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Long Division-With Remainders (1-digit Divisor)

Division (once you get past the division facts that are just the inverse of times tables) is the most complicated whole number operation -- mostly because doing long division requires multiplication and subtraction.  Students who have not learned their times tables or mastered subtraction with borrowing will struggle with long division.

The trickiest part of long division is keeping numbers and place values oriented properly (when you divide into a number you must divide the divisor into each digit of the dividend).

In the US, we use a "house" set-up to organize the numbers. The "house" separates the divisor and the dividend, provides lots of space for multipliation and subtraction operations below the dividend, and segregates the answer safely above the dividend.  This organization helps keep digits lined up, so that the divisor is divided into each digit of the dividend.  It's also helpful once students start using decimals in their answers (or dividends) because it helps keep the decimals lined up.

The problem with the American "house" system is that it switches the order of the divisor and dividend.

So, the problem: $56 \div 4=$ is written: $4\overline{)56}$

It is often difficult for students to get used to the fact that $4\overline{)56}$ is said "56 divided by 4" -- but they should practice because part of understanding division is being able to say the problems correctly (and understanding that when a number is divided by another number, it's cut into that many pieces).

With the American "house" system we follow a common procedure in long division:

  1. Division: Divide the divisor into the first digit(s) of the dividend (write the answer on top of the house).
  2. Multiply: Multiply the answer number times the divisor (write the answer under the portion fo the dividend you are dividing into).
  3. Subtract: Subtract the product of the last step from the portion of the dividend you are working with (write the answer below).
  4. Check: Make sure that answer to your subtraction problem is not greater than your divisor (if it is, then the number you wrote on the top line is too small).
  5. Bring down: Bring the next digit in the dividend down and make it the ones digit of the number you just checked against the divisor.
  6. Repeat: How many times does the divisor go into the new number you just created (write the answer on the top of the house, above the number you just brought down).

Some students use an acronym to remember the steps.  You can try one of the ones below or make up your own.  It's often helpful for begining dividers to write the acronym on the side of their papers and check off the steps as they go along.

Bring downBatheBurgers

So, how do the steps work?

Example: $786\div 5=$

Set the problem up:

$$\eqalign{5&\overline{)786}\qquad&&\text{Write the problem in a house}}$$

Do the first "division" step.  How many times does 5 go into the first digit in the dividend?

$$\eqalign{ \quad & \; \color{red}{1}\qquad && \color{red}{\text{5 goes into 7 one time, write 1 over the 7}}\\5 & \overline{)786}\qquad && \quad \\&\!\! \underline{-5}  \qquad && \text{5 times 1 is five; subtract from 7}\\  &\;\; 2  \qquad && \text{7 minus 5 equals 2}}$$

Now, you bring down and divide into the new number created when you bring a new digit from the dividend and place it next to the remainder left over from the first time you multiplied and subtracted!

$$\eqalign{ \quad & \; \color{red}{1}\color{green}5\qquad && \color{green}{\text{5 goes into 28 five times, write the 5 over the 8}}\\5 & \overline{)786}\qquad && \quad \\&\!\! \underline{-5} \color{green}{\downarrow} \qquad && \\  &\;\; 2\color{green}{8}  \qquad && \text{Bring down the 8 and put it next to the 2}\\&\!\!\underline{-25}\qquad &&\text{5 times 5 is 25}\\&\;\;\;3\qquad&&\text{28 minus 25 is 3}}$$

There is one more digit in the dividend, so you'll bring down one more time, then divide again into the new number.

$$\eqalign{ \quad & \; \color{red}{1}\color{green}5\color{blue}{7}\qquad && \color{blue}{\text{5 goes into 36 seven times, write the 7 over the 5}}\\5 & \overline{)786}\qquad && \quad \\&\!\! \underline{-5} \color{green}{\downarrow} \qquad && \\  &\; 2\color{green}{8}  \qquad && \\&\!\!\!\underline{-25}\color{blue}{\downarrow}\qquad &&\\&\;\;3\color{blue}{6}\qquad&&\text{Bring down the 6}\\&\!\underline{-35}\qquad&&\text{7 times 5 is 35}\\&\quad1\qquad&&\text{36 minus 35 is 1, you have a remainder of 1}}$$

Bring the remainder up to the top.  Most elementary students use a small "r" to denote the remainder.  It's also helpful to show students that remainder can easily be turned into a fraction: $\displaystyle{\frac{\text{remainder}}{\text{divisor}}}$

The answer is now at the top of the "house."  $\color{purple}{785 \div 5 = 157 \text{ r }1}$ or $\color{purple}{785 \div 5 = 157 \displaystyle{\frac{1}{5}}}$

Overall, when we use the "house" to divide, we dividing a divisor into a dividend, place value by place value. By lining the answer up on top of the house, over each digit of the dividend, we help make sure that we get the place value correct (and know where to place the decimal if we have a decimal).

What to look out for?

The most common long division errors involve zeros.  Students tend to leave zeros out.  When practicing division make sure pay attention to the following trouble areas:

  • When the divisor does NOT go into the dividend (or the newly created dividend after a new digit is brought down) a zero must be entered in the answer row before another digit is brought down.
  • When the divisor does NOT go into the last number created when the last digit is brought down, a zero must be entered in the answer row, above the last digit of the dividend, before they write the remainder.

The other common error is that students often forget to check, after they subtract, their difference against the divisor.  Remind students to check.  Also remind students that if they want to enter two digits on the answer row (e.g., if they try to divide 39 by 3 and want to enter 13 in the answer row), they did something wrong in the prior step.  Using the house method, they will only put single digits at a time in the answer row.

Common Core Grade Level/Subject

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