1.10 marking lines and making grids w

Marking Lines and Making Grids

http://www.lahc.edu/math/frankma.htm

http://www.lahc.edu/math/frankma.htm

Marking Lines and Making GridsA line segment is a finite piece of straight line a shown.


We will name line segments as I, J, K, etc.. K


We will name line segments as I, J, K, etc.. If the end points, say A and B, of a line segment are given, we may also address the line as AB.

K

B A



K

B A AB


One of the most useful tools in mathematics is to associate numbers to lengths, or positions, on a line.

A line segment is a finite piece of straight line a shown.


K

B A AB


In this section, we develop some drawing techniques for dividing lines into approximately equal segments.




K

B A AB






K

B A AB

Following are the basic eyeball skills for dividing a line segment.






K

B A AB


* Find the midpoint that cuts the segment in two equal pieces.






K

B A AB








K

B A AB



* Find the two points that cut the segment in three equal pieces.






K

B A AB



* Find the two points that cut the segment in three equal pieces.

Marking Lines and Making GridsTo cut a line segment K into 4 pieces,

K

Marking Lines and Making GridsTo cut a line segment K into 4 pieces, we cut K in half,

K

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

K


K


K

To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.


K


we get 16 pieces. (Practice drawing them).

If we cut each of the 8 pieces into 2


K

To cut a line segment K into 6 pieces,

K





K

To cut a line segment K into 6 pieces, we cut K in half,

K





K

To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.

K





K


K





K


K





K


K




(Or we may cut K into thirds first then cut each into 2 pieces.)


K


K

If we divide each segment above into two again, we would have 12 pieces.






K


K

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.

To cut a line segment K into 5 pieces,

K


To cut a line segment K into 5 pieces, eyeball the midpoint and the 1/3-mark,

K



K



K


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



K



K


This cuts K into two unequal segments.


K


This cuts K into two unequal segments. Cut the short piece into two,


K


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




K

By cutting each fifth into 2 halves, we have ten pieces..

K




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.


K


K




The purpose of the exercises is to make approximate accurate comparative pictures of lengths without too much distortion.


K

Distorted mathematical drawings lead to confusions and false conclusions.


K




Marking Lines


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




K


To draw it, cut K into 2 segments, then divide each into 3 pieces.



K





K




Example A. The length of the line segment K is 24 (units).

mark the dividers, spaced apart by the length that’s equal to

the length K the number of pieces

a. Divide K into 6 pieces and label the dividers. Draw.

K



Starting from the left end point as 0,






K

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




They are 4, 8, 12, 16, 20, and 24.







K0




They are 4, 8, 12, 16, 20, and 24.







K0 4




They are 4, 8, 12, 16, 20, and 24.







K0 4 8




They are 4, 8, 12, 16, 20, and 24.







K240 4 8 1612 20




They are 4, 8, 12, 16, 20, and 24.


b. Divide K into 8 pieces and label the dividers. Draw.

K


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



K



K



K



K


The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.


K0




K30




6K

30




6 9 12 15 18K

21 2430




6 9 12 15 18K

21 2430

We may estimate lengths or distances by inserting dividers.




6 9 12 15 18K

21 2430


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




6 9 12 15 18K

21 2430




A

We may divide K into 6 segments first, then divide each again into 2 pieces.




6 9 12 15 18K

21 2430




A





6 9 12 15 18K

21 2430




A





6 9 12 15 18K

21 2430




A





6 9 12 15 18K

21 2430




A




Each small segment represents 24/12 = 2 inches.


6 9 12 15 18K

21 2430




A


420





6 9 12 15 18K

21 2430




A


1442 160





6 9 12 15 18K

21 2430




A


1442 160



Each small segment represents 24/12 = 2 inches. A is between 14 and 16 and it’s approx. at the15-inch mark.

0

Marking Lines and Making Gridsb. 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

200K

We divide K into 5 segments first, then divide each again into 2 pieces.

0


200K


0


200K


0


200K


0


200K


0


200K


0


so the dividers are at 20, 40, 60, etc..

200K

Each small segment represents 200/10 = 20 miles.


100200



20080K


A


100200



The 90-mile mark is the midpoint 80 and 100.

20080K


A


100200



The 90-mile mark is the midpoint 80 and 100.

20080K

Cutting Cakes

We may apply the technique of dividing line segments to help us to cut rectangular cakes into pieces of approx. equal size.


Marking Lines and Making GridsCutting Cakes

Marking Lines and Making GridsCutting CakesIf we cut the a rectangular cake into 2 rows, then cut it into 3 columns, we would have 2 x 3 = 6 pieces.

Marking Lines and Making GridsCutting CakesIf 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

Marking Lines and Making GridsCutting CakesIf 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


2 rows

3 columns

r rows


2 rows

3 columns

r rows

c columns


2 rows

3 columns

r rows

c columns

r x c pieces


2 rows

3 columns

r rows

c columns

Example C. a. Describe one methodfor cutting a pan-cake into 24 pieces as evenly as possible.

r x c pieces


2 rows

3 columns

r rows

c columns

Example C. a. Describe one methodfor cutting a pan-cake into 24 pieces as evenly as possible.

One way is to cut it into 4 rows and 6 columns.

r x c pieces


2 rows

3 columns

r rows

c columns

Example C. a. Describe one method

Divide the cake into to 2 rows then divide each row into two again to obtain 4 rows.

for cutting a pan-cake into 24 pieces as evenly as possible.


r x c pieces


2 rows

3 columns

r rows

c columns


Divide the cake into to 2 rows then divide each row into two again to obtain 4 rows.



r x c pieces


2 rows

3 columns

r rows

c 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


2 rows

3 columns

r rows

c columns





r x c pieces


2 rows

3 columns

r rows

c columns





r x c pieces


2 rows

3 columns

r rows

c 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. This makes 4 x 6 = 24 approx. equal pieces.



4 x 6 = 24 pieces

r x c pieces

Marking Lines and Making Gridsb. 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.


We may cut it into 1 row and 24 columns, 2 rows x 12 columns,

When we cut a rectangle in to 4 rows and 6 columns, we say that we make a 4 x 6 grid.

3 rows x 8 columns, or 4 rows x 6 columns.





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.





such as cutting a piece of wood or a cake by hand & sight.


The above skills are essential for many basic daily tasks

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.


We find such a scale in quarters and sixths at the base of Da Vinci’s masterpiece shown here.

Due to our ability to extract halves and thirds that we utilize scales based on such divisions.
