The document discusses division and its key concepts. Division is defined as dividing a total amount into equal parts. Key terms in division are: the dividend (total), the divisor (number of parts), the quotient (each part), and the remainder (leftover amount). Examples are provided to demonstrate how to write division statements using these terms and how to interpret them. Important properties of division are also outlined, such as division by 0 not being defined and the relationship between the total, divisor, and remainder.

1. Division I
http://www.lahc.edu/math/frankma.htm

2. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.

3. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.

4. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.

5. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.

6. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.

7. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T÷D=Q

8. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
The total T is
the dividend,
T÷D=Q

9. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
The total T is
the dividend,
The number of parts D
is the divisor.
T÷D=Q

10. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
The total T is
the dividend,
The number of parts D
is the divisor.
T÷D=Q
Q is the quotient.

11. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
The total T is
the dividend,
The number of parts D
is the divisor.
Q is the quotient.
T÷D=Q
says that “if T is divided into D equal parts, then each part is Q.”

12. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
The total T is
the dividend,
The number of parts D
is the divisor.
Q is the quotient.
T÷D=Q
says that “if T is divided into D equal parts, then each part is Q.”
If T ÷ D = Q then T = D x Q or that D and Q are factors of T,

13. Division I
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
The total T is
the dividend,
The number of parts D
is the divisor.
Q is the quotient.
T÷D=Q
says that “if T is divided into D equal parts, then each part is Q.”
If T ÷ D = Q then T = D x Q or that D and Q are factors of T,
e.g. 12 ÷ 3 = 4 so 12 = 3(4), so both 3 and 4 are factors of 12.

14. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:

15. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R

16. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.

17. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”

18. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”

19. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:

20. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:
2 groups

21. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:
2 groups 3 in a group

22. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:
2 groups 3 in a group
1 remains

23. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:
2 groups 3 in a group
1 remains
Note we may recover the total by back tracking:

24. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:
2 groups 3 in a group
1 remains
Note we may recover the total by back tracking: 2 x 3

25. Division I
If four people are to share 11 apples, assuming that no cutting
is allowed, then each person gets two and there are three
apples left over:
the remainder R
We call the three leftover apples the remainder R.
We write this as “11 ÷ 4 = 2 with R = 3, or with remainder 3.”
In general, the expression
“T ÷ D = Q with remainder R”
says that “if the total T is divided into D equal parts or groups,
then each part is Q, with R leftover.”
For example, 7 ÷ 2 = 3 with R = 1 means:
2 groups 3 in a group
1 remains
Note we may recover the total by back tracking: 2 x 3 + 1 = 7

26. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”

27. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.

28. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…

29. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.

30. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0.

31. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0. If you divide nothing into groups,
each group has nothing.

32. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0. If you divide nothing into groups,
each group has nothing.
* T ÷ 1 = T, e.g. 5 ÷ 1 = 5.

33. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0. If you divide nothing into groups,
each group has nothing.
* T ÷ 1 = T, e.g. 5 ÷ 1 = 5.
“T ÷ 1” means to leave the total as one group,
and that one group consists of everyone.

34. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0. If you divide nothing into groups,
each group has nothing.
* T ÷ 1 = T, e.g. 5 ÷ 1 = 5.
“T ÷ 1” means to leave the total as one group,
and that one group consists of everyone.
* Given that T ÷ D = Q with remainder R ,
then the remainder R must be smaller than D.

35. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0. If you divide nothing into groups,
each group has nothing.
* T ÷ 1 = T, e.g. 5 ÷ 1 = 5.
“T ÷ 1” means to leave the total as one group,
and that one group consists of everyone.
* Given that T ÷ D = Q with remainder R ,
then the remainder R must be smaller than D.
e.g. 11 ÷ 4 = 2 has remainder 3, which is smaller than 4.

36. Division I
Following are important observations about the notation
“T ÷ D = Q with remainder R.”
* The expression T ÷ 0 does not make sense.
We may leave the total items as one group “T ÷ 1,” or
separate them into two groups “T ÷ 2,” or three groups, etc…
But we can’t ask people to get in the bus(es) when there is
no bus, we can’t divide something into no group.
* 0 ÷ T = 0, e.g. 0 ÷ 5 = 0. If you divide nothing into groups,
each group has nothing.
* T ÷ 1 = T, e.g. 5 ÷ 1 = 5.
“T ÷ 1” means to leave the total as one group,
and that one group consists of everyone.
* Given that T ÷ D = Q with remainder R ,
then the remainder R must be smaller than D.
e.g. 11 ÷ 4 = 2 has remainder 3, which is smaller than 4.
We could have made the quotient more if there’s more to share.

37. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q,

38. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).

39. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R

40. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1

41. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1
2 groups 3 in a group 1 remains

42. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains

43. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains
Example A.
a. What is 1 ÷ 0?
b. What is 0 ÷1?
c. What is 7 ÷ 1?
d. Write the division 12 ÷ 6 = 2 in the multiplicative form.
e. Write the division “13 ÷ 6 = 2 with remainder 1” in the
multiplication and addition form.

44. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains
Example A.
a. What is 1 ÷ 0?
1 ÷ 0 is undefined.
b. What is 0 ÷1?
c. What is 7 ÷ 1?
d. Write the division 12 ÷ 6 = 2 in the multiplicative form.
e. Write the division “13 ÷ 6 = 2 with remainder 1” in the
multiplication and addition form.

45. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains
Example A.
a. What is 1 ÷ 0?
1 ÷ 0 is undefined.
0 ÷ 1 = 0.
b. What is 0 ÷1?
c. What is 7 ÷ 1?
d. Write the division 12 ÷ 6 = 2 in the multiplicative form.
e. Write the division “13 ÷ 6 = 2 with remainder 1” in the
multiplication and addition form.

46. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains
Example A.
a. What is 1 ÷ 0?
1 ÷ 0 is undefined.
0 ÷ 1 = 0.
b. What is 0 ÷1?
7÷1=7
c. What is 7 ÷ 1?
d. Write the division 12 ÷ 6 = 2 in the multiplicative form.
e. Write the division “13 ÷ 6 = 2 with remainder 1” in the
multiplication and addition form.

47. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains
Example A.
a. What is 1 ÷ 0?
1 ÷ 0 is undefined.
0 ÷ 1 = 0.
b. What is 0 ÷1?
7÷1=7
c. What is 7 ÷ 1?
d. Write the division 12 ÷ 6 = 2 in the multiplicative form.
12 ÷ 6 = 2 in the multiplicative form is 12 = 6 x 2.
e. Write the division “13 ÷ 6 = 2 with remainder 1” in the
multiplication and addition form.

48. Division I
* If T ÷ D = Q, i.e. T may be divided evenly by D,
then T = D x Q, e.g. 12 ÷ 3 = 4 so 12 = 3(4).
* If T ÷ D = Q has remainder R, then T = D x Q + R
e.g. 7 ÷ 2 = 3 with R = 1 so 7 = 2 x 3 + 1
2 groups 3 in a group 1 remains
Example A.
a. What is 1 ÷ 0?
1 ÷ 0 is undefined.
0 ÷ 1 = 0.
b. What is 0 ÷1?
7÷1=7
c. What is 7 ÷ 1?
d. Write the division 12 ÷ 6 = 2 in the multiplicative form.
12 ÷ 6 = 2 in the multiplicative form is 12 = 6 x 2.
e. Write the division “13 ÷ 6 = 2 with remainder 1” in the
multiplication and addition form.
The multiplicative form is “13 = 6 x 2 + 1”.

49. The Vertical Format
Division I

50. Division I
The Vertical Format
We demonstrate the vertical long-division format below.

51. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the frontnumber (the dividend) inside the
scaffold.

52. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
“back-one”
outside
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
2 ) 6

53. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
“back-one”
outside
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
2 ) 6
ii. Enter the quotient on top,

54. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
ii. Enter the quotient on top,

55. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)

56. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
6
3x2
multiply the quotient
back into the scaffold.
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)

57. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
6
3x2
0
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
multiply the quotient
back into the scaffold.
The new dividend is 0,
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)

58. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
6
3x2
0
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
multiply the quotient
back into the scaffold.
The new dividend is 0,
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
iii. If the new dividend is not
enough to be divided by the divisor,
stop. This is the remainder R.
Otherwise, repeat steps i and ii.

59. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
6
3x2
0
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
multiply the quotient
back into the scaffold.
The new dividend is 0, not
enough to be divided again,
stop. This is the remainder R.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
iii. If the new dividend is not
enough to be divided by the divisor,
stop. This is the remainder R.
Otherwise, repeat steps i and ii.

60. Division I
The Vertical Format
We demonstrate the vertical long-division format below.
Example B. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Enter the
quotient on top
“back-one”
outside
3
2 ) 6
6
3x2
0
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front“front-one”
number (the dividend) inside the
inside
scaffold.
multiply the quotient
back into the scaffold.
The new dividend is 0, not
enough to be divided again,
stop. This is the remainder R.
So the remainder R is 0 and
we have that 6 ÷ 2 = 3 evenly.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
iii. If the new dividend is not
enough to be divided by the divisor,
stop. This is the remainder R.
Otherwise, repeat steps i and ii.

61. Division I
b. Carry out the long division 7 ÷ 3.

62. Division I
b. Carry out the long division 7 ÷ 3.
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.

63. Division I
b. Carry out the long division 7 ÷ 3.
“back-one”
outside
“front-one”
inside
3 ) 7
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.

64. Division I
b. Carry out the long division 7 ÷ 3.
Enter the
quotient on top
“back-one”
outside
2
3 ) 7
“front-one”
inside
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.

65. Division I
b. Carry out the long division 7 ÷ 3.
Enter the
quotient on top
“back-one”
outside
2
3 ) 7
“front-one”
inside
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).

66. Division I
b. Carry out the long division 7 ÷ 3.
Enter the
quotient on top
“back-one”
outside
2
3 ) 7
6
2x3
1
multiply the quotient
back into the scaffold.
“front-one”
inside
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).

67. Division I
b. Carry out the long division 7 ÷ 3.
Enter the
quotient on top
“back-one”
outside
2
3 ) 7
6
2x3
1
multiply the quotient
back into the scaffold.
“front-one”
inside
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
iii. If the new dividend is not
enough to be divided by the
divisor, stop. This is the remainder.
Otherwise, repeat steps i and ii.

68. Division I
b. Carry out the long division 7 ÷ 3.
Enter the
quotient on top
“back-one”
outside
2
3 ) 7
6
2x3
1
“front-one”
inside
multiply the quotient
back into the scaffold.
The new dividend is 1, not
enough to be divided again, so
stop. This is the remainder.
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the backnumber (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
iii. If the new dividend is not
enough to be divided by the
divisor, stop. This is the remainder.
Otherwise, repeat steps i and ii.

69. Division I
b. Carry out the long division 7 ÷ 3.
Steps. i. (Front-in Back-out)
Put the problem in the long
“front-one” division format with the backinside
number (the divisor) outside, and
“back-one”
2
the front-number (the dividend)
outside
3 ) 7
inside the scaffold.
6
ii. Enter the quotient on top,
2x3
Multiply the quotient back into the
1
multiply the quotient
problem and subtract the results
back into the scaffold.
from the dividend (and bring down
the rest of the digits, if any. This is
The new dividend is 1, not
enough to be divided again, so the new dividend).
iii. If the new dividend is not
stop. This is the remainder.
enough to be divided by the
So the remainder is 1 and
divisor, stop. This is the remainder.
we have that 7 ÷ 3 = 2 with R = 1. Otherwise, repeat steps i and ii.
Enter the
quotient on top

70. Division I
b. Carry out the long division 7 ÷ 3.
Steps. i. (Front-in Back-out)
Put the problem in the long
“front-one” division format with the backinside
number (the divisor) outside, and
“back-one”
2
the front-number (the dividend)
outside
3 ) 7
inside the scaffold.
6
ii. Enter the quotient on top,
2x3
Multiply the quotient back into the
1
multiply the quotient
problem and subtract the results
back into the scaffold.
from the dividend (and bring down
the rest of the digits, if any. This is
The new dividend is 1, not
enough to be divided again, so the new dividend).
iii. If the new dividend is not
stop. This is the remainder.
enough to be divided by the
So the remainder is 1 and
divisor, stop. This is the remainder.
we have that 7 ÷ 3 = 2 with R = 1. Otherwise, repeat steps i and ii.
Enter the
quotient on top
Put the result in the multiplicative form, we have that
7 = 2 x 3 + 1.