When students struggle with ladder division, many times it is because they learned a procedure and haven’t made sense of the procedure for themselves. In this case students haven’t had enough experiences with division problems that are near friendly numbers so that they can reason about the numbers. Try giving students some problems that are near friendly numbers first if you notice that they aren’t using number sense to form partial quotients. For example, if students continue to subtract only small groups of ten and aren’t able to estimate a larger number for partial quotients, then try giving students numbers that are easier with which to estimate, like in the picture below. In the above picture, I started with some students in intervention who started solving the problem using groups of ten. 2499 divided by 25 is obviously close to 2500. Why not start with 100 groups and reason about taking away one group so that the quotient isn’t too large. When students look at how they could estimate to solve this problem, they have a lightbulb moment. Give them other examples like this to get students in the habit of solving problems with reasoning and number sense as opposed to a procedure.
Well, with large numbers this is something that my fellow colleagues did not feel comfortable teaching. And when that happens…who steps in?? None other than The Mathemagician…ta-da! (which is me of course, but shhh don’t tell anyone!!)…I’ve been off for a few days as I write this, which makes me a little sillier than normal–and probably slightly more interesting! So, on with my lesson! Now just remember when I post these pictures it is not a beautiful, I spent weeks preparing, colorful, lesson. This is a practical lesson anyone could use whether you are savvy with a computer or not. (I may turn this lesson into something more aesthetically pleasing later on.) The part that stumped the teachers was the fact that the standard says “up to 4-digit dividends”.
CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
I had to think myself about how to teach this before I spouted off a lesson idea. (I know I am supposed to be the math expert in my building, but, honestly, I have to look up what exactly common core is expecting before I assume I know what a standard is asking students to do.) I looked at North Carolina Unpacked–my go to document for what common core expects…
After sitting in a week’s worth of meetings about Common Core instruction and best math practices, I feel as if my way of instructing students is gradually morphing into a new beast, which I am not sure I fully can describe yet. In the midst of all of this, I went back to the fifth graders I have been working with on the day after these meetings. I decided to try out some of the teacher practices I had been learning. Sidenote: This set of fifth graders isn’t yet proficient at long division or ladder division for that matter. I posed a long division problem to the students, but instead of giving it to them in “naked numbers” alone, I gave it to them in word problem form as well. In reflecting, I should have started out with the word problem alone and let the students figure out that they were supposed to divide. After watching the students struggle with solving using ladder division or the traditional long division algorithm, I encouraged the students make equal groups to see if that helped them. One student in particular that I worked closely with (whose work is pictured below) managed to find a correct answer using equal grouping, however he could not find his mistake when he used ladder division. Another student…one I might add who typically doesn’t have much perseverance to solve problems…related completely to the contextual situation and persevered through to solve the problem to the end. To get to my MAJOR DIVISION REVELATION, let’s look at this student work below:
So, you ask, what is the big AH-HA? If you notice that the numbers in the student’s circles (equal groups) are the same numbers to the side of the division problem (or the ladder). The numbers to the side of the ladder are actually the chunks that students would naturally break off if they were naturally dividing cookies among people etc.! Now I know according to my PD classes that I shouldn’t have just told and shown the student this relationship, but I should have allowed him to figure it out. I was just a little too excited to hold in my own personal discovery I suppose.
Another discovery I had in experimenting with having students discover their own strategies, I learned that students actually do better and persevere more when the problem is in a context familiar to them and it isn’t just a naked numbers problem. Having a contextual situation actually gives the students an entry point into the problem, and so they don’t give up as quickly. I have always thought backwards. Students MUST UNDERSTAND THE NAKED NUMBERS FIRST and THEN they can SOLVE HARD WORD PROBLEMS. But in reality after my experiment with these fifth graders, they were much better at solving this division problem in their own way when they had a context. I have had a complete paradigm shift, and you were all here to see it above– that is.
Generally it takes students a few weeks initially to learn division– especially long division with multiple digits. Marilyn Burns’ book Extending Division has several helpful hints within her lessons. In place of the traditional method for division, she shows how to teach students with the ladder method in which students divide a dividend with multiples of ten rather than break the dividend apart like the traditional method. The ladder method helps students understand numbers more effectively because they are estimating and thinking of the dividend as a whole. See this example to gain a better understanding of the ladder method.
Add all of the numbers to the right (the ladder) to find the quotient.