Post on 25-May-2015
Marking Lines and Making Grids
http://www.lahc.edu/math/frankma.htm
Marking Lines and Making GridsA line segment is a finite piece of straight line a shown.
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
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.. 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
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.. 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 AB
Marking Lines and Making Grids
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.
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 AB
Marking Lines and Making Grids
In this section, we develop some drawing techniques for dividing lines into approximately equal segments.
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.
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 AB
Marking Lines and Making Grids
In this section, we develop some drawing techniques for dividing lines into approximately equal segments.
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.
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 AB
Following are the basic eyeball skills for dividing a line segment.
Marking Lines and Making Grids
In this section, we develop some drawing techniques for dividing lines into approximately equal segments.
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.
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 AB
Following are the basic eyeball skills for dividing a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
Marking Lines and Making Grids
In this section, we develop some drawing techniques for dividing lines into approximately equal segments.
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.
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 AB
Following are the basic eyeball skills for dividing a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
Marking Lines and Making Grids
In this section, we develop some drawing techniques for dividing lines into approximately equal segments.
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.
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 AB
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.
Marking Lines and Making Grids
In this section, we develop some drawing techniques for dividing lines into approximately equal segments.
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.
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 AB
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.
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
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
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
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
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
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
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
To cut a line segment K into 6 pieces,
K
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
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
To cut a line segment K into 6 pieces, we cut K in half,
K
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
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
To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.
K
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
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
To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.
K
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
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
To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.
K
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
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
To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.
K
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
(Or we may cut K into thirds first then cut each into 2 pieces.)
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
To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.
K
If we divide each segment above into two again, we would have 12 pieces.
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
(Or we may cut K into thirds first then cut each into 2 pieces.)
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
To cut a line segment K into 6 pieces, we cut K in half, then cut each half into 3 pieces.
K
If we divide each segment above into two again, we would have 12 pieces.
To cut a line segment K into 8 pieces, we cut each of the 4 pieces into 2 pieces.
we get 16 pieces. (Practice drawing them).
If we cut each of the 8 pieces into 2
(Or we may cut K into thirds first then cut each into 2 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
Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint and the 1/3-mark,
K
Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint and the 1/3-mark,
K
Marking Lines and Making Grids
To cut a line segment K into 5 pieces, eyeball the midpoint and the 1/3-mark,
K
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
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
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
Marking Lines and Making Grids
This cuts K into two unequal segments.
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
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two,
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
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two, and cut the long piece into three to get 5 pieces.
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
By cutting each fifth into 2 halves, we have ten pieces..
K
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two, and cut the long piece into three to get 5 pieces.
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
By cutting each fifth into 2 halves, we have ten pieces..
K
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two, and cut the long piece into three to get 5 pieces.
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
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.
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two, and cut the long piece into three to get 5 pieces.
The purpose of the exercises is to make approximate accurate comparative pictures of lengths without too much distortion.
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
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.
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two, and cut the long piece into three to get 5 pieces.
The purpose of the exercises is to make approximate accurate comparative pictures of lengths without too much distortion.
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
Distorted mathematical drawings lead to confusions and false conclusions.
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.
Marking Lines and Making Grids
This cuts K into two unequal segments. Cut the short piece into two, and cut the long piece into three to get 5 pieces.
Marking Lines
Marking Lines and Making Grids
Marking Lines We can mark the dividers if the we know the length of the line segment K.
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
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
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
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
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
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
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
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).
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
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
Starting from the left end point as 0,
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).
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
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.
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
Starting from the left end point as 0,
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).
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
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.
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
They are 4, 8, 12, 16, 20, and 24.
Starting from the left end point as 0,
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).
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.
K0
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.
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
They are 4, 8, 12, 16, 20, and 24.
Starting from the left end point as 0,
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).
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.
K0 4
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.
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
They are 4, 8, 12, 16, 20, and 24.
Starting from the left end point as 0,
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).
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.
K0 4 8
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.
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
They are 4, 8, 12, 16, 20, and 24.
Starting from the left end point as 0,
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).
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.
K240 4 8 1612 20
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.
Marking Lines and Making Grids
To draw it, cut K into 2 segments, then divide each into 3 pieces.
They are 4, 8, 12, 16, 20, and 24.
Starting from the left end point as 0,
b. Divide K into 8 pieces and label the dividers. Draw.
K
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
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
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
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
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
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
K0
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
K30
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6K
30
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
We may estimate lengths or distances by inserting dividers.
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
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
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
Each small segment represents 24/12 = 2 inches.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
420
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
Each small segment represents 24/12 = 2 inches.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
1442 160
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
Each small segment represents 24/12 = 2 inches.
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.
6 9 12 15 18K
21 2430
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
We may divide K into 6 segments first, then divide each again into 2 pieces.
1442 160
Marking Lines and Making Grids
The length of each is 24/8 = 3 so the labels are multiples of 3: 3, 6, 9, 12, 15, 18, 21, and 24.
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
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
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
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
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
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
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
so the dividers are at 20, 40, 60, etc..
200K
Each small segment represents 200/10 = 20 miles.
We divide K into 5 segments first, then divide each again into 2 pieces.
100200
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
so the dividers are at 20, 40, 60, etc..
20080K
Each small segment represents 200/10 = 20 miles.
A
We divide K into 5 segments first, then divide each again into 2 pieces.
100200
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
so the dividers are at 20, 40, 60, etc..
The 90-mile mark is the midpoint 80 and 100.
20080K
Each small segment represents 200/10 = 20 miles.
A
We divide K into 5 segments first, then divide each again into 2 pieces.
100200
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
so the dividers are at 20, 40, 60, etc..
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.
Each small segment represents 200/10 = 20 miles.
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
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
r rows
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
r rows
c 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
r rows
c columns
r x c 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.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
Example C. a. Describe one methodfor cutting a pan-cake into 24 pieces as evenly as possible.
r x c 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.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
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
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
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.
One way is to cut it into 4 rows and 6 columns.
r x c 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.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
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.
One way is to cut it into 4 rows and 6 columns.
r x c 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.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
Example C. a. Describe one method
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.
for 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
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
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.Then cut it into 3 columns, followed by cutting each column into 2 to get 6 columns.
for 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
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
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.Then cut it into 3 columns, followed by cutting each column into 2 to get 6 columns.
for 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
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
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.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.
for cutting a pan-cake into 24 pieces as evenly as possible.
One way is to cut it into 4 rows and 6 columns.
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.
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.
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,
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.
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,
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.
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,
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.
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.
When we cut a rectangle in to R rows and C columns, we say that we make a R x C grid.
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,
When we cut a rectangle in to 4 rows and 6 columns, we say that we make a 4 x 6 grid.
such as cutting a piece of wood or a cake by hand & sight.
3 rows x 8 columns, or 4 rows x 6 columns.
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.
When we cut a rectangle in to R rows and C columns, we say that we make a R x C grid.
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.
Marking Lines and Making Grids