HOW TO MAKE SOME INEXPENSIVE VISUAL AIDS [613171]

HOW TO MAKE SOME INEXPENSIVE VISUAL AIDS
TO BE APPLIED TO THE TEACHING OF A UNIT
IN AREA AT THE JUNIOR HIGH SCHOOL LEVEL
A Project
Presented to
the Faculty of the School of Education
The University of Southern California
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Education
by
James Lovell Hancock
January I 950

U M I Number: EP45858
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TABLE OF CONTENTS
CHAPTER PAGE
I. INTRODUCTION AND PROBLEM ……………. 1
• II. THE BLACKBOARD …………………….. 4
III. THE BULLETIN B O A R D …………………. 9
IV. DIAGRAMS ………………………….. 12
V. SPECIAL CONSTRUCTIONS FOR THE AREA'-UNIT. . 28
VI. APPLICATION OF AUDIO-VISUAL AIDS TO THE
r TEACHING OF A R E A …………………. 40
Aims of the u n i t ………………… 43
Introduction of the Area Unit to the
C l a s s ……………………….. 43
VII. THE SQ U A R E ………………………… 45
VIII. THE R E C T A N G L E …………………….. 50
IX. THE PARALLELOGRAM …………………. 53
Optional ……………………….. 55
X. THE TRIANGLE ………………………. 56
XI. THE T R A P E Z O I D ………………… . . 60
XII. THE CIRCLE ………… ……………… 6l
XIII. CONCLUSION ………………………… 65
BIBLIOGRAPHY …………………………… 07

LIST OP FIGURES
FIGURE PAGE
1. Comparison of an Acre, a City Lot, and a
Square Rod ……. ……………. 18
2. Three Major Types of -Angles ………… 19
3. Comparison of a Square Foot and a Square
Inch ……………….. 20
4. Front View of the Trapezoid Diagram Showing
the Proper Labeling . . ………….. 22
5. Diagram of the Square with Appropriate
Labeling …………………………. 23
6. Diagram of the Rectangle with Appropriate
Labeling …………………………. 24
7. Diagram of the Parallelogram with Appropriate
Labeling ……………………. 25
8. Diagram of the Triangle with Appropriate
Labeling …………………………. 26
9. Diagram of the Circle with Appropriate
Labeling …………………………. 27
10. Special Construction Illustrating Equality
of Areas in Parallelograms with Equal
Bases and A l t i t u d e s ……………….. 32
11. Special Construction Illustrating the
Constant P i ……….. 37

H i
FIGURE PAGE
12. Special Construction Illustrating the
Ratio of the Radius to the Diameter
in a C i r c l e ……………………… 38

LIST OF TABLES
TABLE PAGE
I. Common Measures of S u r f a c e ………… 3^
II. Common Measures of L e n g t h ………….. 35

CHAPTER I
INTRODUCTION AND PROBLEM ‘
The teaching of abstractions and subjects that –
embody these abstractions is a source of never ending
trouble for the conscientious teacher. An abstraction
or concept is the end product of a number of direct
experiences. Education involves the classification of
direct experiences into concepts and the formation and
refinement of new concepts. One is not educated by the
mere addition of a quantitative number of concepts, but
must classify and generalize the concepts and experiences
into accurate and usable units.
Learning is an interacting process, an intercommuni­
cation of ideas. The learning process first involves a
problem, then the gathering together of data for the
solution of this problem, the application of the data,
and finally the solution of the problem. Children's
problems are concerned with concrete experiences. From
these concrete experiences the child generalizes or makes
an abstraction usable for a similar situation. The
presentation of a similar situation reverses the procedure-
the child has a generalization or abstraction and applies
it to a concrete situation; thus, learning moves from the
concrete to the abstract and back to the concrete again.

2
The teacher has an important function in the class-
room– that of guiding the building of generalizations and
concepts. Difficulty is encountered when a teacher talks
in glittering generalities and abstractions that are
beyond the grasp of the student. The teacher must start
where the pupil is, in his level of understanding and
guide him to a higher and higher level of abstraction.
One forgets what does not seem important and what is not
understood is less important and is not usable for one1s
own goals.
In mathematics the formulas and concepts involved
are often verbalizations that are meaningless to the
student. These formulas are the summation of a large
number of experiences with numbers but the formulas
can’t mean anything to the student unless they condense
his experiences. Too often a student's experience with
numbers is confined to rote repetition of teacher material
deemed important by the teacher but meaningless to the
student. The teacher must secure pupil participation and
interest by correlating the work as much as possible to
everyday life situations and enriching the child's
experiences with numbers by audio-visual aids and diver­
sity of approach to subject matter.
The question may well be asked why audio-visual
aids in mathematics need to be constructed when the Los

3
Angeles School System has an audio-visual aid department.
The answer is that few audio-visual aids in mathematics
are made for distribution in the schools; consequently,
the teachers who want their students to understand the
material presented must either buy or make the bulk of
their own audio-visual aids. Ingenuity and a little
effort will produce surprising results in visual aid
construction, however, and the constructor neither has
to be adept with tools nor wealthy. Many visual aids
applicable to the classroom experience are near to the
teacher but he either fails to use them properly or fails
to use them at all. The first five chapters of the paper
deal with construction techniques, and the last eight
chapters with the application of these constructions and
other audio-visual aids to a unit on area which is usually
taught formally in the eighth grade. The unit is designed
so that it may be taught within two or three weeks
depending upon the opinion of the teacher as to the
unit's importance.

CHAPTER II
THE BLACKBOARD
The blackboard Is probably the most universally
used visual aid, but few teachers derive the maximum
benefit from its use. This chapter will deal with some
' suggestions that will improve blackboard technique as
well as some devices that are helpful for the mathematics
teacher.
Copying lengthy outlines or lists of subject matter,
such as rules for room control and procedure, is a waste
of time to both the instructor and the student. If such
material should be needed, it should be duplicated and
distributed.
The blackboard should be clear and neat with
extraneous unrelated subject matter erased, so that the
eye will be focused on the Immediate Instruction taking
place. The blackboard is comparable to an advertising
sign. A sign that is overcrowded and untidy attracts
little or no attention. It is often beneficial to leave
an illustration of a problem on the board when under­
taking new work so that the students may have a template
or model to use for comparison; but unless this is the
case, the work not pertaining to the presentation should
be erased.

