Advanced Division {3-6}

With Beans, Cups and Bowls
We have worked with beans, cups and bowls as math manipulatives for a long while. We have used them to explore different base systems, place value, addition and subtractionbeginning multiplication, beginning division, and decimals. Now we use them for some advanced division problems. Sometimes I give them the problems and sometimes I will have them randomly make up problems using a die to roll out the numbers and sometimes I let them make up their own problems. Today I had him roll a 10-sided die to make up the problems. Here is the first one and how we worked through it using beans, cups and bowls. The first roll was for the beans, or ones column and he rolled a 6. The second roll was for the cups, or tens column, and he rolled a 0. The third roll was for the bowls, or hundreds column, and he rolled a 1. The fourth roll determines the number of groups he will be dividing the beans, cups and bowls into. In this case, he rolled a 4. We set up a little chart in his math notebook that holds this information. It looks like the above picture. He draws columns with four lines to them to represent the fact that we are dividing by four.
He sets up the board with the appropriate amount of beans; for this problem, 1 bowl (containing 10 cups with 10 beans in each cup), and 6 beans, equaling 106 beans in total.  For his own explorations, I give him ample time to make his own discoveries by trying to solve the problems in any way he wishes, but for my teaching sessions, I show him the method that I find is the easiest for me, which means starting with the bowls.
It is easy for him to see that the 1 bowl of 100 beans must be broken down into amounts of 10 in order to divide them, so he takes the cups out of the bowl and begins sorting them into four groups.
He notes at the top of the bowls column that 1 bowl cannot be divided without breaking it down by putting a zero above the bowls. He then breaks down the cups of ten into the four rows on the chart and sees that they break down into four groups, each containing two cups of ten and that he has two cups of ten left over, which he notes on the chart. It doesn't show it in this picture, but he also noted it  with a two above the tens column.
He then moves on to the remaining cups, or the tens. He cannot divide that by four, so he must take the  beans out of the cups and put them all in the ones column to be divided there.
He discovers that the two bean cups, once the beans have been taken out and the beans already in the beans column divides into four groups of six beans, with two beans remaining.
He writes the six above the ones column and puts the two for the two remaining beans as the top part of a fraction, with the bottom part as the divisor, or four, in this case, making 2/4. He didn't finish the re-drawing of the beans once they were put in the ones column because he didn't need to. He could see the answer without drawing it, so I did not insist on this for this problem. He may see the need for it in future problems, but he has to see the need for it himself for it to be useful to him. I also don't worry about simplifying the fraction at this point. That is to be done at another time.
Now that he has completed the process, he can begin making up his own problems and working them out, using the chart and board as much as he wants to.

Division with Chips

We started out our working with chips instead of beans by asking him to put five orange chips, four purple chips, three red chips. two blue chips and one green chip on the new trading board.
He has worked with this color scheme before and knows that each color is a multiple of ten of the chip to it's direct right.
We began by dividing this amount (54,321) by three. We began by starting at the ones/green column, making trades from the left, as needed. It wasn't too long before he discovered that this method did not work right.
So, we tried again, dividing the orange/ten-thousands column first into three groups. The leftover chips are converted to purples/thousands. This continues with purples, and all the other colors, moving towards the right. Finally the green/ones chips are divided, with any leftovers places to the right of the trading board, for the remainder fraction.
I also noted what he was doing on the board on the notebook page. I also noted what was happening on the board in the traditional form.
Now, he can roll dice to determine the number of chips to place in each column of the chip trading board, and the number of groups into which they are to be divided. He needs to learn how to draw a mini board on the notebook page with the appropriate number of rows beneath the chip trading board. He also can begin recording the problems in the traditional form, if he wishes, but is not required to at this time.

Long Division
There comes a point in which the problems become so large that the trading board becomes too cumbersome.
"...a time comes in both multiplication and division when the scope of a problem outstrips the practicality of using materials. In multiplication, this situation was alleviated by lattice multiplication. To supplement the students' capabilities in division  the next series of lessons presents an abstract system of dividing that serves the same purpose as the boxes in multiplication." -Bob Baratta-Lorton, Mathematics; a Way of Thinking
We began by his writing the numbers one through 10, putting a circle around each number. I then get him to go by five down the line, so by the 1, he puts 5 and by the 2, he puts 10 and so on.
So now we begin with a division problem, but I want it to be small enough so that we can check his problem with the chip trading boards. This should bridge the gap between them.
4321 divided by 5, and I write it in the traditional manner.
I get him to make a by 5's column as before and I get him to look for a number that could be subtracted from four. When he tells me there are none, I tell him to put a 0 over the 4.
What is the largest number in the fives column that we can take from 43?
40. Can I subtract 40 from 43?
Yes. What about the next number, 45?
No. Okay, we can't use that then. Let's use 40. The circled number next to 40 is what number?
8
Okay, so put the 8 above the 3 in the problem and put the 40 below the 43, and subtract.
Okay. We've used the four and the three. The next number we use is the two. We bring it down beside the three, like this.
Now we have 32. What is the largest number in the fives column we can take from 32?
We continue in this manner, writing the division problem in the usual way, and using the fives column to assist us.
When we get to the last number, which doesn't take any numbers from the fives column, we write it as a fractional remainder.
Advanced Division: Long Division Theory
Now, I get him to check his answer with the chip trading board, (just as I have done with his brother.)
I then give him more problems, with different column numbers to work with.
At some point, I ask him to teach me by telling me the steps, one-by-one as he does the problem.
At this point, he can create his own problems, either from whole cloth, or by rolling a die.
He also is asked to solve word problems...
We are having a party at our homeschool group, and we have been asked to bring peanuts to share. If there are 32 people who are coming to the party and the bag of peanuts we are bringing holds 125 peanuts, how many peanuts would each person get?...


2 comments:

  1. I'm loving looking through your maths posts. I'm still not brave enough to go only hands on but I'm definitely heading in that direction. Thanks so much for sharing your knowledge

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  2. I LOVE the things you do to teach Math. I so wish I had seen these when Keilee was younger. I bet your kids wake up happy and ready to learn most days. :)

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