I came across this post by Andrew Busch and hit the jackpot for all things linear! Within this post were a couple of lessons I have used and some that I have seen other bloggers talk about. The one that caught my eye was the cup stacking. When looking through the new Illustrative Mathematics at this unit, I saw the cup stacking lesson and knew it must be meaningful. I went back to that post and read about how Andrew Stadel and Dan Meyer ran the lesson. In my search, I found where Sarah Carter had also used this lesson for rate of change. This {had} to be my next lesson, or math lab, as I have called them this year!

The math lab started with the simple question, "How many cups would you have to stack to reach the height of me?" After students threw out some guesses on whiteboards to get us started, I stacked five cups beside me for them to start visualizing. I wanted to draw out questions to get them thinking as well as ask for supplies they needed. Students already knew my height because they have been measuring themselves on the growth chart all year. They had to convert inches to centimeters but Siri gave them that information in 2 seconds!

Once they had my height in centimeters (152.4 cm), they tried several different strategies to make another guess at how many cups. Some students looked at how high five cups reached when next to me and then eyeballed how many more it would take. You could actually see their heads bobbing as they counted by fives from my chins up! At this point, I gave each group five cups. You can see below that this group measured the height of the five cups then used that height to make a table.

This group, and a couple others, did not take into account the cup has two parts that have to be considered when calculating the height of the stack. A common misconception was for students to measure how much the cup is increasing and then divide my height by that number.

Many groups recognized the two parts to the cup and figured out how to use them. They did not create an equation but they did exactly what the equation would have done. Once I walked them through writing the equation, they were able to see that we could put any amount of cups in and find the height or find the number of cups by substituting the height. It was amazing watching their thought process to figure out this problem!

I had help stacking the cups {my helper was dying to see if his group's guess was correct so he jumped up to help}! I stopped at each guess we had that was lower and asked if anyone wanted to change their guess. Students were excited watching us stack the cups waiting to see if their guess was correct!

The room was filled with "yessss" and high fives when the cups were stacked to my height! After the stacking, we recorded all of our thoughts in our notebooks. Students were able to see the rate of change and the initial value and how they came together in the equation.

This was a great bridge from rate of change to slope-intercept. It gave students something concrete to reference when talking about initial value, y-intercept, and slope. You could do this lesson as a 3Act with the videos in the links above or use Illustrative Mathematics' take on it as a guide.

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