The purpose of this chapter is twofold: first, to set fortha procedure for introducing materials to students that in-sures proper use, second, to describe in detail the materialsneeded for each lesson.

Students learn more quickly and with greater under-standing when they are able to "see" the concept they areto learn. Use of materials offers students this opportunity.To be truly practical in a classroom setting, however, thematerials must not add to the classroom management prob-lems of the teacher and they must be inexpensive enoughfor classrooms operating on the lowest of budgets. Thematerials described in this chapter have been selected withthese criteria in mind.

Three steps may be taken to insure the ready acceptanceof materials as aids to learning by students in a classroom.

The first step is to use the material as the basis for les-sons involving all students. Many of the students whowould benefit most from using concrete aids will refuse todo so if the material is in any way perceived as a form ofremediation. If manipulative aids are reserved for only theslower students, they are apt to feel they are concedingtheir own ignorance by using the materials. If all studentsin class use the same material, it cannot be identified asremedial.

The second step is to explain to the students the learningvalue of each new material introduced for the first time.The materials the teacher has selected are designed to makethe students learning easier; therefore, they will be able tolearn more this year than in any previous year. Tiles, forexample, are used to help everyone learn about decimals,fractions, area, multiplication, and division. Geoboardshelp everyone learn about geometry, which most peopledon't study until high school. The beans and cups are usedto teach arithmetic in other bases,and some of the materialsmay be used to teach algebra. Most people, again, don'tlearn these things until high school.

The studehts will be able to learn all these things becausethese materials make learning so much easier. Many teachersdo not use them because they feel their stuqents are notready to accept the responsibility that accompanies the useof a material. The students in this class, however, wiIIdecide for themselves if they are ready to accept this respon-sibility, which will help them learn more than they everhave before.

The third step is to ask each student individually if he orshe is ready to learn and therefore ready to accept respon-sibility for the correct use of the material. Before theteacher distributes a material for the first time, all theprestigious uses to which it will be put are described inglowing terms. The students are then told what the respon-sible use of that material entails. For example, the rubberbands on the geoboard are to be used to learn about multi-

plication, fractions, decimals, geometry, and so on. Studentswho use them for snapping or shooting at others have de-cided they are not yet ready to learn using geoboards andrubber bands.

As the teacher hands the new material to each studentindividually, each student must state whether he or she isready to learn using the material and, therefore, ready toaccept responsibility for the correct use of material. If thestudent is ready to learn, he or she is given the material touse. If a student mishandles it, that student has decidedthat he or she is not yet ready to learn, and the material iswithdrawn from that student alone. When the student de-cides that he or she is ready to learn, the material is reissued.

Students are anxious to use materials well if they seethem as prestigious and evidence of a desire to learn.

MATERIALS

Overhead Projector Access to an overhead projector isnot a requirement for any of the lessons in this book. Ablackboard supplemented by a giant pocket chart (describedbelow) serves equally well.

If an overhead projector is available, it is much moreconvenient than a blackboard_ An overhead projectorscreen can be set up so it blocks no one's view, but becausehigher angles of overhead viewing distort the top of thepicture, the screen should be hung at an angle that eliminatesthe distortion. An example of such a set-up can be seen inthis figure. Overhead projector screens may be made bypainting a large piece of cardboard white or covering itwith an old sheet.

Wire to wall

Light path

Counter Table

The overhead has additional advantages over a black-board. First the teacher can face the students while writing,and judge whether the lesson is clear to them. Second, infor-mation may be written on a removable transparency to

save for later use. Third, anytime the students are givena ditto with which to work, an exact copy can be madeinto a transparency and used as a visual aid on the over-head. Fourth, if lines or squares are needed on the over-head screen, a lined or graph paper transparency producesth is effect.

Giant Pocket Chart If the teacher does not have access toan overhead projector, the inadequacies of a blackboardmay be counterbalanced by supplementing it with a giantpocket chart. Instructions for making the pocket chart areincluded in this figure.

IAcetate

ICardboard

IPocket chart

I Materials:Cardboard 61 cm x 80 cmacetate 61 cm x 85 cmmasking tape 8 m

Once the pocket chart has been made, the next task is tomake replicas of all the necessary charts and matrices in·cluded in the black lines for which a transparency wouldhave been made for use on the overhead.

Each chart or matrix is used by inserting it behind theacetate on the giant pocket chart. The pocket chart permitsthe tagboard replicas to be placed behind the acetate sheet.All appropriate markings may then be made on and erasedfrom the acetate sheet without marking on the replica it-self, so it needn't be remade (see drawing above right).

Individual Blackboards Many of the lessons involve thewhole class at once. Use of the individual blackboardspermits everyone the opportunity to respond to questionswithout having to compete with each other for the teacher'sattention.

The blackboards may be made by painting the inside ofnylon stocking box lids with blackboard paint, which can beobtained at most paint stores. The lids make excellentblackboards and the boxes themselves are good containersfor chalk and the towel scraps which may be used as erasers.The stocking box blackboards are durable enough to last anentire school year.

~IIIII

Chart:60 cm x 75 cm

Tagboard chart Tagboard chartin pocket chart

IX IX

Writing on acetate,not on tagboard.

