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Long division, a long battle

The dreaded words that send chills down the backs of students and teachers alike… Long Division! There are so many barriers to students effectively learning how to do long division, and each grade level that has to teach this skill has its own set of difficulties.

When students begin long division in fourth grade, it comes right on the heels of multiplication fluency.  For some students, so close behind that they have not had time to cement the multiplication fact fluency to the point to easily retrieve this information.  Long division becomes an agony of repeated skip counting.

By sixth grade, there are still some students who still struggle with the same issues that plagued them in fourth grade, but new problems crop up.  Students have often been told in early grades, “The big number goes in the house.”  Well, its difficult to convince a preteen that their beloved fourth grade teacher was not exactly correct because now the size of the number has nothing to do with whether the number is the divisor or the dividend.  To complicate this, students are astonished to find out that there will no longer be writing remainders, but instead fractions and decimals.

It took me a long time to learn how to do all types of long division.  In fourth grade was the first time I was taught long division.  I will say taught and not learned because I definitely had not mastered it before I went to the next grade level.  It was so many steps, and I always seemed to mess them up.  The teacher tried to help by giving me mnemonics about famous restaurants and their cheeseburgers.  (Math teachers out there, I’m sure you’re familiar with this one!)  She tried asking me how many times the number “went into” the other.  I have always been incredibly literal in my thinking, so I was picturing the dividend as a little house with a door in it and the divisor would walk up to the door, knock, and ask for a cheeseburger!  Hilarious, but that was not helping me learn to divide.

I finally did learn how to do most long divisions well enough to make it through until I was in high enough math classes to use a calculator.   I have a shocking confession to make though– especially for a math educator– I never learned how to divide when the divisor was two digits!  I finally learned how to divide by two digit numbers about a year ago– after 10 years in the classroom, a successful college education in which I graduated at the top of my class, and three years of high school mathematics.

Because I was so bad at division myself, I feel a special connection with my students who also struggle with this skill.  I can often pick up on little things that are keeping students who are otherwise good math students from being able to divide. (…and also students who are not so great at math as well!)

Here are some of the things that might be tripping your students up and what you can do about it:

  • “The big number goes in the house.”  The big number does NOT go in the house!  While this may be true of the word problems seen at lower grade levels, it is not a mathematically correct statement.  This causes all sorts of issues when students reach grades where they are dividing to find decimal or fraction values.  Rather it would be more appropriate to say that the number being divided goes in the house.
  • “How many times does this number go into that number.”  Students who think very literally may not be able to see the meaning of this statement.  I have asked many students who have not be able to divide and are in higher grade levels if they can explain what that phrase means, and they cannot.  I ask if it would make more sense if I said something like “how many groups of this number can I make out of that number” and that usually makes more sense to them.  They can visualize the number being divided up into portions, but not the divisor repeating to make the dividend.
  • Dirty Monkeys and Cheeseburgers.  Mnemonics are great to help you remember something that you already have at least a basic understanding of, but they are not teaching tools.  Introducing mnemonics too soon can have a negative impact on students who need to cement concrete skills before they try to go to the abstract.  Although the mnemonics are fun and have their place for students who already understand the concept of division and are just needing reinforcement in the order of the steps, going to these devices too early won’t help students who do not understand what each step actually means.  In case you are not familiar with the mnemonics above and you are curious: Dirty Monkeys Smell Bad (Divide, Multiply, Subtract, Bring down) or Does McDonald’s Sell Cheese Burgers (Divide, Multiply, Subtract, Compare, Bring down).  There are of course many others and also variations of these two.
So what to do about this difficult skills?  Don’t worry I won’t leave you without some additional strategies to try!
Top and Bottom Division Strategy
This is the one that I used that helped most for students who were not completely fact fluent or where having difficulty with the number of steps involved in the standard teaching method.
  • Step 1: Make a multiplication table for the divisor.  Label the multiplier column with a “T” for top and the product column with a “B” for bottom.  Begin the table with 0 rather than with 1 and go through 9.
  • Step 2: Ask what is the largest number I can subtract from the dividend.  Have the student identify this number from the “bottom” column.
  • Step 3: Write the top and bottom pair.  Write the “top” above the digit in the “house” and write the “bottom” below.
  • Step 4: Have the student subtract, bring down, and repeat until the entire division is solved.

