Introduction

The advantage of box models over area models is that box models can be used to solve division problems with remainders.

To demonstrate, let's find the quotient and remainder of the following division problem:

83 \div 6

First, we write our division in the usual way.

6

8 3

Filling in our box model by applying our usual method, we get the following:

10 \color{blue}3
6

8 3
- 6 0
\color{red}2 \color{red}3
\color{red}2 \color{red}3
- 1 8
\fbox{5}

Now, note the following:

  • The result of the final subtraction is \fbox{5}, not zero.

  • However, we cannot subtract any further since \fbox{5} is less than our divisor 6.

  • Therefore, \fbox{5} is the remainder!

The quotient is the sum of the numbers on top of the boxes: 10+3=13

Therefore, the solution to our division problem is 83 \div 6 = 13 \,\text{R}\, \fbox{5}.

Let's see another example.

FLAG
Example: Finding a Quotient

Use the box model below to find the quotient of 72 \div 5.

10 \fbox{[math]\,\phantom{0}\,[/math]}
5

7 2
- \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
- \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{2}

EXPLANATION

First, we subtract 5 \times {\color{blue}10} = 50 from 72.

\color{blue}10
5

7 2
- 5 0
\color{red}2 \color{red}2

Next, we bring \color{red}22 to the right.

10
5

7 2
- 5 0
\color{red}2 \color{red}2
\color{red}2 \color{red}2

Finally, since 5 \times {\color{blue}4} = 20 is less than 22, we subtract 5 \times {\color{blue}4} from 22.

10 \color{blue}4
5

7 2
- 5 0
\color{red}2 \color{red}2
\color{red}2 \color{red}2
- 2 0
\fbox{2}

We can't subtract any further since \fbox{2} is less than 5. Hence, 2 is the remainder.

The quotient is the sum of the numbers on top of the boxes: 10 + 4 = 14

FLAG
Practice: Finding a Quotient

Use the box model below to find the quotient of $62 \div 4.$

$10$ $\fbox{$\,\phantom{0}\,$}$
$4$

$6$ $2$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{2}$

a
 $13$
b
 $15$
c
 $14$
d
 $12$
e
 $16$
Practice: Finding a Quotient

Use the box model below to find the quotient of $99 \div 6.$

$10$ $\fbox{$\,\phantom{0}\,$}$
$6$

$9$ $9$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{3}$

a
 $18$
b
 $17$
c
 $16$
d
 $19$
e
 $14$
Example: Finding a Remainder

Use the box model below to find the remainder of 43 \div 3.

10 \fbox{[math]\,\phantom{0}\,[/math]}
3

4 3
- \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
- \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{[math]\phantom{0}[/math]}

EXPLANATION

First, we subtract 3 \times {\color{blue}10} from 43.

\color{blue}10
3

4 3
- 3 0
\color{red}1 \color{red}3

Next, we bring \color{red}13 to the right.

10
3

4 3
- 3 0
\color{red}1 \color{red}3
\color{red}1 \color{red}3

Finally, since 3 \times {\color{blue}4} = 12 is less than 13, we subtract 3 \times {\color{blue}4} from 13.

10 \color{blue}4
3

4 3
- 3 0
\color{red}1 \color{red}3
\color{red}1 \color{red}3
- 1 2
\fbox{1}

We can't subtract any further since \fbox{1} is less than 3. Hence, 1 is the remainder.

FLAG
Practice: Finding a Remainder

Use the box model below to find the remainder of $79 \div 6.$

$10$ $\fbox{$\,\phantom{0}\,$}$
$6$

$7$ $9$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$

a
 $5$
b
 $1$
c
 $4$
d
 $2$
e
 $3$
Practice: Finding a Remainder

Use the box model below to find the remainder of $83 \div 6.$

$10$ $\fbox{$\,\phantom{0}\,$}$
$6$

$8$ $3$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$

a
 $3$
b
 $2$
c
 $1$
d
 $5$
e
 $4$
Example: Finding a Quotient and a Remainder

Suzy baked 97 cookies for a charity sale and distributed them into packages, each containing 8 cookies. Use the box model below to find how many packages she prepared and how many cookies were left unpackaged.

10 \fbox{[math]\,\phantom{0}\,[/math]}
8

9 7

EXPLANATION

To find out how many many packages of cookies Suzy prepared and how many cookies were left unpackaged, we have to divide 97 by 8.

First, we subtract 8 \times {\color{blue}10} from 97.

\color{blue}10
8

9 7
- 8 0
\color{red}1 \color{red}7

Next, we bring \color{red}17 to the right.

10
8

9 7
- 8 0
\color{red}1 \color{red}7
\color{red}1 \color{red}7

Finally, since 8 \times {\color{blue}2} = 16 is less than 17, we subtract 8 \times {\color{blue}2} from 17.

10 \color{blue}2
8

9 7
- 8 0
\color{red}1 \color{red}7
\color{red}1 \color{red}7
- 1 6
\fbox{1}

We can't subtract any further since \fbox{1} is less than 8. Hence, 1 is the remainder.

The quotient is the sum of the numbers on top of the boxes: 10+2=12

Therefore, 97 \div 8 = 12 \,\text{R}\, 1.

So, Suzy prepared 12 packages of cookies, and 1 cookie was left unpackaged.

FLAG
Practice: Finding a Quotient and a Remainder

Use the box model below to find the quotient and the remainder of $75 \div 4.$

$10$ $\fbox{$\,\phantom{0}\,$}$
$4$

$7$ $5$

a
 $18 \,\text{R}\, 1$
b
 $18 \,\text{R}\, 3$
c
 $17 \,\text{R}\, 2$
d
 $17 \,\text{R}\, 4$
e
 $18 \,\text{R}\, 0$
Practice: Finding a Quotient and a Remainder

Sam has $98$ candies and wants to distribute them equally among $8$ children. Use the box model below to find how many candies each child gets and how many candies Sam has left over.

$10$ $\fbox{$\,\phantom{0}\,$}$
$8$

$9$ $8$

a
 $12$ and $5$
b
 $11$ and $7$
c
 $12$ and $3$
d
 $11$ and $6$
e
 $12$ and $2$
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL