Estimating Division Problems: Compatible Numbers

My class is learning how to divide two- and three-digit numbers by one-digit numbers. First, we worked on dividing multiples of 10 and 100, then we moved on to estimating quotients using compatible numbers. Teachers sometimes skip this skill because they find it confusing. In my mind, though, it's an essential process for conceptualizing division and determining reasonableness.

You're probably thinking that I'm back on my estimation and reasonableness soapbox, and you're right about that! Here's how estimating with compatible numbers looks:

If kids learn to ask themselves the right questions, they can become great mathematicians. In this case, the question is simply, "What multiple of [the divisor] is closest to the first one or two digits of [the dividend]? Once they know that, they can quickly estimate the quotient.

At this point, some students may say, "I know the actual answer." And it's true, once a student has conceptualized this process, he or she may be able to take the next step and mentally think, "If 240 divided by 8 is 30, I'll have 17 left (because 257 minus 240 is 17). 17 divided by 8 is 2, remainder 1. Therefore, 257 divided by 8 must be 32, remainder 1."

After working with multiples of 10 and 100, as well as compatible numbers, my students are now ready to learn the common division algorithm. Wait! No, it's more than that! My students are now ready to understand the common division algorithm.