Introduction

Box models allow us to solve division problems by breaking them down into smaller, easier steps.

Let's consider the following division problem:

36 \div 2

The box model for this division problem looks as follows:

2

3 6

We can solve this division problem by following four steps:

Step 1: Pick a multiple of 2 that is easy to compute but is no larger than 36. Let's pick 2 \times {\color{blue}10} = 20.

We write {\color{blue}10} above the box and subtract 2 \times {\color{blue}10} from 36 inside the box.

\color{blue}10
2

3 6
- 2 0
\color{red}1 \color{red}6

Step 2: Repeat the process. We write \color{red}16 in a new box to the right.

10
2

3 6
- 2 0
\color{red}1 \color{red}6
\color{red}1 \color{red}6

Step 3: Pick a multiple of 2 that is easy to compute but is no larger than {\color{red}16}. Let's pick 2 \times {\color{blue}8} = 16.

We write \color{blue}8 above the box and subtract 2 \times {\color{blue}8} from \color{red}16 inside the box.

10 \color{blue}8
2

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

We get \fbox{0} , so the division is done.

Step 4: To find the quotient of 36 \div 2, we add the numbers at the very top: 10+8=18

Therefore, 36 \div 2 = 18.

FLAG
Example: Divide Two-Digit Numbers by One-Digit Numbers With Scaffolding

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

5 1
- \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{0}

Use the box model above to compute the value of 51 \div 3.

EXPLANATION

Step 1: First, we subtract 3 \times {\color{blue}10} = 30 from 51.

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

5 1
-\!\! \fbox{[math]3[/math]} \fbox{[math]0[/math]}
\fbox{[math]\color{red}2[/math]} \fbox{[math]\color{red}1[/math]}
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
-\!\! \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{0}

Step 2: Next, we bring \color{red}21 to the right.

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

5 1
-\!\! \fbox{[math]3[/math]} \fbox{[math]0[/math]}
\fbox{[math]\color{red}2[/math]} \fbox{[math]\color{red}1[/math]}
\fbox{[math]\color{red}2[/math]} \fbox{[math]\color{red}1[/math]}
-\!\! \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
\fbox{0}

Step 3: Pick a multiple of 3 that is easy to compute but is no larger than {\color{red}21}. Let's pick 3 \times {\color{blue}7} = 21.

We write \color{blue}7 above the box and subtract 3 \times {\color{blue}7} from \color{red}21 inside the box.

10 {\color{blue}7}
3

5 1
-\!\! \fbox{[math]3[/math]} \fbox{[math]0[/math]}
\fbox{[math]\color{red}2[/math]} \fbox{[math]\color{red}1[/math]}
\fbox{[math]\color{red}2[/math]} \fbox{[math]\color{red}1[/math]}
-\!\! \fbox{[math]2[/math]} \fbox{[math]1[/math]}
\fbox{0}

We're left with \fbox{0}, so the division is done.

Step 4: The quotient is the sum of the numbers on top of the boxes: 10+7=17

Therefore, 51 \div 3 = 17.

FLAG
Practice: Divide Two-Digit Numbers by One-Digit Numbers With Scaffolding

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

$9$ $6$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{0}$

Use the box model above to compute the value of $96 \div 6.$

a
 $17$
b
 $16$
c
 $20$
d
 $22$
e
 $14$
Practice: Divide Two-Digit Numbers by One-Digit Numbers With Scaffolding

$10$ $\fbox{$\,\phantom{0}\,$}$
$3$

$5$ $4$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$-\!\!$ $\fbox{$\phantom{0}$}$ $\fbox{$\phantom{0}$}$
$\fbox{0}$

Use the box model above to compute the value of $54 \div 3.$

a
 $13$
b
 $17$
c
 $18$
d
 $22$
e
 $15$
Example: Divide Two-Digit Numbers by One-Digit Numbers

Use the box model below to find the value of 91 \div 7.

\,10\, \fbox{[math]\,\phantom{0}\,[/math]}
7

9 1

EXPLANATION

Step 1: First, we subtract 7 \times {\color{blue}10} = 70 from 91.

\color{blue}10
7

9 1
- 7 0
\color{red}2 \color{red}1

Step 2: Next, we bring \color{red}21 to the right.

10
7

9 1
- 7 0
\color{red}2 \color{red}1
\color{red}2 \color{red}1

Step 3: Pick a multiple of 7 that is easy to compute but is no larger than {\color{red}21}. Let's pick 7 \times {\color{blue}3} = 21.

We write \color{blue}3 above the box and subtract 7 \times {\color{blue}3} from \color{red}21 inside the box.

10 \color{blue}3
7

9 1
- 7 0
\color{red}2 \color{red}1
\color{red}2 \color{red}1
- 2 1
\fbox{0}

We get \fbox{0} , so the division is done.

Step 4: The quotient is the sum of the numbers on top of the boxes: 10+3=13

Therefore, 91 \div 7 = 13.

FLAG
Practice: Divide Two-Digit Numbers by One-Digit Numbers

Use the box model below to find the value of $51 \div 3.$

$\,10\,$ $\fbox{$\,\phantom{0}\,$}$
$3$

$5$ $1$

a
 $13$
b
 $17$
c
 $15$
d
 $14$
e
 $16$
Practice: Divide Two-Digit Numbers by One-Digit Numbers

$\fbox{$\,\phantom{0}\,$}$ $\fbox{$\,\phantom{0}\,$}$
$2$

$2$ $8$

Ricky wants to give away half his $28$ marbles to his brother Hank. Using the box model above, find out how many marbles Hank will receive.

a
 $12$
b
 $17$
c
 $14$
d
 $13$
e
 $15$
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