1. Marking Lines and Making Grids
http://www.lahc.edu/math/frankma.htm

2. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.

3. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K

4. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
we may also address the line as AB.
B

5. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
B

6. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
B

7. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
In this section, we develop some drawing techniques for
dividing lines into approximately equal segments.
B

8. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
B
In this section, we develop some drawing techniques for
dividing lines into approximately equal segments.
Following are the basic eyeball skills for dividing a line segment.

9. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
B
In this section, we develop some drawing techniques for
dividing lines into approximately equal segments.
Following are the basic eyeball skills for dividing a line segment.
* Find the midpoint that cuts the segment in two equal pieces.

10. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
B
In this section, we develop some drawing techniques for
dividing lines into approximately equal segments.
Following are the basic eyeball skills for dividing a line segment.
* Find the midpoint that cuts the segment in two equal pieces.

11. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
B
In this section, we develop some drawing techniques for
dividing lines into approximately equal segments.
Following are the basic eyeball skills for dividing a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.

12. Marking Lines and Making Grids
A line segment is a finite piece of straight line a shown.
We will name line segments as I, J, K, etc..
K
If the end points, say A and B,
of a line segment are given,
A
AB
we may also address the line as AB.
One of the most useful tools in mathematics is to associate
numbers to lengths, or positions, on a line.
B
In this section, we develop some drawing techniques for
dividing lines into approximately equal segments.
Following are the basic eyeball skills for dividing a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.

13. Marking Lines and Making Grids
To cut a line segment K into 4 pieces,
K

14. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
K

15. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K

16. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K

17. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces.

18. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).

19. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces,
K

20. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
K

21. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
then cut each half into 3 pieces.
K

22. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
then cut each half into 3 pieces.
K

23. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
then cut each half into 3 pieces.
K

24. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
then cut each half into 3 pieces.
K
(Or we may cut K into thirds first then cut each into 2 pieces.)

25. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
then cut each half into 3 pieces.
K
(Or we may cut K into thirds first then cut each into 2 pieces.)
If we divide each segment above into two again, we would
have 12 pieces.

26. Marking Lines and Making Grids
To cut a line segment K into 4 pieces, we cut K in half,
then cut each half into two.
K
To cut a line segment K into 8 pieces, we cut each of the 4
pieces into 2 pieces. If we cut each of the 8 pieces into 2
we get 16 pieces. (Practice drawing them).
To cut a line segment K into 6 pieces, we cut K in half,
then cut each half into 3 pieces.
K
(Or we may cut K into thirds first then cut each into 2 pieces.)
If we divide each segment above into two again, we would
have 12 pieces.
We can draw a reasonable ruler representing a foot with
inch-marks with the above technique.

27. Marking Lines and Making Grids
To cut a line segment K into 5 pieces,
K

28. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark,
K

29. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark,
K

30. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark,
K

31. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
K

32. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
K

33. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments.
K

34. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two,
K

35. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two, and cut the long piece into three to get 5 pieces.
K

36. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two, and cut the long piece into three to get 5 pieces.
K
By cutting each fifth into 2 halves, we have ten pieces..
K

37. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two, and cut the long piece into three to get 5 pieces.
K
By cutting each fifth into 2 halves, we have ten pieces..
K

38. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two, and cut the long piece into three to get 5 pieces.
K
By cutting each fifth into 2 halves, we have ten pieces.
K
This is useful for the metric distance measurements such as
meters, kilometers etc.. which are scaled by 10’s.

39. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two, and cut the long piece into three to get 5 pieces.
K
By cutting each fifth into 2 halves, we have ten pieces.
K
This is useful for the metric distance measurements such as
meters, kilometers etc.. which are scaled by 10’s.
The purpose of the exercises is to make approximate accurate
comparative pictures of lengths without too much distortion.

40. Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint
and the 1/3-mark, then mark off approximately the midpoint
(actually slightly to the right of the midpoint) between them.
This cuts K into two unequal segments. Cut the short piece
into two, and cut the long piece into three to get 5 pieces.
K
By cutting each fifth into 2 halves, we have ten pieces.
K
This is useful for the metric distance measurements such as
meters, kilometers etc.. which are scaled by 10’s.
The purpose of the exercises is to make approximate accurate
comparative pictures of lengths without too much distortion.
Distorted mathematical drawings lead to confusions and false
conclusions.

41. Marking Lines and Making Grids
Marking Lines

42. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K.

43. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K.
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
K

44. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K.
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
K

45. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K.
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
K

46. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K.
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
K

47. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
K

48. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
The total length is 24 is divided into 6 pieces,
so the length of each piece is 24/6 = 4,
and the labels are multiples of 4.
K

49. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
The total length is 24 is divided into 6 pieces,
so the length of each piece is 24/6 = 4,
and the labels are multiples of 4.
They are 4, 8, 12, 16, 20, and 24.
K

50. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
The total length is 24 is divided into 6 pieces,
so the length of each piece is 24/6 = 4,
and the labels are multiples of 4.
They are 4, 8, 12, 16, 20, and 24.
0
K

51. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
The total length is 24 is divided into 6 pieces,
so the length of each piece is 24/6 = 4,
and the labels are multiples of 4.
They are 4, 8, 12, 16, 20, and 24.
0
K
4

52. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
The total length is 24 is divided into 6 pieces,
so the length of each piece is 24/6 = 4,
and the labels are multiples of 4.
They are 4, 8, 12, 16, 20, and 24.
0
K
4
8

53. Marking Lines and Making Grids
Marking Lines
We can mark the dividers if the we know the length of the line
segment K. Starting from the left end point as 0,
mark the dividers, spaced apart by the length that’s equal to
the length K
the number of pieces
Example A. The length of the line segment K is 24 (units).
a. Divide K into 6 pieces and label the dividers. Draw.
To draw it, cut K into 2 segments, then divide each into 3 pieces.
The total length is 24 is divided into 6 pieces,
so the length of each piece is 24/6 = 4,
and the labels are multiples of 4.
They are 4, 8, 12, 16, 20, and 24.
0
K
4
8
12
16
20
24

54. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
K

55. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
K

56. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
K

57. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
K

58. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
K

59. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
K

60. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
K

61. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
K
3

62. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
K
3
6

63. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
K
3
6
9
12
15
18
21
24

64. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.

65. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
A

66. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces.
A

67. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces.
A

68. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces.
A

69. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces.
A

70. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces. Each small segment represents 24/12 = 2 inches.
A

71. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces. Each small segment represents 24/12 = 2 inches.
0
2
4
A

72. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces. Each small segment represents 24/12 = 2 inches.
0
2
4
14 A 16

73. Marking Lines and Making Grids
b. Divide K into 8 pieces and label the dividers. Draw.
Divide 24 into 8 pieces by dividing K into halves, then into four
pieces, then half each to obtain eight pieces.
The length of each is 24/8 = 3 so the labels are multiples of 3:
3, 6, 9, 12, 15, 18, 21, and 24.
0
3
6
9
12
15
18
21
24
K
We may estimate lengths or distances by inserting dividers.
Example B. The length of the line segment below is 24 inches.
Estimate the position of A to the closest inch-mark by dividing
K into 12 segments.
We may divide K into 6 segments first, then divide each again
into 2 pieces. Each small segment represents 24/12 = 2 inches.
A is between 14 and 16 and it’s approx. at the15-inch mark.
0
2
4
14 A 16

74. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
K
0
200

75. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces.
K
0
200

76. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces.
K
0
200

77. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces.
K
0
200

78. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces.
K
0
200

79. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces.
K
0
200

80. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces. Each small segment represents 200/10 = 20 miles.
so the dividers are at 20, 40, 60, etc..
K
0
200

81. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces. Each small segment represents 200/10 = 20 miles.
so the dividers are at 20, 40, 60, etc..
K
0
20
80
100
200

82. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces. Each small segment represents 200/10 = 20 miles.
so the dividers are at 20, 40, 60, etc..
The 90-mile mark is the midpoint 80 and 100.
K
0
20
80
A
100
200

83. Marking Lines and Making Grids
b. The length of the line segment K represent 200 miles.
Divide K into 10 segments and label the point A at the 90-mile
mark as accurately as possible
We divide K into 5 segments first, then divide each again into
2 pieces. Each small segment represents 200/10 = 20 miles.
so the dividers are at 20, 40, 60, etc..
The 90-mile mark is the midpoint 80 and 100.
K
0
20
80
A
100
200
Cutting Cakes
We may apply the technique of dividing line segments to help
us to cut rectangular cakes into pieces of approx. equal size.

84. Marking Lines and Making Grids
Cutting Cakes

85. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.

86. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
2 rows
3 columns

87. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
2 rows
3 columns

88. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
2 rows
3 columns
r rows

89. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
2 rows
3 columns
r rows
c columns

90. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
2 rows
3 columns
r rows
r x c pieces
c columns

91. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
r x c pieces
c columns

92. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
One way is to cut it into 4 rows and 6 columns.
r x c pieces
c columns

93. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
One way is to cut it into 4 rows and 6 columns.
Divide the cake into to 2 rows then divide
each row into two again to obtain 4 rows.
r x c pieces
c columns

94. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
One way is to cut it into 4 rows and 6 columns.
Divide the cake into to 2 rows then divide
each row into two again to obtain 4 rows.
r x c pieces
c columns

95. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
One way is to cut it into 4 rows and 6 columns.
Divide the cake into to 2 rows then divide
each row into two again to obtain 4 rows.
Then cut it into 3 columns, followed by
cutting each column into 2 to get 6 columns.
r x c pieces
c columns

96. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
One way is to cut it into 4 rows and 6 columns.
Divide the cake into to 2 rows then divide
each row into two again to obtain 4 rows.
Then cut it into 3 columns, followed by
cutting each column into 2 to get 6 columns.
r x c pieces
c columns

97. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
One way is to cut it into 4 rows and 6 columns.
Divide the cake into to 2 rows then divide
each row into two again to obtain 4 rows.
Then cut it into 3 columns, followed by
cutting each column into 2 to get 6 columns.
r x c pieces
c columns

98. Marking Lines and Making Grids
Cutting Cakes
If we cut the a rectangular cake into
2 rows, then cut it into 3 columns,
we would have 2 x 3 = 6 pieces.
In general, if we cut the cake into
R rows and C columns, then we
would obtain r x c pieces.
Example C. a. Describe one method
for cutting a pan-cake into 24 pieces
as evenly as possible.
2 rows
3 columns
r rows
r x c pieces
c columns
One way is to cut it into 4 rows and 6 columns.
Divide the cake into to 2 rows then divide
each row into two again to obtain 4 rows.
Then cut it into 3 columns, followed by
cutting each column into 2 to get 6 columns.
4 x 6 = 24 pieces
This makes 4 x 6 = 24 approx. equal pieces.

99. Marking Lines and Making Grids
b. What are all the different ways to cut the cake into rows
and columns combinations that will produce 24 pieces are
possible.

100. Marking Lines and Making Grids
b. What are all the different ways to cut the cake into rows
and columns combinations that will produce 24 pieces are
possible.
We may cut it into 1 row and 24 columns, 2 rows x 12 columns,
3 rows x 8 columns, or 4 rows x 6 columns.

101. Marking Lines and Making Grids
b. What are all the different ways to cut the cake into rows
and columns combinations that will produce 24 pieces are
possible.
We may cut it into 1 row and 24 columns, 2 rows x 12 columns,
3 rows x 8 columns, or 4 rows x 6 columns.
When we cut a rectangle in to 4 rows and 6 columns, we say
that we make a 4 x 6 grid.

102. Marking Lines and Making Grids
b. What are all the different ways to cut the cake into rows
and columns combinations that will produce 24 pieces are
possible.
We may cut it into 1 row and 24 columns, 2 rows x 12 columns,
3 rows x 8 columns, or 4 rows x 6 columns.
When we cut a rectangle in to 4 rows and 6 columns, we say
that we make a 4 x 6 grid.
When we cut a rectangle in to R rows and C columns, we say
that we make a R x C grid.

103. Marking Lines and Making Grids
b. What are all the different ways to cut the cake into rows
and columns combinations that will produce 24 pieces are
possible.
We may cut it into 1 row and 24 columns, 2 rows x 12 columns,
3 rows x 8 columns, or 4 rows x 6 columns.
When we cut a rectangle in to 4 rows and 6 columns, we say
that we make a 4 x 6 grid.
When we cut a rectangle in to R rows and C columns, we say
that we make a R x C grid.
It’s important to be able to divide a line or make a grid on
papers by hand with reasonable accuracy because accurate
drawings gives us insight into solving geometric problems.

104. Marking Lines and Making Grids
b. What are all the different ways to cut the cake into rows
and columns combinations that will produce 24 pieces are
possible.
We may cut it into 1 row and 24 columns, 2 rows x 12 columns,
3 rows x 8 columns, or 4 rows x 6 columns.
When we cut a rectangle in to 4 rows and 6 columns, we say
that we make a 4 x 6 grid.
When we cut a rectangle in to R rows and C columns, we say
that we make a R x C grid.
It’s important to be able to divide a line or make a grid on
papers by hand with reasonable accuracy because accurate
drawings gives us insight into solving geometric problems.
The above skills are essential for many basic daily tasks
such as cutting a piece of wood or a cake by hand & sight.

105. Marking Lines and Making Grids
Due to our ability to extract halves and
thirds that we utilize scales based on such
divisions.
We find such a scale in quarters and sixths
at the base of Da Vinci’s masterpiece shown
here.