Disclaimer: I am not saying that this method did not exist, I am just saying that prior to me creating, I did not see it anywhere. So I created a model that worked with my students, and decided to give the method a name.
I THOUGHT the time was approaching for me to get ready to teach long division. I definitely wanted to get a head start with teaching the algorithm, because I know the struggle of students remembering all of the steps to solving the problem.
My students were not grasping the conceptual understanding of division and remainders the way that the should. In real-life situations they weren't successfully interpreting remainders. When given situations, some of my students were not able to interpret what happened in the situation because of what was left from the equal group. Due to those reasons, I knew I needed to be very intentional with the language that I used with my students whenever I was talking about dividing.
Traditionally, when teaching long division I've taught: Divide, Multiply, Subtract, Bring Down/Over (if Horizontal Long Division), and repeat. I didn't want to use that language, with the lack of conceptual understanding. I didn't want to throw in any confusion and tell them that when they are dividing, they must multiply, and then subtract. Another thing that I noticed was the language that was used in the process of showing them the traditional algorithm. For instance, when solving 7,568 divided by 3 I would ask "How many times does 3 go into 7?" That isn't necessarily wrong, but I wanted to be very intentional that my language aligned with building my students concepts of division, quotient, and remainders. So I started to ask them, "What is the quotient of 7 divided by 3?" When my student answered with the quotient, I realized there wasn't a need for them to multiply and then subtract. I could simply just ask them what the remainder was and explain to them they needed to unbundle the remainder by moving it to the previous place so that it could be joined with the digit in that place. I didn't like the fact that multiply and subtract were mandatory in the process. I wanted kids to understand that for each digit they were finding the quotient, and moving the remainder. IF they needed to subtract to find the remainder. OK. If they found the remainder by counting on. Ok. However, I didn't want them to miss the fact that they those additional steps in long division were only in place only to help them find the remainder.
I explained to my students about the quotient and remainder. I told them that that I would be coming up with a model to show them that I believe they would love. On my planning period one day, I played around with many models, and finally decided on the model below! And I named it, Quotient-Remainder Short Division, QRS Division for short.
My students LOVED it. Since I had previously shown them long division, I did emphasize that Vertical or Horizontal Long Division was an option. However, they loved the fact that in many cases they did not have to multiply and subtract, because they were able to find the remainder by counting on. However, for the sake of progression, I emphasize that in some cases they may find the need to subtract to in order to find the remainder. They viewed it as quicker and all of the steps were not overwhelming to them. Many of my students felt like they were doing less and that problems were easier, because it didn't require all of the steps. However, the biggest win to me was that they were constantly viewing each step for what it is...Dividing to find quotients and remainders.
I created QRS at a time when I thought I had to teach long division. I am guilty of not fully reading my standards. Yes, my students should be finding quotients and remainders of 3-4 digit numbers. However, fourth graders should be doing that using different strategies based on place value. Per the common core standards and many state standards, students are NOT expected to do the standard algorithm for long division until 6th-grade. There are many great place value strategies that students can use to divide. QRS is an algorithm because it is digit-based. However, it is still a much better alternative to long division in my opinion. It isn't a place value strategy, but it does help build concept by constantly reinforcing division as quotients and remainders, because in reality that is what dividing is. Sometime the remainder is left as a whole. Sometimes the remainder is a decimal. Regardless, of how the remainder is represented...it is still a remainder.