I just LOVE the MTBoS Search Engine! I want to send out a public thank-you to all who contributed to making this happen.

I was planning my lesson on systems of linear equations, and within an hour, I had a nice progression . . .

I started with Andrew Stadel‘s stacking cups Act 1 video. I proceeded a la Dan Meyer by asking – what questions do you have? This resulted in all sorts of answers, but soon, students began asking about the measurements . . . I decided to feed the hungry and tell them these cups were pre-measured for them!

This was great! I got them into their #vnps formation with a deck of cards, and off they went at their collective question of – *how many cups until they are the same height?*

Some groups expressed some frustration with understanding the problem so I brought in some stacked cups and suggested they try to find their heights. This helped.

Over the next 20 minutes, interesting discussions were generated within the groups:

- One was around
*rate of change*as the steady increment of cup height. - The other was around
*defining variables*.

My favorite question was when a student asked:

“Why is it that at 0 we have a height? Shouldn’t 0 cups have no height?”

My response was:

“What does your x mean?”

Somehow, this question brought light into the situation, and the group revised their equation.

Being inspired by this group’s realization, I began asking other groups about what their variables mean. Some explained their x refers to the number of cups in the stack, while others explained their x refers to the number of cups in the stack *after the first cup*.

I chose to bring attention to this to the class as a whole. Once they sat down, I asked them what the equation should be. I had mixed answers. Then I asked what x means for each. As a whole class, we noted that y = 1.3 x + 9.2 is an equation for height y of x cups after the first cup, and y = 1.3 (x-1) + 9.2 is an equation for height y of x cups total. We also noticed that 1.3 was the rate of change in both cases, and it was the slope of the graph of that line.

Because of this discrepancy, the class as a whole has now established that *defining variables* is VERY IMPORTANT! This has become part of the classroom culture and discourse.

If I had told them to begin with, “you must define your variables,” it would have not made it into their practice – BUT, since it really mattered in this problem, it stuck! More importantly, a general focus on *meaning* was introduced.

This was followed by a general discussion about systems of equations, possible results to a system of equations, and approaches to solving.

Although this is the dry part, I find that the time we took to systematically have them work through solving the following systems was time well spent because it brought them into their comfort zone of algebra:

After this formality, I sensed the desire for more problems! Time to get them back out of their comfort zone and into ambiguity again!

The scenarios we explored included the following #MTBoS inspired ideas:

Double Stuff! (from Christopher Danielson’s “My Oreo Manifesto“)

Glue Sticks! (from Kyle Pearce’s “Piling Up Systems of Linear Equations“)

If we had more time, I would ask them to create their own scenarios.