Box models are powerful because they provide a general method for dividing large numbers. And, as we've seen, they can also handle cases with remainders.
Let's consider the following three-digit by one-digit division problem.
We set up our box model in the usual way.
We now follow our usual four steps:
Step 1: Pick a (fairly large) multiple of that is easy to compute but is no larger than Let's pick
We write on top of the box and subtract from inside the box.
Step 2: Next, we bring to the right.
Step 3: Pick a multiple of that is easy to compute but is no larger than Let's pick
We write above the box and subtract from inside the box.
We get so the division is done.
Step 4: To find the quotient of we add the numbers on top of the boxes:
Therefore,
Let's now look at an example with a remainder.
Use the box model above to compute the quotient of
First, we subtract from
Next, we bring to the right.
Then, we pick a multiple of that is easy to compute but is no larger than Let's pick
We write above the box and subtract from inside the box.
We can't subtract any further since is less than Hence, is the remainder.
The quotient is the sum of the numbers on top of the boxes:
Therefore,
Use the box model above to compute the value of $954 \div 9.$
|
a
|
$104$ |
|
b
|
$105$ |
|
c
|
$110$ |
|
d
|
$108$ |
|
e
|
$106$ |
Use the box model above to compute the quotient of $809 \div 4.$
|
a
|
$200$ |
|
b
|
$204$ |
|
c
|
$203$ |
|
d
|
$202$ |
|
e
|
$201$ |
Laura paid dollars for concert tickets. Using the box model above, find out how much Laura paid for each ticket.
To find out how much Laura paid for each ticket, we need to divide by
First, we subtract from
Next, we bring to the right.
Then, we pick a multiple of that is easy to compute but is no larger than Let's pick
We write above the box and subtract from inside the box.
We get so the division is done.
The quotient is the sum of the numbers on top of the boxes:
So,
Therefore, Laura paid for each ticket.
Use the box model above to find the value of $617 \div 6.$
|
a
|
$104\,\textrm{R}\,3$ |
|
b
|
$103\,\textrm{R}\,1$ |
|
c
|
$102\,\textrm{R}\,5$ |
|
d
|
$102\,\textrm{R}\,1$ |
|
e
|
$104\,\textrm{R}\,5$ |
Henry cuts a $255\,\textrm{in}$ long rope into $5$ pieces of equal length. Using the box model above, find the length of each piece of rope.
|
a
|
$53\,\textrm{in}$ |
|
b
|
$55\,\textrm{in}$ |
|
c
|
$57\,\textrm{in}$ |
|
d
|
$51\,\textrm{in}$ |
|
e
|
$65\,\textrm{in}$ |
Sometimes, we might want to use three (or more) boxes to solve a division problem. This often happens when the dividend in our division problem is large.
Although there is no limit to how many boxes we can use, we'll stick to a maximum of three boxes in this lesson.
Let's see an example.
Use the box model above to find the value of
First, we subtract from
Next, we bring to the right and subtract from it.
Next, we bring to the right.
Finally, we pick a multiple of that is easy to compute but is no larger than Let's pick
We write above the box and subtract from inside the box.
We can't subtract any further since is less than Hence, is the remainder.
The quotient is the sum of the numbers on top of the boxes:
Therefore,
Use the box model above to find the value of $733 \div 6.$
|
a
|
$121 \,\text{R}\, 3$ |
|
b
|
$121$ |
|
c
|
$122 \,\text{R}\, 3$ |
|
d
|
$122 \,\text{R}\, 1$ |
|
e
|
$121 \,\text{R}\, 1$ |
Maggy used $612\,\textrm{g}$ of all-purpose flour to make $4$ apple pies. Assuming Maggy used the same weight of flour for each pie, use the box model above to find how much flour she used for each pie.
|
a
|
$158\,\textrm{g}$ |
|
b
|
$151\,\textrm{g}$ |
|
c
|
$153\,\textrm{g}$ |
|
d
|
$148\,\textrm{g}$ |
|
e
|
$157\,\textrm{g}$ |