The Division Algorithm as Mental Math
mental math series, part 14
Did you know the division algorithm is already a mental math process?
As we’ve seen before, left-to-right algorithms tend to be easier to do mentally. Fortunately we needn’t change our thinking to divide mentally because it already is a left-to-right algorithm.
All we have to do is think about each step’s meaning and practice it mentally.
Let me show you.
Terminology Review
The dividend is the number we are dividing into. The divisor is the number we are dividing by and the quotient is the answer.
Mental Division
Suppose we want to solve 256 ÷ 8.
Step 1: Begin by finding an acceptable range for how many times 8 goes into 256.
Because 8 x 10 = 80 and 8 x 100 = 800, we know 8 will go into 256 between 10 and 100 times. So we can safely assume our answer will be a two-digit number.
Step 2: Determine what multiple of 10 multiplied with 8 gets you closest to 256. Because 8 x 30 = 240, 8 goes into 256 30-something times.
Step 3: Count how many eights are between 240 to 256.
Two more eights will do the trick. So our final answer is 256 ÷ 8 = 32.
Traditional Algorithm
Let’s check this process against the pencil and paper method. We begin by finding how many eights go into 250 followed by how many eights go into 16.
This turns out to be identical to the mental math process we used above.
Another Example
This time let’s try 1012 ÷ 7.
Step 1: Find an acceptable range.
Because 1,012 lies between 700 and 7000, seven must go into 1,012 between 100 and 1000 times. This implies our answer will be a three-digit number.
Step 2: Since the answer is a three-digit number ask yourself, “seven times how many hundreds will fit into 1012?”
The closest we can get is 7 x 100 = 700. So 7 goes into 1,012 one hundred and something times.
Step 3: Take the difference between 700 and 1012, which is 312, and ask yourself, “seven times how many tens will fit into 312?”
Because 7 x 40 = 280, 4 tens is the closest we can get to 312 without going over. Therefore 7 goes into 1012 one hundred and forty-something times.
Step 4: Determine how many more 7’s we need to get close to 1012. So far we have found that 7 times 140 = 980.
Therefore, we need to find how many 7’s are in the difference of 1012 and 980, which is 32.
Since 7 x 4 = 28, we conclude 7 x 144 = 1008.
This means 7 goes into 1,012 one hundred and forty-four times with 4 left over.
Dividing With Larger Divisors
As with most things, as the numbers get larger solving problems mentally becomes more difficult. Feel free to implement this same process with larger divisors and dividends.
If that task seems daunting, try utilizing other division tricks to simplify the problem, such as these:
- Lesson Fifteen: Starter Tricks for Mental Math Division: I cover strategies for dividing by multiples of 10 and values such as 5, 50, 500
- Lesson Seventeen: Strategic Division: I show how to use factors of numbers to break division into a series of smaller problems.
Next Lesson: Prime Numbers and the Sieve of Eratosthenes
Thanks for reading!
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