For the time being we're going to talk about the problem 575 ÷ 4
The most basic definition of division is splitting a number into smaller, equal sized groups, so 575 ÷ 4 really means I have 575 things and I am going to split them into 4 smaller, equal groups . Initially students tend to show this with beans counting out one large group and then dealing them one by one into each group until they run out or cannot give each group the same amount - accurate but not the most time effective.
Base Ten Blocks
Students soon discover that they can use base ten blocks to represent the same numbers and piles of beans are replaced by blocks allowing students to county by tens - a little more efficient. Students start by building their starting number (dividend) and imagining four groups to sort the blocks evenly into.
Then they put as many hundred blocks as possible into each group.
If they end up with extra hundred blocks, they break each hundred block into tens rods - this doesn't change the number of blocks they are sorting, just the versatility of putting less than 100 at a time into each group.
Then they put as many ten rods as possible into each group.
If they end up with extra ten rods, they break each ten rod into ones blocks - again this doesn't change the total number of blocks, just how they are grouped.
Then they put as many one blocks as possible into each group.
Counting up the number of blocks in each group gives you the quotient and the number of blocks that could not be divided is the remainder (or left over). So 575 blocks divided into 4 groups equals 143 blocks in each group and a remainder of 3
Groups Picture Using Numbers
Students move on from blocks to drawing the same ideas with numbers replacing the blocks. We try to hold onto the idea of division as a story, so they imagine dividing sharks or balloons or marshmallows - whatever helps them to remember what they are doing instead of just memorizing steps. They practice keeping track of how many they can put into each group, how many items they have used, and how many they have left to still divide. In this picture representation the number in the box represents how many items remain to be divided and the circles represent the four groups.
We start by putting 100 into each group. That means we used up 400 of our beginning 575 and now only have 175 items left to divide.
If I put 40 more into each group then I used 160 items (40 + 40 + 40 + 40) and I only have 15 left to divide.
If I put 3 more into each group then I used 12 and I only have 3 left to divide.
I can't divide 3 items into 4 groups (without going into fractions or decimals which we don't cover in 4th grade division) so I have a remainder of 3 and 143 in each group.
The partial quotients is the closest strategy to the traditional method we have covered so far. It is really just a numbers representation of the ideas above, so I put the pictures side by side so you can see where the ideas fit in our mental picture.
I start by deciding how many items I can put into each of my four groups. Since the four groups will all end up identical at the end of my division I don't need to keep track of them separately. My blue numbers along the left side tell me how much I put into each group. since I put 100 into each group I used 400 of my beginning 575 and I only have 175 left to still divide.
Now I put 40 more into each group, used 160 of the 175 and have 15 left over to still divide.
3 more put into each group means I used 12 items and only have 3 more left.
Since I cannot divide 3 items equally into 4 groups, the 3 becomes my remainder. I add the blue numbers on the side to determine how much ended up in each of my four groups. I put 100 into each group, then 40 into each group, then 3 into each group, so I have 143 total in each group.
The biggest benefit of the Partial Quotient method over the traditional method for younger students is it's flexibility. The method still works even if you don't put nice round number into the groups each time. Below are several examples of ways students could find the answer using different Partial Quotients methods. Though they put a different amount in each group, they are all still keeping track of the same ideas and they still find the correct final quotient. This allows them to work at their own pace and understanding level.
Math Help >