5
The material to he presented should he as brief and
simple as. possible–short suecinet statements or diagrams
are much easier to follow.
– -Since the blackboard is a visual aid, the writing
must be legible and capable of visibility from the back
of the room, the effectiveness of the blackboard is
reduced if small illegible writing is used. Student
concentration is wasted on deciphering handwriting
rather^than following the presentation.
The use of colored chalk-often makes the difference
between success and failure in classroom instruction.
When bright colors are used to indicate an altitude, for
example, and bright contrasting colors are used to indi­
cate the base and label component parts, the eye is
attracted to the color and the pupil discerns the parts
indicated more readily. Soft colored chalk is preferred
to hard chalk because it is easier to use and is more
brilliant in hue. A box of about fifteen colors may be
obtained for sixty-five cents at the Stationers Inc. at
Fifth and Spring St., Los Angeles, California. This seems
expensive but it must be remembered that a box will last
many years, depending upon the use.
In erasing colored chalk, a rag is often useful.
Colored chalk dust clogs the ordinary school eraser so

that it smears the hoard with subsequent use. The hand
should never be used as an eraser. The rag or the eraser
should be kept nearby at all times.
An eight to fourteen inch piece of wrapping twine,
such as is used to tie laundry boxes, serves as a good
expedient for drawing circles. Make a few turns around
a piece of chalk with one end of the string. Grasp the
free end of the string between the thumb and first finger
of the left hand. Press the left hand firmly against the
board and using this as center, draw a circle by rotating
the right hand one revolution. The radius of the circle
is determined by the distance between the two hands. If
the class is small enough, this idea can be extended by
securing a piece of string for each student in the group,
thus allowing group board work with circles. Such indi­
vidual work with the string would help implant the idea
of the locus of the circle.
It is very important to make drawings as accurate
and realistic as possible so that the student ean actually
see, for instance, that the opposite sides of a parallel­
ogram are equal without using his or her imagination.
Straight lines must be straight; equal lines, equal;
parallel lines, parallel; circles, true; right angles
must have perpendicular sides; equilateral triangles must
have equal sides and angles; Isosceles triangles, equal
base angles, etc.

7
Good drawing technique requires the making of
straight lines which is only possible with the use of a
straight edge. The yardstick is the most convenient
instrument for drawing straight lines; however at the
blackboard, the yardstick is awkward and cumbersome to
use because it is difficult to remove from the flat
surface. This is readily overcome by going to Woolworth
and Co. or any hardware store and buying two one-quarter
inch length wood screws and a five cent plastic handle
to be fastened to the yardstick. If the screws protrude
through the wood, file the points until they are level
with the surface of the wood or the blackboard will be
scratched. If a yardstick is not available in the class­
room, cut a flat piece of wood two or two and a half
inches wide by one yard long, and mark off the one foot,
two foot, three foot and two inch gradations between the
consecutive foot marks. The handle may then be applied
as explained previously.
During the presentation of the subject, the teacher
should use a positive approach and place the illustrations
on the board at the right moment to clinch the discussion.
Only simple Illustrations should be made while the dis­
cussion is being made, the more complex illustrations
should be constructed before class on some form of drawing
paper. The teacher should never stand in front of the

8
illustration, but should use a pointer when referring to
details.
The lighting conditions should be adequate to
insure proper sight. Care should be taken to avoid any
glare which may result from sunlight or poor placement
of the lights in the room. Glare prevents a portion of
the glass from viewing the material clearly.
The practice of pasting material on the black­
board should be reduced to a minimum because it limits
the usable space and usually leaves a paste mark on the
board after removal of the material. Nails, pins, and
tacks disfigure the board permanently; use the bulletin
boards for hanging up material.
It is essential to plan the work to be presented
at the blackboard, or at least to have a vague idea of
the drawings and illustrations which are going to be
used so that the materials may be gathered together before
the period begins. Nothing looks quite so absurd as
the sight of a teacher looking frantically about the room
for a piece of string, colored chalk, or some other
material, while the class interest is impaired, and
control is momentarily lost.

CHAPTER III
THE BULLETIN BOARD
The bulletin board Is extremely useful as a follow-
up device to extend and repeat work previously undertaken.
For example, a detailed drawing of a trapezoid may be
reproduced and put on the bulletin board for review.
Another important use of the bulletin board is the pre­
sentation of current editorials and events that are
related to the class work. The teacher may select com­
mittees of students for each unit of work that is under­
taken and make them responsible for that particular unit
of work. Pictures and information should be replaced as
often as possible to avoid monotony. If the students are
given the full responsibility and the right to exercise
their imagination, the teacher will find that the bulletin
board will become the center of attraction with little
or no [.effort on his or her part.
Two types of bulletin boards are in use, the
permanent and the portable. There are distinct advantages
and disadvantages entailed In the use of both types. The
permanent type should be so placed that it is in the flow
of traffic, is well lighted, attractive and readable. It
Is limited in use because It is not always easily acces­
sible to the teacher while class is in progress. Special

10
constructions are impossible if the teacher has to walk
to the back of the room. It is unhandy to put posters
and charts that are used during the class session on a
bulletin board that is not conveniently near the instruc­
tor. . Some rooms have permanent bulletin boards both in
the back and at the front of the. room. If this is the
case, the teacher has access to one of them at all times.
Not all rooms, however, have bulletin boards; for this
reason, it is advisable to have a portable bulletin board
that may be easily moved to meet the needs of the situation.
A portable bulletin board may be made from celotex
very cheaply. A piece of celotex three feet by two feet
costs about fifty or sixty cents and is obtained at almost
any lumber yard. If an attractive appearance, is desired,
a frame of quarter-round stripping may be purchased for
thirty or forty cents and applied to the edges. It is
useful to fasten a picture hook and wire on the back of
the bulletin board so that it may be hung on the wall
when desired. Another inexpensive material to use is
linoleum. A three foot by four fobt piece costs about
fifty cents at any regular linoleum shop, ask for ends or
remnants of rolls. Follow the above procedure for
framing and use the back of the linoleum for the front
of the bulletin board. The board may be made more trans­
portable by splitting the material in half and joining

11
the tvo halves with hinges, enabling the teacher to fold
the board for easy, carrying. This board may then be used
for yarn construction to be described later, and may be
used as an expedient permanent bulletin board when there
is no other board in the room.
The most important point to remember is that the
bulletin board must be related to pupil activity and
interest, and not be used as an ornamental tool.

