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Two-way table inquiry

 The minimum amount of data required to complete an m by n two-way table is given by the product of m and n.  The questions and observations below come from students aged between 11 and 14 in small-group exploratory discussions with a teacher.Is the prompt saying that to complete a 3 by 2 two-way table, you need six pieces of data?What is a two-way table? (When shown an example of a two-way table table, the student asked "Do the dimensions of the table include the 'Total' row and column?")What values for m and n can we use?How many ways are there to arrange mn pieces of data in an m by n two-way table?How many ways are there to complete a two-way table table if you are given the required amount of data?Is it possible to create an incomplete two-way table that can be completed in one way only?Can you complete an m by n two-way table with less than mn pieces of data?Note: The prompt is simpler if the dimensions (m, n) are assumed to exclude the headings and the 'Total' row and column. However, some students found the prompt easier to understand when all the data cells are included in the dimensions. This would make the prompt:The minimum amount of data required to complete an (m + 1) by (n + 1) two-way table is given by the product of m and n. Reasoning through inquiry In the two-way table, each dot represents a piece of data. There are two chains of reasoning that could be used to complete the table (key: 'Tot' stands for Total):Chain 1: cell YA ⇒ YB ⇒ TotB ⇒ TotC ⇒ XC ⇒ XTotChain 2: XTot ⇒ XC ⇒ TotC ⇒ TotB ⇒ YB ⇒ YAIt is noticeable that one chain is the reverse of the other. The teacher might direct students to consider this as one solution. Students can start to complete the table if there are two pieces of data in a column or three in a row. If, for example, there were four pieces of data in row X and three in column A, then no solution is possible. Line of inquiry There are 924 arrangements of six dots in 12 cells, which are too many to find in most classrooms. However, the idea of combinations (and permutations) is a line of inquiry that could develop from the prompt. It is possible to list systematically the number of arrangements of, for example, three dots in five cells (10) and four dots in seven cells (35).