Cups and Downs

Pattern recognition and making smart conjectures was an overarching theme that ran through the different course activities. We planned the next activity, Cups and Downs, with this in mind. Instead of cups (due to resource constraints), we used coloured number tokens that had a printed number on one face while the other face was blank. A photograph of one of our students working on a problem using pink coloured tokens is shown below.

The goal of the task was simple - our students were to identify the processes and patterns behind turning ‘n’ number of tokens from an inverted position (blank face) to a straight position (number face) by turning exactly ‘m’ number of tokens in each move where m < n. After experimenting with small values of m and n that we chose based on the mathematical reasoning behind the activities, we asked our kids to make conjectures about whether a solution will always exist and, if not, what were the conditions under which a solution did not exist. Based on their reasoning, we had them predict outcomes as the values of m and n increased.

Below is an example - there are 7 tokens numbered 1 to 7 (n = 7) and you are allowed to invert exactly 3 tokens at a time (m = 3). The upward arrow (↑) shows a numbered faced and the downward arrow (↓) shows a blank face. All tokens start by facing downward with the goal of making each them face upwards in the minimum number of moves.

Thus, this task can be accomplished in three steps. How about 8 tokens inverting 3 at a time? Are there combinations of m and n that make a solution impossible? These were the kind of tasks that we engaged our students in!

One of our students explaining his thought process to Jayasree Ma'am

After each child had experimented individually with different combinations for a while, we asked them to work in teams of 2-3 to come up with conjectures that they could generalise for different values of m and n. Thereafter, the teams presented their solutions to the class with an analysis of the patterns they had observed along with their reasoning behind why some cases work while others do not. They showed a willingness to correct their conjectures if convincingly disproved by another member of the class and worked to modify them for more rigorous general solutions.