Benefits to this method are that once a student as practiced a few times it is three steps: Top and Bottom, Subtract, Bring Down.  Also, the insistence on the fact table to begin the work is useful to students who are not fluent yet.  Also, in this process, the emphasized skills are writing the multiplication facts and subtraction.  The student does not have to be able to visualize the dividing of the number.  Because of this, though, I like this method for older students who have been otherwise unsuccessful with the traditional method rather than this being go to method for a whole class.

Partial Quotients
This is the wonderful method that finally let me be able to divide by two digit divisors!  This method, while not used much before, is now becoming more and more common because of the emphasis on alternative methods by new state standards as well as by the common core standards.
In this method, you do not have to use the largest possible quotient, but you get to pick the quotient.  It is similar to a repeated subtraction method of division.  The main stumbling block to this method is the user has to have some number sense about compatible numbers and multiplication to use it effectively.
  • Step 1: Choose a number to multiply by that is easy for you to compute in your head.  This will often be numbers that end in zeros, or are multiples or 1, 2, 5, or 10 because using number sense these can be done mentally.  Write this number above the “house”.  Line up place values.
  • Step 2: Multiply the divisor by the number mentally.  Write this number under the number in the “house”.  Line up place values
  • Step 3: Subtract making sure your place values are lined up.  You will not be “bringing down” in this method.  The difference is now your new dividend for the next repetition.
  • Step 4: Repeat with another compatible quotient.  Stack it above the other, lining up the place values.  Continue through with the steps until your division is complete
  • Step 5: Add up all of the partial quotients to get your final answer.

What do you do to help your students learn long division?


4 thoughts on “Long division, a long battle”

  1. Thanks for making this homeschool parent feel better! I was getting a little bummed teaching my 4th grader long division with & without remainders. It’s good to know I’m not the only teacher trying not to pull my hair out. While doing flashcards, he’s quick to answer. When discussing the “why” an answer is the correct product, he can truly explain(ex. “28 divided by 7 = 4 because you have 4 in each set of 7”) But when presented with 29 divided by 7, and asked what can I multiply with 7 to get a product closest to 29 without going over, he replies, “3”. This may very well be asked RIGHT AFTER working 28 divided by 7.
    Sometimes I forget how my previous classroom students (20 yrs ago) would get a little confused while doing long division and begin to think that it must be me. It does seem that it would be easy for them to learn it right after multiplication facts, but if there are any “holes” there will be clear difficulties. I never heard of those mnemonics, I’d have probably gotten confused with them. But I do agree; I became better at long division with remainders when my fluency improved. Thanks again, I’ll try your techniques.

  2. Having struggled to grasp the basics of fractions and how to reduce them to their smallest equivalent fraction they are often not shown that division is just another way of representing a fraction. This can be a light bulb moment with single and double digit division because if students look at the number they may see ways of reducing the dividend & divisor down just as they do with fractions where they divide the numerator and the denominator by the same number. Even if they cannot reduce the number all the way down (because one of the numbers is prime or they don’t share any more common divisors) the division problem is now a lot easier to do because you are dealing with smaller numbers. I personally think that the partial quotient division is a lot easier for students to grasp as you only need to deduct multiples of 1, 2 or 5 to arrive at the correct answer. However I find that students benefit from practising left to right subtraction and using 10’s complement addition. For advanced students venturing into 3 digit divisors on large numbers it is worth using the Sum of Digits to check that the (divisor x quotient) + remainder = Dividend.


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