Thursday, May 9, 2013

Using Partial Quotients to Teach Long Division

Do you know how to use partial quotients to teach long division?

Let's take a look at the common core standards for long division

Grade 4
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.

Grade 5
CCSS.Math.Content.5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-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.

Grade 6
CCSS.Math.Content.6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm

There seems to be a lot about alternative strategies and using what kids know about place value and the relationship between multiplication and division to develop long division ideas.  We have been committed to this idea in my school for the last 5-6 years.  It has really helped kids understand division and be much better at estimating an answer and knowing when an answer is unreasonable.  It took a lot of convincing but all the teachers in my school have now agreed that we do not teach the standard algorithm for long division until grade 6.  Now the common core supports this as well.

So now in our school we start talking about partial quotients in grade 4.  We use rectangular arrays and area models to support the development of this model and we get kids to a pretty good place by the end of grade 4.  The best part is that kids can use numbers that are friendly and fluent to them to solve these problems.  

So here is some evidence of this strategy at work from today's fourth grade math class.  We got a new student last week who has no concept of division or any strategies for long division so when I do a booster group with them next week, I will post more ideas about how to develop the partial quotients using arrays and area models.  But for now, here are the different ways kids used partial quotients to solve this problem.  

This student took more steps than any other student in the class.  They did 5 partial quotients but used numbers that were very friendly known facts.  

This is by far the most popular was this problem was solved today.  Can you see how it leads right into the traditional algorithm?

Another very efficient strategy
This student used 4 partial quotients to solve the problem

What strategies do your students use for long division?


  1. I'm curious to see how you relate the partial products algorithm to arrays and area models.

  2. I would love to show you! I have been getting a lot of emails and questions about partial quotients and will be working on a new blog post that really she's how I teach these ideas in grades 4 and 5. Stay tuned! I hope to have it completed soon!

  3. What's wrong with teaching them to do it using place value the way I learned in school. My child can do it and get the correct answer in half the steps.

    1. Many students can be successful with learning steps to a procedure but some students don't remember the steps because they never really understand what they are doing. This method really uses what they already know about multiplication and builds up from there. Teaching a procedure only works for some kids and does not connect to any previous learning.

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  5. I understand how partial quotient helps children who have a hard time learning the steps way. What about the students who grasp the "old" way the way we all learned? In testing? Does the new common core testing cover both areas, is there a middle ground for the students who can learn the "old" way or is it only the "new" way? In your experience and opinion?

    1. In my opinion as well as my understanding of how the Common Core testing is supposed to work the goal is to make sure kids have an efficient method for solving division problems as well as strong conceptual understanding. If I felt a kid already had a solid understanding of the traditional algorithm I would not go back and do partial quotients. However, as an adult who was very successful with the old way learning partial quotients has helped me to better estimate quotients and solve quite complicated division problems in my head.

  6. I am so glad I found this! I teach fourth grade, and while I do enjoy the new way of thinking in math, teaching division has stumped me this year. It has been my experience over 25 years, that more students struggle with the algorithm than understand it easily. I think this will really help. AND I was one of those students who struggled. I wish my teachers would have had strategies like this in their arsenal when I was in grade school! :-) Thanks for posting!