CHAPTER IV
DIAGRAMS
Diagrams are extremely useful in mathematics
because they clarify work that may be otherwise difficult
to explain. The teacher has the choice, however, of
putting the diagrams on the blackboard or on paper. Any
diagram that may be put on paper may also be put on the
blackboard, but there are obvious limitations of each
technique. The teacher may use a diagram at the black­
board to illustrate a specific point in the discussion,
but often this type of drawing is made freehand with
sketchy labeling. Great care has to be taken that the
diagram approximates reality; if not, the visual concep­
tion is lost and.the diagram should not have been used
in the first place. It is preferable to use the straight
edge for straight lines anc colored chalk as outlined
in the chapter on the blackboard because a square should
look like a square, for example, and the circle should
look like a circle. The student will spend less time
using his imagination and more time using his mind for
concentration if a little more care is exercised at the
blackboard for diagrams.
Diagrams that are made on cardboard or drawing
paper overcome many objections that may be raised against

13
the diagram drawn at the blackboard. If, for example,
a detailed drawing is to be made at the blackboard, the
entire class has to wait for the teacher to complete the
diagram. Time is wasted that might otherwise be bene­
ficially used for instruction; furthermore, class control
and discipline tend to get out of hand while the diagram
is being made. This objection is eradicated if the
teacher puts the complex diagram on drawing paper and at
the appropriate time holds or pins it up. Continuity in
teaching procedure is maintained, and pupil interest is
sustained and often increased by the timely introduction
of the ready-made diagram. The complaint is often made
that drawing paper diagrams consume too much construction
time to be practical. Actually, time is saved when one
considers that the blackboard diagram is erased and has
to be remade each time the occasion arises. The paper
diagram on the other hand, can be used repeatedly from
year to year and may be used very effectively for review
purposes.
When selecting the drawing material for the diagram,
several factors should be kept in mind. It is preferable
to select a heavy duty paper or light poster board rather
than light drawing paper, because the visual aid is used
repeatedly and is handled many times. A heavy duty
material will not split, bend, or crease as easily as the

14
lightweight material. White Is usually a poor color to
choose for the material because it soils too easily and
looks unclean after two or three demonstrations. An
orange or yellow faced material is suggested because it
does not soil as easily and offers a bright visual sen­
sation that attracts attention, but yet is light enough
to allow clear visibility. The texture of the material
is also important. If the material is too coarse, the
ink or poster paint will not absorb evenly, but will run
and make blurry lines.
Material that combines the qualities of color,
texture, and thickness mentioned, may be obtained at any
stationers store. Among others, Webberaft Stationers and
Printers at 2 5 0 7 West Washington Boulevard, sells poster
board suitable Tor this work in twenty-eight inch by
forty-four inch cards at ten cents each.
There are several techniques involved in making
colored diagrams. One method of making colored lines is
to buy an inking pen and use waterproof drawing ink.
Higgins Company manufactures a wide assortment of colors
costing twenty-five cents a bottle. The colors included
are carmine red, blue, leafgreen, russet, red orange,
red violet, turquoise, neutral tint, brown, and indigo,
plus indigo ink and white Ink. A variety of colors is
available in inks allowing unlimited color combinations.

15
There are several disadvantages to the use of ink; one
Is that the Inking compass for drawing circles costs
about two or three dollars; secondly, the Ink colors
aren’t as deep and rich In hue, thus tending to give a
thin appearance to the lines. Colored pencils afford
another excellent methodibr drawing colored lines. The
pencils may be purchased for ten cents apiece at any
art store. Care must be exercised in selecting the
pencils that are to be used. If one seleets a hard lead,
It is too difficult to make a wide line, and If one uses
a very soft lead, the fine lines are difficult to make.
The solution to the problem is to buy the bulk of the
pencils with soft lead and buy two or three hard lead
colored pencils for detailed work. When colored pencils
are used for drawing circles, an ordinary school compass
may be purchased for fifteen cents that is adequate for
the work. Craftint Manufacturing Company sells a large
variety of show card paints that may be used for coloring
In solid figures. The price of the show card coloring is
twenty cents a jar, which is enough to do twenty to
thirty diagrams or posters. This material may be pur­
chased at the Webbcraft Store mentioned previously. A
brush is needed to apply the paint, and costs as little
as fifteen cents or as mueh as a dollar. The better
quality brushes more than compensate for their higher

16
cost by the quality work that is produced, and the speed
with which the work may be done. Once a good brush is
purchased it should last twenty or thirty years with
proper care. It is advisable to buy at least two brushes,
one pointed detail brush for fine work, and a broad
lipped brustihfor painting the large surfaces. The colored
pencil is more economical than ink and is cleaner to work
with because there is -practically no chance for smearing.
It is recommended that colored pencils be used for outline
work and show card paint be used for solid figures.
Lettering should preferably be done with india ink
because it is the easiest to use and offers the greatest
contrast for a large number of colors. An ordinary pen
holder may be purchased for five cents and the broad
nosed speedball points may be purchased for eight to ten
cents apiece. It is advisable to purchase at least two
points, one one-eighth inch point for small work and one
one-quarter inch point for large work. India ink should
not be used for drawing straight lines unless the teacher
is experienced with Inking because it is very easy to
let ink seep under the straight edge and smear when the
ruler is pulled away; thus ruining several hours work.
The colored pencil is much better for the novice when
drawing straight lines. The main consideration with all
diagram or poster construction must be visibility of

17
details and clarity of understanding.
Another useful diagram may he constructed by draw­
ing an acre. Place an average size city lot in one corner
of the acre, and a square rod in the opposite corner. An
average size city lot is fifty feet by one hundred and
fifty feet. Figure One is an example of the completed
diagram.
A diagram showing the three types of angles, acute,
right, and obtuse, as shown in Figure Two is also
helpful.
The size of a square foot may be shown more force­
fully by drawing a square one foot by one foot and sub­
dividing it into one hundred and forty-four square inehes.
The contrast between the size of the square inch and the
square foot may be emphasized by coloring in one square
inch as in Figure Three.
A series of six diagrams showing the square, circle,
trapezoid, parallelogram, triangle, and rectangle is
very important to have for the area unit. Pieces of
poster material approximately elemen inches by fourteen
inches, orange colored on one side and light grey on the
other side, may be purchased for about three cents apiece
for this work from the Webbcraft Stationers and Printers
mentioned previously. The labelled figures should be
drawn on the orange colored side with the type of figure

208; 718
HOW LARGE IS AN ACRE?
208.7 FT,
1 SQ. ROD
16& FT. x 16| FT.
O N E
A C R E5 Q . FT,
Figure 1.
Comparison of an Acre, a City Lot, and a Square Rod

TYPES OP ANGLES
ACUTE RIGHT OBTUSE
Figure 2.
Three Major Types of Angles

20
1 SQ.HOW LARGE IS A SQ. FT.?
1 FOOT
(r-»O
Figure 3.
Comparison of a Square Foot and a Square Inch

21
at the top, the formula at the bottom, and the figure
itself labelled with the parts of the formula in between.
It is better not to color in the figure on the orange
side because it may impair the labeling. The same
figure should be placed on the back or light grey side
with no labeling or formula, but should be colored in.
The back side, then, serves as a flash card type of aid
for recognition of the figure, and the front side gives
the appropriate formula and the figure labelled with
the parts of the formula. When one particular figure
has been studied, it may be hung up over the bulletin
boards for reference purposes, the front side out.
Figure Four shows the front view of the trapezoid.
Illustrations Five, Six, Seven, Eight and HIne, show the
other five figures studied in area. The figures are
all shown so that the proper formula and labeling will
be used. All too often the teacher confuses the child
by inconsistency in labeling the figures. Here, an
attempt has been made to maintain the same nomenclature
throughout.

TRAPEZOID
b2
M
bl
A * | a (b-^ bg)
Figure 4.
Front View of the Trapezoid Diagram Showing
the Proper Labeling

23
SQUARE
S
A = S 2
Figure 5.
Diagram of the Square with Appropriate Labeling

ALTITUDE24
RECTANGLE
&
ii
b = L
A s a b * L W
Figure 6.
Diagram of the Rectangle with Appropriate Labeling

25
PARALLELOGRAM
w
b
A s a.b
Figure 7*
Diagram of the Parallelogram with Appropriate Labeling

26
TRIANGLE
M
Figure 8.
Diagram of the Triangle with Appropriate Labeling

27
CIRCLE
A * pi R 2
Figure 9.
Diagram of the Circle vith Appropriate Labeling

CHAPTER V
SPECIAL CONSTRUCTIONS FOR THE AREA UNIT
There are many models and constructions to illus­
trate specifics in the area unit. All of the constructions
presented here are simple, inexpensive, and require
practically no special skill with tools.
A first hand experience can he given the children
to illustrate the size of the inch, foot and yard hy cut­
ting a piece of clothes line into one-inch strips, one-
foot strips, and one-yard strips. One set may he made for
each row or just two sets may he made and passed around
the entire room. A slight variation of this may he to
have the lengths cut from wood, hut thin strips of wood
that are a foot or more in length would he too "breakable.
The permanent bulletin board, or preferably the
portable bulletin board, may be used to illustrate all the
shapes of the quadrilaterals studied in area. Take four
thumb tacks and press them into the bulletin board at the
places where the vertices of the figure will occur.
Obtain a small amount of bright blue yarn and bright
red yarn and make a few turns on one tack. Stretch the
yarn .to another tack and complete one side. Repeat this
process until all four sides are constructed. Use the
blue yarn for one pair of opposite sides and the red yarn

29
for the other pair of opposite sides. A third color' may­
be used to illustrate the diagonals if desired*; The
teacher may ask the students to bring a little piece of
linoleum or some other fibrous material, thumb tacks,
and yarn, so that each student may do these constructions
individually.
The six figures studied in area may be cut from a
piece of plywood one-quarter inch thick. A three to five
inch solid square and proportionally scaled figures for
the other, five would cost about sixty to seventy-five
cents depending upon the quality of the plywood. The
same thing may be done with ordinary cardboard boxing
but this is not very durable and is easily broken.
Movable angles may be made from wood by nailing a
piece one-half to one inch in width and about eight inches
long to another piece of wood the same width and any length.
End is nailed to end, and the pivotal point is the vertex
of the angle. A duplicate movable angle should be made
to illustrate the principle of equal angles. The same
result may be accomplished by using cardboard boxing
instead of wood. The cardboard may be joined at the
vertex by a wire brad that is commonly used to fasten
notebooks. The latter method is much cheaper and the
teacher may have every student in the room make himself
a movable angle from cardboard.

30
Variable parallelograms may be made in a similar
manner. Cut four strips of w§od or cardboard, one pair
equal to about eight inches and the other pair equal to
about twelve Inches. Join the pieces at the vertices as
described above. The parallelogram may also be moved to
the position of a rectangle and its characteristics
indicated. Two parallelograms should be made to show
what is meant by equal quadrilaterals.
A movable triangle may be made of wood or cardboard
by cutting three strips of unequal length and joining
only two of the vertcies, leaving the third vertex un­
fastened; thus, the third side can move in any direction,
allowing the teacher to demonstrate the right angle as
a special type of triangle.
A special construction may be made to illustrate
the principle that if two parallelograms have the same
sized base and the same sized altitude, their areas are
equal. Buy a piece of one-quarter to one-half inch ply­
wood about one foot wide by eighteen to twenty-four inches
in length. This should cost about thirty-five cents.
Secure a pair of tongue and groove boards the same length
as the plywood base and one-half to one inch in width.
Hail the grooved board to thetop part of the plywood base
and cut about a ten inch piece of the Jbongued board and

fit it in the groove. Obtain a strip of wood about one-
half inch wide and the same length as the plywood and
nail it to the plywood base underneath the tongued board
to act as a frame or guide. The ten inch piece is then
free to move along the entire length of the plywood.
Secure a ten inch piece of wood and nail it near the
bottom of the plywood for the opposite side of the paral­
lelogram. Cut two eight inch strips of inner tubing the
same width as the wooden sides. Nail one end of the
rubber strip to the edge of the tongued board, and the
other end to the edge of the board fastened at the bottom.
Follow the same procedure with the other piece of inner
tubing, forming the fourth side of the parallelogram.
Paint the four sides black so that they will be more
visible to the eye. The tongued board may then be moved
to many positions with corresponding parallelograms that
all have the same area. If desired, the plywood board
may be marked off in square inches so that the area of
dach new parallelogram may be determined by the student.
The teacher should explain that the non-base sides are
rubber so that the student won’t get the idea that all
movable parallelograms maintain a constant altitude.
The completed construction should look like Figure 10.
The measures of surface and length should be in
tabular form on posterboarding. One of two methods may

32
TONGUE AND GROOVE BOARD
MOVABLE BOARD
RUBBER
SIDES
WOOD STRIP-
Figure 10.
Special Construction Illustrating Equality
of Areas in Parallelograms with. Equal
Bases and Altitudes

33
be used, either both groupings may be put on the same
chart or each may be put on a separate piece. It is
preferable to use two eleven by fourteen inch pieces of
cardboarding, the same size and material as was used for
the figure diagrams. Label one "Measures of Surface",
and the other "Measures of Length". Tables One and Two
show what should be included in the tables.
Two useful devices may" be used to illustrate the
properties of the circle. Cut two nine to twelve inch
circles out of one-eighth to one-quarter inch plywood
stock. The circles may be marked on the plywood by an
ordinary fifteen cent compass and cut out with any flexible
type saw at one's disposal. Cut a diameter and radius
for the circles from wood stripping one-quarter inch
wide by approximately one-eighth inch thick. Nail the
diameter to one circle so placed that the center of the
diameter coincides with the center of the circle, and
allow enough clearance so that the diameter may rotate.
Paint or color the diameter a bright color so that it may
be easily seen. Buy a cheap ten or fifteen cent tape
measure and tack one end securely to the circumference of
the circle that has the diameter on it. Allow enough
tape to measure the distance around the circumference and
cut the remainder off and discard it. Place a thumb tack
on the free end of the tape and stick it in the circumference

34
TABLE I
MEASURES OP SURFACE
14# SQ. IN. = 1 SQ. FT.
9 SQ. FT. 5 1 SQ. YD.
30i SQ. YD. = 1 SQ. RD.
160 SQ. RD. = 1 ACRE
640 SQ. A. = 1 SQ. MI.

TABLE II
MEASURES OF LENGTH
12 IN. = 1 FT.
5 FT. = 1 YD.
YD. = 1 RD.
520 RD. – 1 MI.
1760 YD. = 1 MI.
5280 FT. – 1 MI.

so that the free end will stay in place when not in use.
Draw a diameter on the second circle one-quarter inch wide.
Color half the diameter a bright blue and the other half
a bright red. Paint or color the wooden radius the same
bright red color as was used for the radius in the second
circle. Kail one end of the radius to the center of the
circle and allow enough clearance for movement. The
completed instructions should look like Figures Eleven
and Twelve. The circle with the diameter and tape
illustrates the ratio of pi ahd allows the student to
measure both the circumference and diameter, enabling
him to compute the value of pi. This figure also empha­
sizes that there is more than one diameter in any one
circle. The circle with the movable radius illustrates
the principle that the radius is one-half the diameter
and that there Is more than one radius in any given
circle. If the circles are made the same size, the teacher
may indicate what is meant by equal circles. Some teachers
may prefer to combine both circles in one by putting
the tape on the circle with the movable radius, but this
is less favorable because too much is crowded into one
visual aid and the result may be confusion on the part
of the student rather than enlightenment.
Where school workshop facilities are available,
the teacher may gain cooperation and assistance inrmaking

37
MEASURING TAPE
ON CIRCUMFERENCE
DIAMETER
Figure 11.
Special Construction Illustrating the Constant Pi

38
•PAINT BLUE— \ /-PAINT RED—
MOVABLE RADIUS
PAINTED RED
F igure 12.
Special Construction Illustrating the Ratio
of the Radius to the Diameter in a Circle

these, constructions from the shop instructor and his
students. This will not only benefit the classroom
teacher, but will provide project material for the shop
classes as well.

CHAPTER VI
APPLICATION OF AUDIO-VISUAL AIDS TO THE
TEACHING OF AREA
Beginning with this chapter of the paper, an
attempt is made to make mathematics more meaningful and
functional to the student, particularly in the field of
area which is formally taught in the eighth grade of
Junior High School and occupies about two or three weeks
time in the curriculum.
The method of instruction is to break down the
mathematical processes involved as far as possible; thus,
the work has a directional sequence, is more easily
comprehended by the student, and each step leads pro­
gressively into the next. This break-down is extremely
important because it brings mathematics to the student's
level of understanding rather than teach abstractions
that are incomprehensible and unmeaningful to the student.
It is in the break-down that audio-visual aids serve
teaching purposes most adequately. The audio-visual aids
are used to illustrate and clarify specific details and
problems that are confronted in the break-down rather
than to use them in a haphazard random fashion, or merely
as entertainment; therefore, the aids are integrated with
the work and are subsequent to the over-all objectives of

41
the course.
The basic plan of teaching is three-fold–to
motivate, to teach the theory, and finally, to give func­
tional applications as much as possible. In an effort to
make the work more meaningful, angles, measurement, sides,
parallel lines, etc., are taken up as they are needed in
the presentation rather than teach them as isolated
elements unrelated to any work in particular. It is
hoped that the student will secure better integration of
the material and see the interrelationships of all the
factors involved by the use of this technique.
A disproportionate amount of time is spent with
the first figures studied because the entire groundwork
for square measure is being laid. The feeling of square
measure, the differentiation of units, new vocabulary,
and the manipulation of the formulas are but a few of
the new concepts that must be learned by the student.
The majority of this work has to be undertaken in the
first figures studied so that the material taken later
will be properly understood.
There are six chapters consisting of the following:
square, rectangle, parallelogram, triangle, trapezoid,
and finally, the circle. The formula for finding the
area of each figure is not stated outright at the begin­
ning of the study of that figure, but is a consequence

of work that leads progressively to the formula; for
example, the rectangle is constructed three or four times
on graph paper, then the students and the teacher talk
about the properties of the rectangle before the formula
is introduced, rather than give the formula first and
follow it as a guide for teaching. The formulas are not
proved theoretically, but are proved emperically. It Is
doubtful if much is to be gained by engaging in lengthy
theoretical proofs of formulas; generally, the student
is more confused than if he has a statement of the formula
and an opportunity to test its validity by experimentation.
This viewpoint is substantiated when one remembers that a
large number of facts and phenomena in this world are not
directly understood by the lay public, but nevertheless,
the facts are used. The electric light is a good example.
Its basic properties are not understood by the layman,
but it can be demonstrated to work. The student at the
junior high school level must accept the teacher's word
for some things which science and mathematics have proved
to be true, the formulas for area being no exception.
The appropriate addio-visual aids to be used are
indicated in parentheses in the body of the paper. – A
conversational tone is employed such as the author would
use if talking to the teacher and demonstrating how the
unit would be taught if the class were before them.

43
AIMS OF THE UNIT:
1. To correlate area to everyday life with
functional applications.
2. To form a basis for future work in mathematics.
3. To instill appreciation for the basic
principles of mathematics underlying the
study of area.
4. To demonstrate, if possible, the use of area
in fields other than mathematics.
5. To follow the prescribed course of study as
outlined by the district, using it as a
skeleton outline for the unit involved.
6. To maintain skills and fundamental processes.
INTRODUCTION OF THE AREA UNIT TO THE CLASS:
The overall picture or Gestalt of area should be
made before the formal presentation of the area unit.
Ask the class how area is used and illustrate by finding
the size of a football field, size of a farm, real estate
lots, how much paint is heeded for a given wall area,
and in sewing, the yardage is measured in area units, etc.
A bulletin board will help motivate the student if they
are encouraged to cut out various examples of form in
area found in magazine illustrations such as pictures
of room interiors that have circular mirrors, square

44
designs in the rug, rectangular table tops, etc. The
teacher should guide and direct questions that will lead
to such topics as how area is determined, and what units
of measure are used in area. The film "AREAS'* produced
by Knowledge Builders Classroom Films, which is twelve
minutes in length may be used to answer some of the
questions raised by the discussion and also raises new
ones that will be answered by the presentation of the unit.
This film was chosen because it is felt that a different
approach may stimulate learning, actually the film strips
and motion pictures that are available for this unit are
extremely limited. The film material that is available
is not too good, and in many cases more could be gained
by a teacherrdemonstration or discussion with the aid of
colored chalk than by viewing the particular film or
filmstrip. The reason for the inclusion of the film is
that the more senses used in the learning situation, and
the more varied the approach, the more permanent the
learning.

CHAPTER VII
THE SQUARE
The square is chosen as the Introductory figure
because It forms the basis of denominate numbers used In
area.. Ask the students for a general description of
the square (either draw a square on the blackboard or
hold up the back side of the diagram of the square).
Define the square as a four sided figure whose sides are
equal in length and whose sides are perpendicular to
each other. The angle concept should be introduced and
defined as the space between two lines that meet at a
point called the angle vertex (use angle illustrator here).
A right angle is a special kind of angle in which the
sides are perpendicular to each other (change movable
angle to .the right angle position). The square, then,
has four right angles formed by the perpendicular sides.
Ask the students to look around the room and
indicate any squares that they see .(the teacher should
hand up or place a few square shaped objects in the room
before class). Pass out graph paper, which itself is
composed of a large number of squares ș , and ask the students
to draw first a one inch square and then a two inch
square. Introduce the concept of area as the space on
the inside of• the figure, or the cover of the figure

46
(illustrate by drawing several squares on the blackboard
and coloring in with red chalk; have the students color
in the squares they have drawn on the graph paper). Ask
the students to draw another one inch and two inch square
on the reverse side of the graph paper. Indicate that
there are a large number of little graph squares that lie
within the figure drawn. Have them count the number of
graph squares contained in the one inch figure and put the
result on the board. Next, have them count the total
number of graph squares whose sides lie on one side of the
one inch square, and the number of graph squares whose
sides lie on the side perpendicular to the first. Put
the results on the board. Ask if anyone can see a
shorter method for finding area than counting each indi­
vidual graph square that is contained in the larger
square. Show that the same area is obtained if the number
of graph squares whose sides lie on one side is multi­
plied by the number of graph squares whose sides lie on
the perpendicular side. So that the student doesnit
think that the sides of the larger square are measured
in square units, emphasize that only the sides of the
small graph squares touch the side of the larger square,
and since the side of the graph square is in linear units,
the side of the larger sqaure is composed of the totality
of these small linear units; the side of the larger square

must therefore bemmeasured in linear units. Have the
students count the number of graph squares whose sides
lie on the sides of the two inch square to find the area
by the shorter process. Explain that these graph squares
are too small to measure most of the areas that one uses
in living,' so there are units adopted that are larger in
size that don't take as much time to use. Some of these
units are the square inch, the square foot, and the
square yard. These units are almost the same as the
graph squares in that they are used to find area, the
only difference is that a square inch has sides one inch
in length, a square foot has sides one foot in length,
and a square yard has sides that are one yard in length.
Show that the relationship between square inches, square
feet and square yards, is similar to the relationship
between linear inches, feet and yards (use the tables
of linear and square measure). Obtain several rulers,
yardsticks, and tape measures, so that the class may
measure linear distances such as the width of table tops
and the height of the door, etc., (pass out the pieces of
rope cut in one inch, one foot, and one yard lengths to
show the difference in size of the length units). Compare
a square inch and a square foot by use of the diagram
showing one hundred and forty-four square inches in a
square foot. Ask the class if one would use square feet

to measure a book, or square inches to measure a black­
board, stressing the places where each unit has its best
use. How does one find the area of any square? Simply
multiply the length of one side by the length of another
side which is perpendicular to it. This is a long way of
telling how to find the area of a square, so peoplb have
devised a short method of saying the same thing; this
shortcut is called a formula. The formula for the area
of the square is – A = S 2 where A stands for area, S stands
for side, the little two above the S means to multiply
S by itself, and the two horizontal lines mean that
everything to the left of it equals everything to the
right of it; thus, it is called an equal sign (hold up
the front side of the square diagram showing the formula
and labelled parts). Now the area of any square may be
found by use of the formula shown; the figure doesn’t have
to be drawn before one's eyes; all that the student has
to know is the length of each side to find the area.
Emphasize that area is always in square units; if the
sides are measured in feet, the area will be in square
feet; if the sides are measured in square yards, the area
will be in square yards, etc. The teacher should demon­
strate this more abstract phase by working several
examples at the blackboard and then giving the students
several problems to solve at their seats to see if the

49
idea is generally understood. Review the vocabulary used
in the work, such as square, area, square inch, square
feet, square yard, perpendicular, right angle, formula,
etc.

CHAPTER VIII
THE RECTANGLE
The rectangle should he introduced in a similar
manner as the square. Ask the class to try to explain
the differences between a square and a rectangle (hold
up the hack side of the rectangle diagram and the hack
side of the square diagram). It should he clear that the
sides of the rectangle are unequal in length, hut other­
wise the rectangle is the same as the square (show the
movable parallelogram in the position of a rectangle).
Define the rectangle as a four sided figure whose sides
are perpendicular and whose opposite sides are equal in
length. All the angles formed by the sides are right
angles, the same as in the square. Have the students
point out examples of rectangles that are in the room,
such as the blackboard,ceiling, floor, walls, textbooks,
table tops, etc.
Pass out graph paper and have the class draw a four
inch by one inch rectangle. Use the method of counting
the -total number of little graph squares to find the area.
Have them count the number of graph squares whose sides
touch each side of the rectangle and see if any of the
students can arrive at a shorter way of finding the area
of the rectangle. Next, ask the class to draw a rectangle

three Inches by one-eighth Inch and have them find the
area. This last problem will Illustrate that it is
possible to have parts or fractions of an area unit.
Make the transition to the standard units, emphasizing
that fractions of a square inch, foot, or yard, may also
occur the same as it did for the problem on the graph
paper.
The side of the rectangle upon whieh it rests is
called the base; some people call It the length, but it
is better.to call . I t the base because all the figures
studied will have bases that will be defined in the same
way. Similarly, the distance between the base and the
opposite side is called the width, height or altitude*
but it is best to call it the altitude because it will
always be defined as the perpendicular distance between
the base and the opposite side in a four sided figure.
It just happens that In the case of the rectangle, the
altitude Is the same as the side (less confusion will
be caused later if the area is always found by multiplying
the altitude by the base as in other area figures, rather
than the special case for the rectangle of multiplying
the length by the width). Generalize from the problems
and have the students assist in stating the rectangle
area formula A = ab, where A stands for area, a stands
for altitude, and b stands for base (show front side of

52
rectangle diagram with the labelled parts). Problem to
be worked by teacher at the board: How much wood is
there In a rectangular door three feet by five feet?
Have the students work several similar problems at their
seats where direct substitution is employed.
Define a denominate number as any number which has
a unit attached to it, as five inches, two feet, one
gallon, one pound, etc. Emphasize that it is not possible
to find the area of a rectangle or any other figure unless
the denominate numbers are the same. If the altitude is
in feet, the base must be in feet and not inches; if the
base is in yards, the altitude must be in yards. When a
problem is given that has one dimension in feet and one
dimension in inches, for example, change both dimensions
either to feet or both to inches before using the area
formula. The teacher should work several mixed denominate
number problems at the board and assign several to the
students to be worked at their seats to be aure that the
idea is understood. Several practical problems are
available such as finding the area of a football field,
the area of a rectangular lawn, area of the blackboard,
area of rectangular rugs, tabletops, etc. Review the new
vocabulary used–opposite sides, rectangle, altitude,
base, and denominate number.

CHAPTER IX
THE PARALLELOGRAM
The class should be Introduced to the parallelogram
by holding up the back side of the parallelogram diagram
and asking how It differs from the others studied. The
one pair of opposite sides Is slanted, but how would the
parallelogram be defined? Define the parallelogram as a
four sided figure in which the pairs of opposite sides
are parallel' and equal (use the movable parallelograms
here). What is parallel? If two lines are parallel, they
never cross each other but always are the same distance
apart. Are the angles formed by the sides the same type
as the angles formed by the sides of a rectangle? Are
the angles less or greater than a right angle? After
asking these questions, the teacher should explain that
an acute angle is any angle less than a right angle, and
an obtuse angle is any angle greater than a right angle
(use angle illustrators here and also put up the chart
showing the types of angles). The teacher should empha­
size at this point what is meant by equal angles. If
two angles are equal, one angle may be placed on top the
other in such a manner that all the parts will coincide–
the vertex of one will fall on the vertex of the other,
and the sides will lie on top each other (use the pair of

5^
angle Illustrators here). There are two equal obtuse
angles and two equal acute angles formed by the sides of
the parallelogram.
The teacher can give an emperical proof for the
parallelogram formula by having the students draw a
parallelogram on graph paper and cutting out the figure
with scissors. Have the students draw a perpendicular
line from one corner of the base to the other side and
cut along this line. The piece that is cut off will
form a rectangle when placed alongside the other slanted
side. . Since the class already knows the formula for the
area of a rectangle as A = a b, they now have the emperical
proof that the area formula for the rectangle and paral­
lelogram is the same. In the case of the rectangle, one
of the sides was the altitude, but here, the side is not
the altitude. In a parallelogram it must be emphasized
that the sides are not perpendicular, and an altitude
is the perpendicular distance, not the slant distance
(refer to the front part of the parallelogram diagram).
Several direct substitution problems should be worked
at the board, and if the class is not too large, several
members of the group may go to the board to work the
problems. Use some problems, using the acre and the square
rod, explaining that these are merely different measures
of area (show the table of surface measure; hang up the

comparison diagram of the acre, city lot, and square rod).
OPTIONAL:
Have the students prove that if two parallelograms
have equal bases and altitudes, they have equal areas.
This may be proved by the formula very easily when it is
shown that in the formula A = a b, the altitude is the
same in both cases, and the base is the same. If equals
are multiplied by equals, then the results are equal.
Use the special construction described in Chapter V to
illustrate this principle visually.

CHAPTER X
THE TRIANGLE
The introduction of the triangle follows the same
general technique as described previously. Hold up the
triangle chart and ask the class the distinguishing
features of"this figure. The teacher should try to guide
the class into a formation of some type of definition.
After the class has had a chance to deliberate, define the
triangle as any three sided figure that is closed, and
by closed is meant that three vertices are formed (use
movable triangle to illustrate).
OPTIONAL: The teacher may take up the classifica­
tion of triangles according to angles and sides. Refer to
the chart on angles and draw the triangles at the board
with colored chalk to illustrate. This is included
because some teachers feel that the classification of
triangles is imperative, but actually, the classification
has no bearing on the solution of area problems since the
formula for the area is the same for all triangles
irrespective of type. This would be better if Included .
in some later course.
Pass out graph paper to the class and askthem to
draw a parallelogram with a three inch base and a one
inch altitude. Have the students draw the diagonal of the

57
parallelogram. The class should then cut out the figures
and cut along the diagonal forming two triangles that are
equal, this may be tested by placing one on top of the
other. Ask the class how the area of the parallelogram
is found, and after obtaining the answer A = a b, ask
what the area of one of the triangles would be. Since
there are two equal triangles, and the total area of the
two figures must be equal in area to the parallelogram,
the area of one triangle must be one-half that of the
parallelogram, or A = l/2ab. The letter "b" still stands
for the base, which is the side the figure rests upon,
and the letter "a" is the altitude (use the labelled
side of the triangle diagram for illustration). There is
a slight difference in the definition of the altitude in
the triangle, here, it is the height or perpendicular
distance from the base to the highest point of the
triangle, which is the vertex opposite the base. The
teacher should work several direct substitution problems
at the board before assigning any seat work.
The right triangle is a special triangle In which
one of the-angles is a right angle (use movable triangle
here). The teacher may show the derivation of the right
triangle by taking the rectangle and dividing it diagonally
as mentioned above. The instructor should emphasize that
the area of the triangle is always the same no matter

58
which side of the figure is used as a base. Have the
students use each side as a base and find the correspond­
ing area for each case.

CHAPTER XI
THE TRAPEZOID
The trapezoid is not as important as the other
figures studied, some school courses omit it entirely.
The class should be exposed to the trapezoid even if it
is not possible to spend a great deal of time with it.
Hold up the back side of the trapezoid diagram and ask
how this figure differs from the others studied thus far.
Define the trapezoid as a four sided figure in which two
sides are parallel and two are non-parallel. The parallel
sides are called the bases, and the perpendicular distance
between the two bases is called the altitude.
The derivation of the trapezoid formula can be
proved quite easily, but it is a little beyond the grasp
of the ordinary student. Have the students drop perpen­
diculars from the vertices of the top base to the bottom
base. The base is then divided into three parts, and the
figure is divided into two triangles and one rectangle.
The area of eaGh of the two triangles is one-half of the
altitude multiplied by its portion of the base, and the
area of the rectangle is the altitude of the trapezoid
multiplied by its portion of the base. Combine and
collect terms, bringing the quantity within the parentheses
to a common denominator and factor out l/ 2a from the

6o
parentheses. The formula Is A = l/2a (b^ + b 2) where
b^ is the lower base and b 2 is the upper base, and "A"
is the altitude (show the labelled side of the trapezoid
diagram). Most of the student's difficulty occurs with
the substitution in the formula. The teacher must empha­
size that the bases must be added and their sum multi­
plied by l/2a. Several examples may be worked by the
teacher to clarify this point, then have the class take
turns going to the board. A practical problem is to find
the area of a trapezoidal arch that is found under the
roof of many houses.

CHAPTER XII
THE CIRCLE
The study of the circle should be broken up into
two parts; one finding the circumference, and the other,
finding the area. Ho formal work with circles has been
done in the usual course of study preceding this unit;
hence, the teacher can assume little or no knowledge
of the circle.
Pass out compasses to the class and have them draw
three or four circles on their paper. After this is done,
ask for a definition of the circle (Hold up the back side
of the circle diagram). Define the circle as a closed
figure that is everywhere equidistant from a point called
the center of the circle (illustrate by using the string
to draw a circle, emphasizing that the hands are always
held the same distance apart; therefore, every point on
the circumference is the same distance from the center).
Equidistant means equal or same length. The curved line
is called the circumference; a straight line drawn from
the center to the circumference, is a radius; a straight
line originating on the circumference and passing through
the center of the circle to a point on the circumference
on the other side, is the diameter (show marked chart of
circle). The fact that all radii in the same or equal

circles are equal, that there are an infinite number of
radii in any circle, should he mentioned (use the special
construction of the circle with the movable radius).
Similarly, it should be shown that all diameters in the
same or equal circles are equal, and that there are an
infinite number in any one circle (use the special con­
struction with the movable diameter). The diameter is
twice the radius, or conversely, the radius is one-half
the diameter (use the special construction with the
radius and the painted blue and red radii).
In any circle, the number of times the diameter
divides into the circumference is a constant number,
and is equal to 3-14; this constant is given the Greek
letter name pi. Explain that a constant is a number that
has the same value at all times. The teacher measures
the circumferences and diameters of several circular
objects in the room and divides "C" by "D" to get pi
(use the special construction of the circle with the
tape on the circumference). Assign homework in which
the students measure the circumference and diameter of
four or five circular objects at.home. Have them divide
the circumference by the diameter in each case to obtain
3.14; this will implant the idea of a constant more
firmly in their minds. If then, it is known that in any
circle the circumference divided by the diameter is always

equal to the same number it is the same as saying that
the circumference is always this constant pi, multiplied
by the diameter, or C “ pi D. This means that if the
diameter of any circle I t s known, the circumference may
be found by multiplying the diameter by the constant,
pi, or 3.14. It should be pointed out that circumference
is similar to perimeter, it is the distance around the
circle. The teacher should work a problem of direct
substitution and have the class do several similar types
using C = pi D. Since D » 2 R, the circumference formula
can also be written as C = 2 pi R. Have the students
solve a few problems using this new formula. Ask the
pupils to find the circumference of all the circular
objects they can find by measuring the radius and sub­
stituting in the formula.
The area formula may be stated directly or Intro­
duced by the following proof. If one visualizes cutting
a circle along a radius and then straightening the circle
out, a rectangle is formed. The base or length of the
rectangle is equal to the circumference, and the width
or altitude is equal to the radius of the circle. One
can find the area of the rectangle by multiplying the
altitude by the base, or circumference times the radius,
A * C R. Since C « 2 pi R, A = 2 pi R H or A = 2 pi B 2
which is the formula for the area of the circle. This

formula simply states that If one knows the radius of any
circle, the area may be found by squaring the radius and
multiplying the answer by pi, or 3*1^. The teacher should
explain a direct substitution problem and assign several
similar problems to the class. Ask the class to find the
area of the same circular objects that they measured for
the circumference problems.

CHAPTER XIII
CONCLUSION
Some teachers may not wish to prove the formulas
as indicated due to a retarded group. If such is the case,
less is to be gained by forcing these proofs on the group'
as it would add rather than detract from the confusion.
The proofs are included for the average or above classes
that the teacher feels are capable of mastering the steps
of reasoning.
The work in the area unit is so designed that it
progresses directly into the unit on volume. The volume
unit is taught quickly and easily if the same nomenclature
is used consistently as in the work on area.
After the conclusion of the unit on volume, a field
trip to some local housing project would be advantageous
if time permits. Here, real concrete life situations
can be integrated with the previous knowledge learned in
the classroom pertaining to area and volume. There are
many visible examples of the work studied, a few such as
the following should be Indicated by the instructor:
1. What is the surveyor trying to measure?
2. What type of figure is the two by four?
3. What is the area of some of the lots within
the building project?

4. How many different types of figures can be
found in the buildings being constructed, and
in those that are completed?
5. How is concrete measured?
6. What type of figure is a telephone pole?
7. What type figure is the water heater that will
be installed in the houses? What do they
measure?
8. Where can perpendicular and parallel lines
be found?
9. How are the windows shaped?
10. What types of angles can be seen?
When the group is back in the classroom, the teacher
should have a follow-up discussion in which the students
should participate freely, the instructor merely acting
as director and guide. The discussion should center
around the practicality and usability of the units studied.
Point out that area and volume are continually used in our
everyday life, and there is a need for them to know
something about these processes if they are going to
adequately live in our modern society.

BIBLIOGRAPHY

BIBLIOGRAPHY
Dale, Edgar, Audio-Visual Methods in Teaching. New York;
The Dryden Press, Inc., 1946. 546 pp.
Haas, Kenneth B., and Harry Q. Packer, Preparation and
Use of Visual Aids. New York: Prentice-Hall, Inc.,
T 9 ? 6 7 " 22TTpp. –
Hoban, Charles F., and Charles F. Hoban, Jr., and Samuel
B. Zisman, Visualizing the Curriculum. New York:
The Dryden Press, Inc., 1'94T1 300 pp.
McKown, Harry C., and Alvin B. Roberts, Audio-Visual
Aids to Instruction. New York: McGraw-Hill Book
Company, Inc., 1949. 608 pp.
"Multi-Sensory Aids in the Teaching of Mathematics,"
Eighteenth Yearbook of the National Council of
Teachers of~MathematTcs. New York: Bureau oT~
Publications, Teachers College, Columbia University,
1945. 455 PP-
U niversity of S o u th e rn California Library