Introduction

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

In this approach, we repeatedly subtract easy multiples of the divisor from the dividend. Each multiple we subtract gives a partial quotient, and the sum of all partial quotients is the final quotient.

This strategy is called partial quotients, and the box model helps us visually track the 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
 $22$
b
 $14$
c
 $17$
d
 $20$
e
 $16$
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
 $15$
b
 $22$
c
 $17$
d
 $18$
e
 $13$
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
 $14$
b
 $16$
c
 $13$
d
 $15$
e
 $17$
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
 $15$
b
 $12$
c
 $17$
d
 $13$
e
 $14$
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