1) Sharing division. In it, you think that 16 / 2 means "If there are 16 beads divided between 2 people, how many does each one get?
2) Quotative division. In it, you think how many groups of the same size you can form. Or, "how many times does the divisor fit into the dividend?" For example, 16 / 2 is interpreted as: "If you have 16 beads, how many groups of 2 beads can you make?"
Sharing division is easy to understand as a concept, but it's hard to do without the knowledge of multiplication tables. Just imagine, if a child who doesn't know their times tables is trying to find the answer to the question, "If there are 56 strawberries and 8 people, how many will each one get?" It'll take some guesswork, trying and checking.
But quotative division is easy for children to do with the help of manipulatives (or by drawing pictures). Just make a group of 56, and start forming groups of 8 out of it. Once you're done, check how many groups you got. There's no guessing. It's just repeated subtraction - each group the child forms is "subtracted" or set apart.
For this reason I have decided to start off with quotative division in my Division 1 book. Children can first just partition off groups of certain size from the picture problems and count how many groups they got.
And this activitity lends itself perfectly to the 100-bead abacus, as well.
For example, we have 20 / 4.
Take 20 beads:
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
Group them in groups of 4.
| 0000 0000 00 |
| 0000 0000 00 |
Note the one group is formed from 2 beads on one bar and 2 on the other. Answer: 5 groups.
Another example: 27 / 9
First take 27 beads.
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
Form one group of 9 from the first bar (green), another from the second bar (blue). The third group is the purple ones.
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
Groups of fives are particularly interesting since they look so "orderly" on the abacus.
Ask your student, like I did today, what's 90 / 5?
Well, you organize your 90 beads like this, and answer is easily counted as 18 groups.
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
You can, of course, use abacus for sharing division, too. But make easy problems where the sharing is easy to do, such as 20 / 2 or 24 / 2 or 15 / 3.
Once the concepts of quotative division and sharing division are clear, you need to spend a lot of time with multiplication/division connection, because that is what children will use to solve most division problems - they'll need to get away from using manipulatives after the initial stages.
Math Mammoth Add & Subtract 4 covers the following addition and subtraction related topics which are approximately for the third and fourth grade level: