Directions and Angles

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Adler Directions and angles 1494487

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Adler Directions and angles

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The "Reason Why' Books

Directions and Angles

Irving and

Illustrated

Ruth Adler

by Ellen Viereck

The John Day Company

New

York

The "Reason Why" Books by Irving and Ruth Adler .\IR

ATOMS AND MOLECULES COAL COMMUNICATION THE CALENDAR DIRECTIONS AND ANGLES E\'OLUTION FIBERS

HEAT HOUSES INSECTS IRRIGATION: CHANGING DESERTS TO GARDENS

LEARNING ABOUT STEEL THROUGH THE STORY OF A NAIL MACHINES MAGNETS NUMBERS OLD AND NEW NUMERALS: NEW DRESSES FOR OLD NUMBERS OCEANS RIVERS SETS

SHADOWS STORMS TASTE, TOUCH AND SMELL THE EARTH'S CRUST

THINGS THAT SPIN TREE PRODUCTS WHY? A BOOK OF REASONS

WHY AND HOW?

YOUR EARS YOUR EYES

A SECOND BOOK OF REASONS

© 1969 by No

Ir\

ing Adler

part of this book

may be

reprinted, or repromechanical or other means, now known or hereafter invented, including photocopying and recording, or in any infonnation storage and retrieval system, without permission in writing from the Publisher, The John Day Company, Inc., 62 West 45th Street, New York, N.Y. 10036. Published on the same day in Canada by Longmans Canada Limited. All rights reserved.

duceu or

utilized in

any form or

b\"

any

electronic,

Library of Congress Catalogue Card Number: 69-10489

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS Directions and Angles

^

^

AQd/l^^

Opposite Directions

6

Directions on a Straight Line

7

Making

a Straight Line

Up and Down Where You

~

Are

8

9

Directions on a Plane

10

Directions in Space

12

Change

14

of Direction

Angles

16

Half Planes

18

The Interior and Exterior of an Angle Comparing Angles for Size

18

20

Right Angles

22

Rectangles

24

Squares

25

Horizontal Lines and Planes

26

Angles That Are Not Right Angles

28

Two

29

Right Angles Together

Greater than

Any Angle

Flag Signals

The Measure

30 32

of

an Angle

34

Angles on the Face of a Clock

36

Making a Protractor

38

Using a Protractor

40

Up and Down

42

The Spinning

at Different Places

of the Earth

North, South, East,

Finding North

West

43 44 46

Directions

What

is it

that

is

where? The answer

A

hand

of a clock

and Angles

always moving but doesn't go anyto this riddle

is

"a

hand

of a clock."

moves by turning around one

place,

instead of going from one place to another.

There are many things that move by turning. Some, like the turntable of a

phonograph, stay

in

one place

while they turn. Others, like the wheel of a bicycle,

from place

to place as they turn.

when you unwind its ning like a top. You about on the earth.

string.

The

A

move

top turns or spins

earth also turns, spin-

often turn yourself as you

move

A hand

of a clock points in a certain direction.

turns,

is

to teach

changes

it

it

directions

we

its

direction.

The purpose

book

of this

you some useful and interesting things about

and changes

how and how we

of direction. It will explain

use an arrow as a picture of a direction

use an angle as a picture of a change of direction. explain the

meaning

south, east,

and west.

angles for

When

of the directions up,

down, north,

show you how to compare explain what a right angle is and

It will

size. It will

how you can make

It will

one. It will also

and use an instrument

for

tell

you how

measuring angles.

to

make

Opposite Directions If

two people stand back

directions. In the

direction in

to back, they face in opposite

drawing above, an arrow points

which each person

is

facing.

to the

Each person

word forward for the direction in which he faces and the word backward for the opposite of that direction. The picture shows that what is forward for one person may be backward for another person. Left and right are also opposite directions. In the picture below showing two people back to back, what is

uses the

left for

one person

All the directions in a plane can

drawn from P,

be shown by arrows

a single point, P, of the plane. If

direction on the plane,

through

BA

and the

line

CD

CD

is

does not pass

then you can draw an arrow from P that

parallel to the line

the direction

CD

a

and shows the same direction

is

as

CD.

11

Directions in Space

A

tiny speck of dust floating in the air

what we mean by a point pictured by the dust different point

Many

is

Each

The

point in space

the place where

it

Every

is.

a different place in space.

straight lines

in space.

is

in space.

shows roughly

can be drawn through any point

of these lines has

two

on

directions

All

it.

drawn

these directions are directions in space. Arrows

from that point to show these directions surround the point like the thorns of a nettle or like the quills of a

frightened porcupine.

make

point in space, point,

and

To

picture the directions from a

a small ball of clay to stand for the

stick toothpicks into the ball all

Then each toothpick

is

like

around

it.

an arrow showing a different

direction in space.

may meet at a point, or they may never meet no matter how far the lines are drawn in either direction. What do we call If

two

straight lines are

lines that

drawn

in space, they

meet?

Pairs of lines in space that never different kinds.

One

are not together in pair like this finger

12

meet may be

made up of two any one plane. You may kind

is

by holding one index

above the other so that

it

of

two

lines that

picture a

finger or pointing

crosses over

it.

Each

index finger pictures a straight

but there

is

line.

The Hnes never meet,

no plane that contains both

lines like this are called

The other kind

is

lines.

skew (SKYOU)

made up

of

two

gether in one plane and never meet.

directions

directions

lines that are to-

We

know

If

we

already

and that the

on one of them are the same

on the other one.

pair of

lines.

that lines like these are called parallel lines

two

A

as the

two

use lollypop sticks to

picture straight lines in space, a bundle of sticks lying side

by

side, as in the

drawing below, pictures many

lines that are parallel to

same two

each other, each showing the

directions as the other.

All the directions in space can

drawn from

a single point, P, in space. If

tion in space,

then there the line that

is

is

CD.

be shown by arrows

and the

line

CD

CD is

a direc-

does not pass through P,

P and plane you can draw an arrow from P

a plane that contains both the point

In that

parallel to the line

tion as the direction

CD and shows

the same direc-

CD.

13

change

When another,

a

hand

it

changes

of Direction

of a clock turns its

direction.

from one position

You can

to

picture this

change of direction by drawing two arrows from one point in this way:

same direction

Draw one arrow

so that

it

shows the

draw

as the starting position of the hand;

the other arrow so that

shows the same direction

it

stopping position of the hand.

as the

To remind you which

of

the two arrows shows the direction of the stopping position,

draw a curved arrow between the two,

as in the

diagram below, showing the change of direction when a

hand

of a clock turns

from 12

to 2.

A clockwise tara If

the

from 2

hand

to 12,

of a clock

is

turned back the other

you can picture the change

way

of direction

by

using the same two straight arrows, with the curved

arrow between them pointing the other way.

tA 14

coaatercLockwlse tura

A

turn that might be

the clock

is

running

way

that goes the other

Which

is

of the turns

made

hand

b\' a

of a clock while

called a clockwise turn. is

A

turn

called a counterclockwise turn.

shown below

are clockwise?

Which

are counterclockwise?

Not every turn tion. If

is

hand

you turn the hand

stopping position there

of a

is

no change

of a clock changes

all

the

the same as in

its

way around

its

direc-

so that

its

starting position, then

direction.

The

turning that brings the hand back to is

its

its

least

amount

of

starting position

called one complete rotation.

One compLete

rot at to rv 15

Angles

I

On page tion b\'

14

we

pictured a turn or a change of direc-

drawing two arrows from one point and drawing

a curved arrow

changes

between them.

in this picture to get

We

shall

now make two

another kind of picture of

a turn kno\\Ti as an angle.

we do

Sometimes

not care whether a turn

or counterclockwise.

arrow

Then we mav

in the picture. This

is

the

is

clockwise

leave out the cur\ed

first

change.

we draw an arrow to show a direction, it makes no diEerence how long we make the arrow. Even if the arrow is made longer or shorter, it still points in the same direction. Then we might as well leave the length of the \\'hen

arrow out of the picture. This can be done by picturing each direction as an arrow that has no length but goes on

and on away from is

its tail

the second change

A

straight

The

we make

tail

its tail

without coming to an end

point of the ray

tex).

Vertex

I 16

in the picture of a turn.

arrow that has no length but goes on and

on awa>" from a ray.

without coming to an end. This

A

ra.

y

is

called

its

called

is

vertex

(

\T^R-

We can now describe the new picture of a turn in this way: An angle

is

the set of

all

points of

two rays that are

on different straight lines hut have the same vertex. Each ray

called a side of the angle.

is

An.

angle

vertex Draw point

One

P

a straight line,

point, P,

the other side of P.

on one

Each

P separates the two

side of P.

of these sets

The other is

it.

The

two

sets.

on

di\"ides the other points of the line into

set of points is

one. If

and choose a

set

is

on

called a half line.

half lines, but does not He in either

you add the point P

to either half line,

you get a

ray.

KaLf-Llne

^

kal-f-llae

To name an angle, we use three capital letters in this way: The middle letter is the name of the vertex of the angle; the other letters name other points on the sides of the angle, one on each side. For example, the

first

angle

shown below may be called either angle RST or angle TSR. What are the names of the second angle below?

ha(.f -

pLcLae

A-

F

K i

P

G

H

/ L

-J Q

V

Serria^

A

O

5

T

•—

IU

J

I

lore


/

>r

^v

/

0/

y

/."

3.

FoLdLecL

-f

Lot oujalrt

35

Angles on the Face of a Clock

Imagine a ray drawn from the center of a clock face to each

number on the

clock.

These rays form twelve

congruent angles that add up to one complete rotation.

The measure

of each of these angles

is

30 degrees, be-

cause 12 times 30 degrees equals 360 degrees. The angle

formed by the hands of a clock

one o'clock

is

one of

these angles. So, at one o'clock, the angle formed

by the

at

hands of a clock has a measure of 30 degrees.

What is

the measure of the angle formed by the hands

of a clock at

36

two

o'clock? At three o'clock?

what is the measure

of the angle

fonned by the hands

of a clock at four o'clock? At five o'clock? At nine o'clock?

I What is the measure of the of a clock at eight o'clock?

angle fonned by the hands

At ten o'clock? At eleven

o'clock?

Through how many degrees does the minute hand a clock turn in an hour? Through

does

it

how many

degrees

turn in a quarter of an hour?

Through how many degrees does the hour hand clock turn in an hour? it

of

of a

Through how many degrees does

turn in half an hour?

37

Making a Protractor

An

instrument for measuring angles

tractor.

Here are the

is

called a pro-

making one out

directions for

of

paper or cardboard.

Use a

stiff

paper or a thin cardboard that can bend

down on

without breaking. Put a plate upside

and draw a plate.

The

line

on the paper

The cur\ed

line

By

around the edge of the

you make

part of the paper that

called a disk.

all

is

the paper,

in this

way

is

surrounded by the

a circle. circle

is

cutting along the circle with scissors,

cut out the paper disk.

Fold the disk exactly to separate the will

in half.

two halves

Then

of the disk.

be made out of one of these

disk so that the

two ends

of

together. Press the crease

cut along the crease

its

flat.

The

protractor

halves. Fold the half

straight

When

disk, the crease will look like the line

edge are brought

you unfold the half

PC

in the

drawing

below. The point, P, where the crease meets the straight

edge of the half disk

Draw

38

will

be the center of your protractor.

a line along the crease, like the line

PC

below.

On the right-hand disk

up

to the position

way

this in

such a

in the

drawing

edge

at

side of PC, fold the

PB

to

make

that the angles

will

a

edge of the half

new

crease, PA.

Do

CPB and EPA shown

be congruent. Draw a

along the

line

PB. Unfold the paper, and draw a line along the

crease PA.

Do will

p p the same thing on the left-hand side of PC.

then have

lines

and the

five lines

straight

where they form

six

each of these angles

There

is

drawn on the

edge of the half disk

all

These

meet

at P,

congruent angles. The measure of is

30 degrees.

a piece of the curved edge of the half disk in

the interior of each of these it

dot to the center P.

You

coming from

P,

Put two dots on

six angles.

into three equal parts. Join each

each piece to divide

grees.

half disk.

You

will then

have a

series of rays

forming angles whose measure

Write the numbers

10, 20, 30,

and

so on,

is

10 de-

up

to 170,

next to these rays, as shown in the drawing above. Your protractor will then be complete. learn

how

to use

Turn

this

page over

to

it.

39

Using a Protractor

d

If

you have made the protractor described on page

\ou can use

it

to find the

measure of

39,

an\- angle to the

nearest 10 degrees.

To measure tracing

it

the angle

shown below,

first

copy

it

by

on a piece of paper. Then extend the sides of

\our cop\' until they are longer than the distance from the center of >"our protractor to the curved edge.

Then

follow

these steps to measure \our copy of the angle: Put the protractor on top of the angle so that

the protractor straight (

3

)

the

is

(

1

)

the center of

on the vertex of the angle; (2) the

edge near the 10

is

on one side

of the angle;

the protractor rests on the interior of the angle.

number on

the protractor that

side of the angle

is

is

the measure of the angle.

The as

picture it

on \our cop\' of the angle. The measure of the angle

40

Then

nearest to the other

below shows how )our protractor should look

degrees.

and

rests is

40

Trace angle

XYZ shown below,

and measure

same thing with angles RST, ABC, and

it.

Do the

1

UVW.

R

W In the diagram below there are three angles. Trace

only the angle

KLM whose vertex

is

at L,

LKM, and measure

Then

trace angle

angle

KML, and measure

and measure

it.

it.

Finally illy, trace

^Km

it.

M

J 41

Up and Down At each point on the rections called

at Different Places

earth's surface there are

up and down. Down

the center of the earth. center of the earth.

Up

The

is

is

two

di-

the direction to

the direction

away from

the

picture below shows people

standing at different places on the surface of the earth.

The up

direction

is

these directions are

shown all

for each person. Notice that

that intersect at the center of the earth. directions are the

on

lines

Remember

that

different because they are

same only

if

they are shown by arrows

that are on parallel lines. If

two people are on opposite

direction that

is

up

for

VV^

42

sides of the earth, the

one of them

is

down

for the other.

The Spinning

The spins

of the Earth

The Hne around which it earth. The points on the sur-

earth spins Hke a top.

is

called the axis of the

face of the earth that are on

its

axis are called the

North

Pole and the South Pole.

Because the earth

ground

spins, the

at the

North Pole

turns around the pole like a phonograph turntable.

It

turns counterclockwise as seen from above the ground.

A horizontal

arrow drawn on the ground from the North

Pole would turn with the ground.

arrow drawn, and ture

can imagine an

we can use the imaginary arrow to

and measure the turning

A horizontal

We

of the earth.

arrow drawn on the ground

at the

Pole makes one complete rotation in a day. So

Through

earth turn in half a day?

how many Through how

hours? Through

degrees does the earth turn in

many

North

we say that

the earth turns through 360 degrees in a day.

how many degrees does the What part of a day is six

pic-

six

hours?

degrees does the earth turn in one hour?

43

Nortk Pole

North. Pole

^ULtk

SoutK Pole North, South, East,

Through every point on the

West

earth's surface that

the North Pole or the South Pole there that joins circle

is

it

Pole

is

just

one

is

not

circle

North Pole and the South Pole. This

to the

called a meridian

(muh-RID-ee-uhn)

circle.

The

drawing above on the right shows many of them.

At each point on the tal plane.

have

earth's surface there

a horizon-

In that plane the direction in which you would

to start in order to

move toward

along a meridian from that point rection in

is

which you would have

is

the North Pole

The diorder to move

called north.

to start in

toward the South Pole along a meridian from that point is

called south.

At a point that Pole, north line.

is

neither the North Pole nor the South

and south are opposite

This line

is

directions

on the same

called the north-south line in the hori-

zontal plane through that point. If you face north at that point, the direction in the horizontal plane that points to

your right left is

44

is

The direction that points to your East and west are opposite directions

called east.

called west.

To tke NortK,1i)Le-

Dlrectloas La tKe korlzorutoLL pLctae at .west

ecx^t.

CL

poinjt tKcdt

th-e,

not

Is

Nortk or SoutkPoLe.,

To the SouilvPoLe.

on a

line called the east-west line.

The

east-west

hne

through a point makes right angles with the north-south line

through that point.

Directions in the horizontal plane at the North Pole are very different from those at a point that

not the

is

North Pole or the South Pole. At the North Pole every direction in the horizontal plane points

North Pole. So there

no direction there that

you move

north. In fact,

if

from the north

pole,

that

is

it

you could follow

away from

starts

to the

in

is

the

called

any horizontal direction

you along a meridian

circle

South Pole. So, at the north

pole, every horizontal direction

is

south.

At the north

pole there are no directions called east and west.

Where on tion north?

the earth's surface

is

every horizontal direc-

Finding North It is

easy to find out which direction

shadows

cast

by a

stick

on a sunny day.

north by using

is

If

you hve

in the

United States or Canada or Europe, follow these directions:

Drive a stake into level ground so that the stake vertical.

On

a sunny

day the stake

will cast a

shadow on

the ground. As the sun moves across the sky, the will turn

morning

and change it

will

grow longer

grow

again.

in length at the

shorter,

When

and

the

same

shadow

time. In the

in the afternoon

shadow

is

is

shortest,

it

will

it

will

be pointing north. Th.e sKortest sKcLcLow polats

aorth.

It is

not easy to

shortest.

tell

But you can

shortest in this way.

when tell

the

shadow

where

Take a length

it

will

of the stake

is

be when

is

of rope that

is

it

longer

than the shortest shadow cast by the stake. Tie one end of the rope to the

bottom of the

around a small pointed swing

46

it

stake. Tie the other

stick. Pull

end

the rope taut, and

around the stake while you press the point of

the stick against the ground. circle

on the ground.

around a

vertical post

of a pointed stick.

when

)

Draw

of the

shadow

stick will scratch a

you are making the

circle

on a pavement, use chalk instead

In the morning, watch for the time

the end of the

circle.

(If

The

shadow

of the stake falls

on the

a line on the ground showing the position at this time.

Do

the same thing in the

You then have two lines on the ground that are the same length. These lines lie on two rays whose vertex is at the stake. These rays form an angle. The afternoon.

direction north

between

is

in the interior of this angle

halfway

its sides.

North ^nd, of

"th-e

Is

Kaifwau

between- ilT>e equoL

shaAow, end. of the shcudow

47

ABOUT THE AUTHOR Irving and

Ruth Adler have

more than

written

sixty

books about science and mathematics. Dr. Adler has

been an instructor sity

and

of the

at

in

mathematics

at

Columbia Univer-

Bennington College, and was formerly head

mathematics department of a

New York

City high

school. Mrs. Adler taught mathematics, science in schools in the

New

York area, and

and

art

later also taught at

Bennington. In addition to working with her husband writing this book, she joined with in the

Reason

most of them

Why

series

as well as for

him on 29 other

and drew the

many

titles

illustrations for

other books written by

him.

Books by Irving Adler alone and books by him

in col-

laboration with Ruth Adler have been printed in 85 different foreign editions, in 15 languages editions.

and

in

10 reprint

The

WHY

REASON

Books

by

Irving

and Ruth Adier

"They are excellent"— New York Herald Tribune "The best of the matter is that, with authors like the Adlers, their name is a guarantee. One can be certain that not only is the exposition clear and logical, but that the scientific —The Horn Book Magazine matters presented are correct and up-to-date."

EVOLUTION books with great scientific accuracy and have the ability to Well presented and interesting." matters for young readers —Catholic Library World

"The Adlers present simplify difiBcult

their

all

.

.

.

COAL "Described in this interesting, well-written text are the uses, origin, mining processes, and chemistry of coal. Pictures of methods and equipment are particularly useful." —Library Journal

THINGS THAT SPIN "A

helpful

—The Horn Book Magazine

and stimulating book."

SHADOWS "An

easily

understood explanation of the causes and uses of shadows."

—ALA Booklist

NUMBERS OLD AND NEW "A

fascinating book for the student interested in mathematics."

—American Library Association "Exceptional book about

how we came to count as we do." —Child Study Association

WHY? A "I'd suggest that

it

be given

to a child

collecting unrelated facts."

interrelation of plants

trations in

two

and

America

Book of Reasons

with an inquiring mind and acquisitive inj^tinct for —Virginia Kirkus

INSECTS "The

of

AND PLANTS

insects, for the

colors."

middle grades. Attractive and useful iUus—The Horn Book Magazine

ATOMS AND MOLECULES "Successfully aiming at the eight-to-ten-year-olds, the Adlers introduce chemical symbols —The Horn Book Magazine and chemical formulas even a bit of nuclear chemistry." .

.

.

MAGNETS "An

excellent primer for potential .scientists

of a

wide range

of age groups."

and engineers,

it

should excite the imagination

— " "American Association for the Advancement of Science HOUSES

"In keeping with others in the series, simple. Recommended."

tliis

survey of dwellings

'.JOHNOAT^

''.

V

is

comprehensive, clear and —Library Journal