Cup Stacking Part 2

Today we tackled the challenge of building a triangle stack of cups that’s as tall as me. I’m 5’6” which we converted to 66”. Since the table is 30” tall the stack needed to be 36” tall.

We used our prior knowledge that the cup is 4.3 inches tall and divided 36” by 4.3” to know that there should be 8 rows of cups in a tower as tall as me.

We also noticed from experimenting that a triangle stack of cups has the same number of cups across the base as the number of rows high.

There will be 8 in one row, then 7, then 6, then 5 all the way to 1. Some knew a quick way to add these numbers up.

The next challenge was to build a triangle tower from the floor to be as tall as me. There were some quick calculations by some, and some intense stacking by others. It turns out that we needed way more than the initial estimate of double the number of the desk tower.

While the building was going on in one corner of the room, the rest of us were exploring what Desmos could tell us about the pattern we noticed.

We used x to represent how many rows tall our triangle would be, and y represents the number of cups used in the triangle. We could tell by looking at the 1st differences that it is not linear. The dots also do not make a straight line. We noticed that the 1st differences, or the “change” changes by 1 each time. When we see a second difference that is the same, the graph is quadratic so we can do a quadratic regression. Desmos has made this much easier now with a button at the top left of the table window.

We need to choose quadratic regression, and then the equation is given to us. We can use the graph as well to solve problems. If I want to make a triangle stack that is 15 rows tall, I can make x=15 then check to see the intersection of that vertical line and the graph.

We need 120 cups to make a 15 row tall triangle stack.

We will use Desmos often this year as we explore different relationships. It is a very powerful free tool.

Today we had fewer minutes off task, and less breakage of the cups. There was a pinky swear contract for good behaviour at the start of class which was respected, for which I am very thankful.