Long Division Help (or not):

Remember: Division is multiplication in reverse. What works for one, works for the other. Everything else is just icing.

Definitions:

Divisor: the number being “divided by”, the number “outside” the bracket (later, this will become the denominator, the number on the bottom of a fraction)

Dividend: the number being divided, the number “under” the bracket (later, this will become the numerator, the number on the top of a fraction)

Remainder: the amount “left over” after division is completed

The Steps:

1. Write the divisor. Draw the division bracket over it and write in the dividend. I suggest using graph paper for early division teaching to guarantee that all the lines and columns remain straight.

2. Many books now say something like “take the first partial dividend the least number of digits at the left of the dividend that will contain the divisor”. This overly complex wording is simply a short-cut to make certain you get your columns lined up at the right starting point. If you are confused, ignore this and go to step 3. If you do mess with this, skip step 3. The concept (i.e., really easy way to do it) behind this instruction is that you count the number of digits in the divisor then add one to that number. That number is how many columns you need to count across (left to right) to determine which column over the dividend you can begin to record numbers in. Example: 123456 divided by 13. 2 digits in the divisor plus one (3). Division begins above the digit 3 in the dividend. Personally, I think this is all too complicated for a new student of division and will be learned naturally through practice. I also would not bother with teaching or enforcing the “initial estimation” step.

3. Evaluate division by the column. The columns above the bracket, in line with the digits of the dividend, will guarantee that the results end up in the right positions. Look at the first digit—Can this digit be divided by the divisor? If not, put a ‘0’ in the column over that digit. This ‘0’ is just a place-holder to make sure the student learns to line everything up. Once this procedure becomes rote the student will automatically begin skipping it. Look at the next digit—Can this new, two-digit number be divided by the divisor? If not, put in the 0 and go to the next digit; continue this until the answer is yes. This column is where you will write the first number. This should become intuitive soon and common sense usually confirms what the student will find. If the rest of the number is covered-up, the division should make sense (i.e. 123456 divided by 13. 0 over the 1. 0 over the 2. Start in the column over the 3. If the rest of the number is covered, the division becomes 123 divided by 13 and looks reasonable to the experienced student). Once the answer is yes, go to 4.

4. Everything you do from here on can be done with the unused columns covered by a piece of paper. I use this technique and graph paper to help students get over their initial fear of large numbers and learn to tackle the problem systematically.

5. Use only the “uncovered” columns. Guess: How many times will the divisor go into the dividend? Just guess. Right or wrong, we make progress and the most wrong you can possibly be will require only that the problem has to be rewritten.

6. Take the number guessed. Multiply it by the divisor. Take the value calculated and write it in the columns with the lowest digit in the rightmost column determined in either step 2 or 3. Is this new number greater than the dividend? If so, revise your estimate to a lower number and repeat this step (Mattie must use an eraser. Other students may strike-over but Mattie must either erase or recopy. She is developing a very bad habit very early. It is for her own good if she stops this early. I know. I do the same thing). Otherwise, subtract the number. Is the remainder less than the divisor? If not, revise your estimate to a higher value and repeat this step. (There is a short-cut here, if you wish to use it. If the remainder from the estimate is such that you can easily guess the correct value, you can put in the correct value on top and simply subtract the difference. For example, if the remainder is 14 and the divisor is 13, simply increase the number on top and subtract 13 more from the remainder. I would recommend that students not be allowed to do this until they are clear on the procedure even though I have a bad habit of teaching this too early.)

7. Once the remainder is less than the divisor, increase your attention by one column (if we’re out of numbers, we’re done). Using this new, additional column, go back to step 5 and keep looping until done (help make sure that students are not confused by using 0 if necessary). The final remainder is, of course, the remainder.

Check Your Work:

1. Since division is the reverse of multiplication, the communicative property of mathematics guarantees reciprocity. In other words, if you check and it doesn’t pass, there must be an error somewhere.

2. Multiply the divisor by the “answer” from the division.

3. Add the remainder.

4. If this total is not exactly the same as the dividend, find the calculation error and correct from there.

Future Warnings:

Just a few warning for future reference:

1. Division will become fractions during the next two years in a good math program. Remainders are expressed as fractions with the remainder as the numerator and the divisor as denominator.

2. If you need an “answer” in decimals, put a 0 on the end of the remainder, a decimal point in the answer, and follow the same steps. Most teachers I know just love to make this more complicated than it needs to be.

3. If there is a decimal in the divisor, the easiest way to remove it is to multiply both divisor and dividend by 10 until the decimal is gone. Example: 0.25 divided by 1.333 becomes 2.5 by 13.33 becomes 25 by 133.3 becomes 250 by 1333.

The Cheats:

There are several different, non-standard approaches to division to try if the student (or teacher) is having trouble (or just finds it easier).

1. Obviously, if division is a way of expressing fractions, you can always convert the problem to a fraction and solve it that way. 6 divided by 3 or 6/3—either way, the answer (after reduction of the fraction) is 2.

2. Some people have trouble with the concept of division. Feel free to reverse the problem. Instead of 6 divided by 3, try “What number, when multiplied by 3, gives a result of 6”. Welcome to basic algebra. It seems more complicated but I’ve had success with this tact before.

3. Always remember the communicative property of mathematics. As long as you do the same thing to both divisor and dividend, you can do any math trick you want. If both numbers end in 0, divide both by ten (drop the 0) and then start dividing (130 by 40 is 13 by 4). If both numbers are even numbers, divide both by 2 and then start dividing (130 by 40 is also 65 by 20—hey, let’s take out a 5 and make it 13 by 4). Having trouble, double both numbers if it makes the problem easier to look at. As long as the adjustments are in ratio, do what is easiest for you.

4. Numbers divisible by 9 have a numeric sum of 9. Doubled digits are divisible by 11. All the old multiplication tricks you have learned still work, just the other way.

5. Do more homework. No one born after 1965 has been assigned enough math homework for the procedures to become ingrained. Unlike other studies, math must be practiced or it will be lost. As a student re-performs procedures they already know, they learn additional skills and pragmatic short-cuts they would not otherwise have learned (try estimating without a solid background in division, practice makes closer). In addition math teaches logic and a systematic approach to problem-solving. I strongly believe that if we taught less ‘self-esteem’ and humanities and more math, we would be a more humane people. Math builds confidence through repeated success and these skills have a large carry-over into other areas. Ignore the complaints and make up more problems, no child (or adult) ever died from too much homework (that I know of).

6. Show your work. All students should show every step of their work. This allows the teacher and the student to pin-point the exact problem. I’ve spent half my life cursing this requirement and the other half demanding it from other people.

7. Relate the work to the real world. Many students lose interest in math because they do not see any need for it. Truth be told, most adults I know believe that if they can balance their checkbooks, they have no more need of math. Most adults also would trade ignorance for bliss and call the deal a bargain. I think I have side-tracked from long division into teaching theory, sorry.

8. Longest Division: This is the emergency method I save for those who are completely stuck. It is very time consuming but it does help drive the concepts home is all else fails. Take the dividend and subtract the divisor. Subtract the divisor from the remainder. Repeat until exhausted and remainder is smaller than divisor. Count the number of times you had to subtract the divisor. There is your answer.