Cup Stacking & Linear Equations

This week in my teacher education mathematics class we visited a problem of stacking cups. The problem was created in a few levels. First our instructor showed us one cup and posed the question of how many cups tall she was. We all provided our estimations. These were recorded to be revisited later.

Next we were asked what information we would need to find out an exact answer. We decided that we would need the height of the cup and the height of our instructor. From here it wasn't very difficult to solve for the number of cups. We chose to stack the cups in alternation of inversion. All we needed to do was divid the height of the instructor by the height of one cup. This would give us the height of our instructor with cups as the unit of measure. We compared this to the estimations to see which groups were close. This provides opportunity to discuss errors.

Our instructor then posed the next level of the problem; we had to stack the cups all the same way (how they came in their packaging). Now she asked what new information we might need to solve the question. We decided that we needed to know the height of the "lip" of the cup. From here the solve was a little more difficult than before but still easily comprehended. You would have the height of one cup (the top cup) plus x number of cups to reach the height of the instructor.

At this point the math gets a little interesting and the connection to linear equations becomes evident. With a group of students you can have them develop the equation that represents this linear relationship from only the information given so far. The y-intercept would be the height of one cup and the slope of the line would be the height of the "lip" of the cup.

Another approach to this problem could be as follows. Provide the students with two district heights of stacked cup (i.e., 5 cups is 20cm high and 15 cups is 32cm high.) From here the students could use the two points as coordinate points and develop a linear equation to represent it from here. This would involve them deriving the slope of the relationship with ∆Y/∆X.

Overall, this is engaging activity and provides easy and practical connection to linear equations. Furthermore, it is easily modified to allow for differentiated instruction which as educators know is fundamentally important to a good less.