Desmos to explore y=mx+b

What's the difference between a variable and a constant?

On Monday, I asked my Algebra 1 students this question with regard to $y=mx+b$. They weren't sure. To be fair, it can be a tricky question. Aren't they all variables? I can plug numbers into all the letters, can't I? I asked about situations involving money. What stays the same:

• the amount per bottle return or the number of bottles people bring back?
• the price of a candy bar or the number of candy bars purchased?
• the price per gallon of gas or the number of gallons purchased? 

I especially like the last paring because both things can change, but one of them is usually constant at least long enough to fill a tank!

Because they were having trouble seeing this when we get into the more abstract $y=mx+b$, let's explore! 

On Tuesday, I created this Google Document for students to use to explore linear equations in Desmos. For those that don't want to type in the information as the students did, you can use this to get started (just change the m and b values using the sliders to hit all the points).

Some students were slow to get started--I helped them find how to start. As they were exploring, one student said "Mr. Conrad, my brain hurts." I loved this comment because even though the kid was thinking hard, he wasn't quitting. 

During discussions, about where to find the m (slope) and b (y-intercept) values in the table, some students were shocked ("I wish I knew this before") at the connection to rate of change to m and when \(x=0\) in the table to the b value. I know that they've seen this concept before, but they've never yet made that connection. We'll see how well it sticks after Thanksgiving.

I am thankful that my district has 1:1 technology that makes it easier to explore concepts. I am thankful for students that are willing to try new things and learn from them.