### Abacus and basic division concept

Let's be reminded first of all that there are two basic "interpretations" of division:

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.

`| 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

`| 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.

`| 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.

Anonymous said…
You can do 'sharing division' by repeated subtraction. 56 divided among 8 people, give 1 to each, have 48 left ...
Stacty said…
I like the quotative division idea, but I think you have it backwards. 16/2 is not 8 groups of 2 it's 2 groups of 8. These two concepts are drastically different. While asking the question, how many group of two can you make DOES get you the answer it gets you the answer with the wrong concept. If I were splitting 16 beads between two people (16/2) they would each have 8 beads. Maybe the question should be "how many beads will I have in each group if I make two groups?" This also sets up the idea of the remainder.
Maria Miller said…
Stacty,
I think you confused the concepts of quotative and sharing division. 16/2 can mean: "16 divided between 2 people", and that is sharing division. But it can also mean "How many groups of 2 are in 16?"

We can use the latter concept with many contexts, such as fraction division (3/4 : 1/4 is interpreted as "How many 1/4s are there in 3/4?" Or in word problems such as, "The class has 45 kids and they are organized into groups of 5. How many groups are there?"