This activity is another one that stems from one by Al Overwijk, but a bit more directed than his. In Al’s activity, he opens it up to any model of cup stacking which results in relations that are linear, quadratic & beyond. Maybe my class can come back to this later in the course & do it that way. For now, we stuck to a nested cup model that I knew would result in a linear pattern since that was my learning goal for this activity.Also different from Al is we used a stool simply because we don’t have enough solo cups to get to the height of a person when they’re nested inside each other 😦
The Scenario:
Act 1: Notice, Wonder & Estimate
What do you notice (facts)? 
What do you wonder (Q’s)? 

What is the height of the chair 
The cup is way smaller then the stool 
How many cups do you need to get the same height 
Each picture has more cups 
How many cups does each picture have 
there more than 1 cup. 
why do you have only 3 chairs? 
Each picture has different number of cup 
How many cups that can be same hight with the chair 
there’s more cups in each picture 
how much cups does it 
Stool and cups 
How many cups we need to put together so that they have the same height with the stool? 

What’s the height of the chair 
The number of cups increase 
Why are there cups 
– There are a different number of cups in each picture.
– The stool is much taller than the stacked cups.
– The only thing that changed is the number of cups 
– How many cups will it take to stack it as high as the stool? 
More cups in each picture 
How many cups do we need to cover the hight of the chair 
cups 
what is the volume of each cups? 
The number of cups is increasing 
How many cups would it take to be the same height as the stool 
3 stools, cups is increasing 
How many cups it will take to reach the height of the stool 
In each picture there is a certain amount of cups
The chair is always the same 
How many cups it increases at a time 
Act 2: Collect data & solve
Each group received 8 cups and were instructed not to share their cups with any other group. They quickly grabbed rulers & began measuring. They made tables of values on their boards to record their measurements. But quickly most groups began to look for patterns after taking 2 measurements & then continued their tables based on their assumptions of how it will continue to grow. But I let them continue with that thinking for now. Here were there boards:
So every group had an answer but the strategy was lower level (tables and making assumptions about rate of change). Next I had them log in to a Desmos activity I had built to get them to perform a regression on their data. I expected them to input heights for 1, 2, 3, …, 7, 8 cups. But of course they input the tables they’d made on their boards that were full of inaccurate assumptions about how the stack would grow. So we had a discussion about whether their tables were data or assumptions. I asked which one would we rather base our regression on? They decided data. So I handed 8 cups back out to each group (this was now onto day 2 & so their groups had changed since we do “visibly random grouping”) and asked them to measure the height for 1, 2, 3, …, 7, 8 cups and adjust their tables in Desmos. The activity asked them to decide if the relation would be linear or not (they agreed yes) and then asked them to perform a linear regression. I love the recent Desmos feature where it carries their current graph & work forward to the next slide/question! Here are some samples of their Desmos work:
I then asked them to take one of the equations for line of best fit that they’d found in their group and go to their boards to solve for the number of cups needed to match the stool’s height:
Act 3
We stacked the cups as a group next to the school and the actual answer was 73! We discussed possible reasons for differences between our first answers, our answers after using Desmos & the real answer.
A third day was spent practicing two skills that had been assigned via Khan Academy over the previous days of work:
Graph from slopeintercept form
Slopeintercept equation from graph
The entire activity is available here.
– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)