Division
Tips for Division
1. There are two distinct concepts of division, the idea of dividing into equal groups and the idea of repeated subtraction. Since all students visualize and understand things differently, be sure to allow your students to use both concepts to model division.
Look at these expressions:
4 / 0
0 / 4
Many students incorrectly evaluate one or both expressions. Tell your students to check their answers using multiplication. Is 4 / 0 = 0? If it is, then 0 x 0 = 4. Since this is incorrect, then 4 / 0 does not equal 0. Is 0 / 4 = 0? If it is,
then 0 x 4 = 0. Since this is correct, 0 / 4 = 0.
2. Dividing by 3
Add up the digits: if the sum is divisible by three, then the number is as well. Examples: 111111: the digits add to 6 so the whole number is divisible by three. 87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three.
3. Dividing by 11
Let's look at 352, which is divisible by 11; the answer is 32. 3+2 is 5; another way to say this is that 35 2 is 33. Now look at 3531, which is also divisible by 11. It is not a coincidence that 3531 is 352 and 11 ?321 is 3531.
Here is a generalization of this system. Let's look at the number 94186565.
First we want to find whether it is divisible by 11, but on the way we are going to save the numbers that we use: in every step we will subtract the last digit from the other digits, then save the subtracted amount in order. Start with
9418656  5 = 9418651 SAVE 5
Then 941865  1 = 941864 SAVE 1
Then 94186  4 = 94182 SAVE 4
Then 9418  2 = 9416 SAVE 2
Then 941  6 = 935 SAVE 6
Then 93  5 = 88 SAVE 5
Then 8  8 = 0 SAVE 8
Now write the numbers we saved in reverse order, and we have 8562415, which multiplied by 11 is 94186565.
