# The prompt Mathematical inquiry processes: Verify a general statement with particular cases; reason about necessary conditions. Conceptual field of inquiry: Two-way tables; categorical data; permutations.

The contention in the prompt is that the minimum amount of data required to complete a two-way table is the product of two less than the length (m) and two less than the height (n). So, if the length and height were five - with both including column or rows for the categories and totals (see below) - the minimum amount of data would be nine. ## Students' questions and observations

These questions and observations come from students aged between 11 and 14 in small-group exploratory discussions with a teacher.

• 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 do m and n stand for?

• Is the prompt saying that to complete a 4-by-3 two-way table, you need just two pieces of data?

• What values for m and n can we use?

• Is four the smallest value for m and n? Or could it be three?

• How many ways are there to arrange 6 pieces of data in 5-by-4 two-way table?

• How many ways are there to arrange (m - 2)(n - 2) 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 (m - 2)(n - 2) pieces of data?

# Reasoning through inquiry The contention in the prompt is that a minimum of six pieces of data are required to complete a 5-by-4 two-way table. In the table above, each dot represents a piece of data. There are two chains of reasoning that could be used to complete it (key: 'Tot' stands for Total):

• Chain 1: cell YA ⇒ YB ⇒ TotB ⇒ TotC ⇒ XC ⇒ XTot

• Chain 2: XTot ⇒ XC ⇒ TotC ⇒ TotB ⇒ YB ⇒ YA

It 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).