Rules that Expire
As students explore math they begin to make rules, and they expect their work to obey these rules. As students progress they bump into math where their rules does not work anymore. This is an opportunity to explore how to alter the rule to accommodate this new information.
A rule that expires this week is that division always makes your dividend smaller. Students will discover that 5 ÷ 1/3 = 15. This quotient will initially be a bit disorienting as it breaks a rule they have created.
Students can make sense of this by exploring what 5 ÷ 1/3 is asking them to do. This expression is asking them to find out how many 1/3s there are in 5 wholes. This connects to the work you did with them last week converting fractions. An example of a model you may see is below:
In this example 5 was broken into thirds, and then the thirds were counted.
At this point we are not yet teaching an algorithm, but we can ask students to start looking for patterns they see. The algorithm will be further explored in coming posts.