How to write the equation of a line through two points, without graphing.

There is always some subset of students who recall the notation-heavy version of the slope formula.  There is a larger subset of students who can almost recall the formula before getting bogged down in where "x-one" and "y-two" go.  When the inevitable conversation is raised about why we're not using the formula, I ask, "What does that formula mean?"  I give students a few moments to chew on that, and maybe we make progress, but I'm also wary of moving quickly here, because talk of formulas is precisely what causes another subset of my kids to tune out.

Instead, as I've outlined above, I reiterate what we're doing when we calculate slope.  We are using a ratio to compare the amount that y changes to the amount that x changes.  I provide the visual.  Here, there's the inevitable exclamation - this way is so much easier! - at which point it's my job to swing the pendulum back again and show students that whether you use the formula or draw arrows, you're really doing the same thing!

So while I'm thinking of this, here are some side notes that came to mind as I taught the lesson today:

Note #1: I'm not doing away with mathematical notation or vocabulary.  I want students to understand the "delta" symbol for change, and at the same time that I replace the subscripted formula with labeled arrows, I'm using that symbol accordingly.  I do not shy away from using the word ratio to talk about slope, because that's what slope is.

Note #2: In general, why does subtraction feel so hard for so many kids, but telling the distance between two numbers feels so much easier?  Does the way that students get tripped up on the formula make it feel harder to subtract?

Note #3: The graph that results from today's activity - and the "steps" that many students will draw on it - is similar in flavor to the arrows I use to denote slope.  It's fertile ground for an extension to talking about reducing fractions.  I don't specifically set out to do that here, but to do so is useful!

What is a "Formula," Anyway?
Developing a Conceptual Understanding: What is a "Formula," Anyway?

# A Graph is a Set of Points

Unit 7: Lines
Lesson 5 of 10

## Big Idea: Students practice writing the equation of a line through two points, and then they see that different pairs of points can result in the same linear equation, which results in a big breakthrough for many kids.

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43 minutes

### James Dunseith

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