Long Division the Easy Way
Long division. Those two little words evoke more groans out of students and parents than any other mathematical algorithm. It can be painful, that's for sure. Those of us who grew up before calculators remember doing those gawd-awful problems, trying to figure out how many times this guzinto that. We ended up erasing our work when we guessed wrong until we had nothing but a big black smudge where once there had been a division problem. And heaven forbid that we should have any scratch work around the problem.
Now that everybody has a calculator, many people say division is obsolete. I lost count of the number of students and parents who simply couldn't understand why it was even in the curriculum. There is a lot of institutional resistance to teaching and practicing long division, so much so that I've known some math teachers in the primary grades who didn't teach it. I got many students in the 6th grade who couldn't do a long division problem if their life had depended on it. By the time they left me in the 8th grade, they were experts. Here's how I did it.
The difference in this algorithm is that you don't have to nail the number the first time, then go back and erase it if you guessed wrong. Instead, we use a series of estimates to whittle the problem down to size. These estimates are written down the right side of the problem. At the end, those estimates are added together along with any remainders and the final answer can be written on top of the box. The scratch work is done along side the problem, keeping it clean for the answer. The only thing written on the top of the box is the final answer.
First, I'll show a real simple one to demonstrate the mechanics. By using small numbers that we already know the answer to, we are using a problem solving strategy called Modeling.
Here's the problem:
Step 1: Think - How many 4's are there in 9? Let's say the student says there's 1 of them.
Step 2: Write - The student writes a 1 off to the right of the problem indicating that they have estimated that there is one 4 in 9.
Step 3: Multiply 1 x 4, which equals 4
Step 4: Subtract the 4 from 9 to get 5.
Step 5: Think - I can take another 4 out of the 5? Yes
Step 6: Write - The student writes a 1 off to the right of the problem indicating that they have estimated that there is one more 4 in 5.
Step 7: Multiply 1 x 4, which equals 4
Step 8: Subtract the 4 from 5 to get 1
Step 9: Think - 1 is less than 4, so I have a piece of a 4 but not the whole thing. I have a remainder, which we can express a number of ways. We'll use fractions for remainders right now. So I have a remainder of 1 out of the 4 or ¼. (NOT 1/9th)
Step 10: Add the estimates. 1 + 1 = 2 plus the leftover ¼ = 2¼
Step 11: Write the final answer in the assigned space
Here's the problem using the above steps.
5 ------ 1
1 ------ less than 4 = a fraction = a remainder of 1 out of 4 or ¼
Add 1 +1 + 1/4 = 2 ¼
= 2 1/4 or
This looks like a lot of steps for a simple problem but if you look, there's a pattern.
THINK > WRITE > MULTIPLY > SUBTRACT
You keep doing that until the remainder inside the problem is less than the divisor outside the problem. In this case, we went until we ended up with a 1 inside the box and 4 outside. Then we calculate the remainder, put it in proper format, add the estimates to the remainder then write the final answer where indicated.
When we get to the remainder, this is the sequence. You only do it one time.
THINK > CONVERT > ADD > FINISH
This system is flexible and self-correcting. It's also scalable for different grades and academic levels. With a little number sense, anyone can do this, even if they don't have their times tables or place value down cold. Eventually they will. If they don't hit the numbers right the first time, they make another estimate instead of erasing everything and starting over. If a student had recognized right off the bat that there were two 4's in 9, their first estimate would have been 2 and they would have gone straight to the remainder.
This is also a great tool for diagnosing error patterns. They have to show their work as part of doing the problem. Arithmetic errors? Clerical errors? Basic facts? Number concepts? Place value? Fractions? Decimals? Terminology? Algorithms? I don't need a big day long standardized test to tell me where a student is with their math. I can do it in a couple of long division problems, especially if they do them while I'm watching. Are they talking to themselves? Counting on their fingers? Facial expression? Body language? A math problem like this with an observant teacher will tell all.
Here's another problem with bigger numbers. Now there are some place value considerations. Instead of worrying about starting with 3 or 30 or 301, just use the whole number. Start estimating how many 14's there are in 301 right off the bat. That's how we estimate in real life anyway. They'll get there eventually, some faster than others.
Problem: Write it out in words so they understand what they are to do: three hundred one divided by fourteen or fourteen goes into 301 how many times.
------ 5 (THINK: Student estimates there are five 14's in 301. WRITE: 5 )
-70 ------ (MULTIPLY: 5 x 14 = 70. SUBTRACT 70 from 301 leaving 231)
231 ----- 10 (THINK: Realizes 5 was way low. Estimates another 10. WRITE: 10 )
-140 ----- (MULTIPLY: 10 x 14 = 140. SUBTRACT 140 from 231 leaving 91)
91 ----- 5 (THINK: Used before was 70. Use it again. WRITE: 5 )
-70 ----- (MULTIPLY: 5 x 14 = 70. SUBTRACT: 70 from 91 leaving 21)
21 ----- 1 (THINK: There's one more 14 in 21. WRITE: 1)
-14 ----- (MULTIPLY: 1 x 14 = 14. SUBTRACT: 14 from 21 leaving 7)
7 ----- ½ (THINK: 7 is less than 14 but not zero so I have a remainder.
(CONVERT: I have 7 leftover out of 14 which equals 7/14 = 1/2)
(ADD: Estimates and the remainder. 5 + 10 + 5 + 1 + ½ = 21 ½)
(FINISH: Clearly write the final answer in the correct place.)
This is a great system. Not only does it simplify long division, it exercises every other math operation. Long division problems were a staple in my classroom and it really paid off. Confidence went up, test scores went up and when my 8th graders got to the high school, they ran circles around everyone else.
For assessments, warm ups, bonus points, quick classroom fillers and review, you can't beat long division problems. Just a few at a time will work wonders. Give this system a try. You'll never go back.
Si facile, omnes esset facere....Mister L.