This inquiry has tended to be shorter than others on the website, perhaps lasting two hours of classroom time. Moreover, it has also required more guidance (see Levels of inquiry). Often comments about the prompt from younger students relate to the difference between the terms. These can generate valid questions for inquiry:
- The difference between the two amounts you add horizontally is one, the same as for the vertical amounts.
- Is it possible to find other numbers that have the same horizontal and vertical differences?
- The differences create a sequence: 1.5, 2, 2.5, 3.
- Is it possible to create a linear sequence from the differences of other terms?
- What about other types of sequences?
However, the mathematical focus of the inquiry should be on the multiplicative relations if students are to understand the concept of ratio. It is at this point that the teacher might need to guide the inquiry with pertinent questions or by stipulating that there can only be one horizontal 'rule' and one vertical 'rule'. If fractional multipliers are likely to confuse the class, the teacher could start with a prompt that has whole number multipliers (without forgetting that prompts should be set above the class's current level of knowledge to encourage curiosity and speculation). At this point in the inquiry, students will often decide to create their own examples (on template 1 below).
Three interesting changes to the prompt that have been suggested by students are presented below: