The scene: a room of 20 sixth-graders, giddy for the start of the new school year and well rested from their summer break. Each are handed a paper cup filled to the brim with Martinelli’s Sparkling Cider. Sixth Grade Teacher Karen Zaidberg, who will take them through a year of heavy-duty mathematics, implores them to raise their cups and proposes a salute to the new year. She demonstrates how to perform the perfect “cup clink,” which will officially usher them into sixth grade. They comply, and good cheer is shared as the students crisscross the room, making sure they have touched everybody’s cup once and only once.
Hidden in this merriment is a mathematics problem: assuming that all 20 students exchange toasts with one another, how many “clinks” should have been heard? In the preceding chaos, nobody had the foresight to keep count. Even if one member of the class had kept track of his or her own activity, would everyone agree that was the correct solution?
Welcome to the opening unit of sixth grade mathematics, where Manhattan Country School students embark on a three-week investigation to sketch networks, investigate patterns, graph triangular numbers and solve algebraic equations.
But why this kind of excitement, when many teachers use the first days of school to review previously taught skills or conduct assessments to see what their students forgot over the summer? The philosophy at MCS is that all these things can take place in a context where students are working in an environment that is both stimulating and dynamic, where answers are never obvious, and where the math takes place in the process of unlocking new truths.
Students approached the “clinking glass” problem using a number of strategies, such as drawing a “network” of dots to represent individual students and lines to record the glass clinks that were exchanged. As the number of people increased, the class sees that creating the network and recording the number of clinks gets more and more complicated, especially when you consider that making a network of 10 people would require 45 connections, each of which would have to be checked and counted to make sure each glass had clinked with each other one exactly once.
To confirm the results, the class models the problem by acting it out in slow motion, the first student clinking with 19 others, the second student clinking with 18 others, until the final student, the last to go, realizes that there is no one left with whom to clink. The equation to solve the problem works out to be 19 + 18 + 17 + 16….. all the way down to 1, which, upon first look, would appear quite cumbersome to solve.
Luckily, we have an 18th century German mathematician to thank for a shortcut to this problem. Karl Friedrich Gauss invented a technique to pair up numbers to create equal sums, thus finding the sum quickly and accurately. In this case, we can rewrite the equation 1 + 2 + 3 + 4…. + 19 as (1 + 19) + (2 +18) + (3 + 17) all the way up to (9 + 11), which creates nine pairs of 20, which, including the unpaired 10 in the middle, yields a result of 190 clinks.
But the fun doesn’t stop there: the glass clink problem offers up interesting patterns in the form of triangular numbers which can be graphed to create strange curves known as parabolas, which is an entirely new experience for our sixth-graders, and one they will encounter once again when they study graphing quadratic equations in eighth grade.
The “glass clink” investigation is based on an activity I taught at the Bank Street School for Children back in the mid-1990s. At the request of Upper School Director Maiya Jackson, I spent many hours of my summer break digging through my voluminous archives from three decades in math education to document, refine and polish this curriculum. Special props should go out to Karen, who patiently learned about the ins and outs of each activity, and was willing to go out on a limb to try something new. Future sixth grade classes can now look forward to clinking many cups of sparkling cider for the beginning of the new school year.
In this poster, the student compares two algebraic equations that will yield the solution to the glass clink problem with different numbers of participants.
This student used the equation N x (N - 1) ÷ 2 to calculate 190 clinks heard when 20 people tap cups.
This student explained how to use Gaussien Sums to solve the glass clink problem by pairing up numbers from the beginning and end of the addition problem.
This student created a table to show how adding people to the glass clink problem increased the number of clinks by a regular pattern, which resulted in the study of triangular numbers.
An example of an activity sheet where students used Gaussien Sums to calculate the sum of an arithmetic progression.
In this activity, the students graphed the first 10 square and rectangular numbers and compared the parabola that was created by each.