Nylon Stocking BoxBlackboard

Bottom-forstorage of chalkand towel scrapused as eraser.

Top-insidepainted withblackboardpaint.

When a group answer is expected, each student writes ananswer on the blackboard and turns it face down on thedesk. The teacher gives the signal to hold up the boardswhen all have been turned face down. This gives everyonea chance to answer the teacher's questions.

When each student knows he or she will have a chanceto respond to the teacher's questions, it becomes muchmore worthwhile to follow the lesson.

The blackboards are also the teacher's source of a classanswer, the one appearing most frequently as the boardsare held up.

Ceramic Tiles (2.5 cm square) The ceramic tiles are thekind used to tile floors or walls, most often coming insheets of 144. It is often possible to obtain incompletesheets free at construction sites or at a reduced price fromtile wholesalers.

MATERIALS 294

Tile wholesalers (listed in the yellow pages under Tile,ceramic) may allow teachers to purchase at discount oddlots of tiles or tiles from incomplete sheets. It is importantwhen asking for a discount for the teacher to identify him-self or herself as a teacher and explain what use will bemade of the materials. Builders or sellers will sometimesprovide a certain amount of materials free if they know thegoods are for use in teaching.

If possible, between 100 and 144 tiles should be obtainedfor each student in class. These amounts are not mandatory,however, and if a lesser number of tiles is available, thesame activities may be explored by having the studentswork together in teams.

The students may remove the tiles from their backings.The tiles should be stored in stocking boxes, cloth bags, orplastic containers to make them easier to transport andstore.

If there are no funds available for the purchase of tilesand none have been donated, an adequate substitute maybe made by cutting out 2.5 cm squares of cardboard.Shirt cardboard from a laundry is usually thick enough towork well.

The ti les double as their own visual aid on the overheadprojector, since they silhouette nicely on the screen. If nooverhead projector is available, visual aids may be cut frompaper. These square shapes should be large enough for allstudents to see easily. They may be pinned to the bulletinboard or used on a flannel board.

When the teacher is showing students what to do with amaterial, the visual aid supplementing the words shouldresemble the material as closely as possible, to avoid con-fusion. The aid needn't be made of exactly the same mate-rial the students use, but it must have as many of the sameproperties as possible. Squares of paper on a bulletin boardor flannel board are closely akin to tiles on a student'sdesk. Drawings of squares on a blackboard are not. Al-though students who abstract easily will be able to make theconnection between drawings on a blackboard and tiles ontheir desks, those who have difficulty in learning or who donot understand English become lost and confused.

Unifix Cubes Unifix cubes are the only material recom-mended for use for which an adequate substitute cannoteasily be made. Unifix cubes are plastic shapes 1.7 cmhigh which snap together to form columns. They may bepurchased from a number of mathematical supply houses, asample listing of which is found on page 299.

An important supplement to the Unifix cubes are theUnifix grids, which are used to permit individual columns ofcubes to be linked together to form solid walls or blocks.

A good starting number of cubes for a classroom is be-tween 500 and 1,000. The long-range goal, however, shouldbe to acquire as many as 3,000 cubes over a period of years.

The cubes may be stored in a plastic bucket or cardboardbox. In many schools, the cleaning solvents and waxes usedby the custodian come in durable plastic buckets, usuallydiscarded when empty. Most custodians will be happy tosave these buckets for use in the classrooms.

The Unifix cubes may be used as visual aids on the over-head projector as long as their attribute of color is not animportant component of the lesson. If color for the cubesneeds to be represented, then acetate squares five cm byfive em, outlined in black to make their boundaries visible,may be used. Each acetate square should be labeled A, B,C, D, or E. The letters are used to represent colors, be-cause no two students will have the same colors with whichto work. Ten or fifteen squares for each letter should besufficient.

If no overhead projector is available, the teacher mayuse the same paper squares used to represent tiles. Whencolors need to be represented, the squares may be lettered.

Beans, Cups, Bowls, and Tin Cans The lessons that usebeans and cups, and later add bowls and tin cans, may bedone with a variety of materials. Beans are used becausethey are relatively cheap and available in large supply. Any

295 MATERIALS

other small objects readily available would do as well:pebbles, shells, seeds, bottle tops, small washers, cut-upbits of cardboard, paper clips, brass fasteners.

The cups, bowls and tin cans are meant to be containersfor the beans or other small counting objects. When the cupsalone are used, the only criterion for the containers is thatthey all be the same size. Cut-open milk cartons, baby foodjars, empty match boxes, shaving lather lids or any othercontainers available in quantity would serve as well.

As bowls and tin cans are introduced, the new criterionis that each new size container must be larger than thecontainer in use before it. If small milk cartons were usedin place of cups, cut-off half gallon milk cartons might beused instead of bowls, and gallon milk cartons instead ofcans. The containers used should be a function of whatmaterials are readily available in the school or in the stu-dents' homes.

Any counter will also be a good visual aid for itself onthe overhead. If no overhead is available, the teacher shouldcut paper shapes similar in outline to the counters. Theseshapes may then be used either on the bulletin board orflannel board to supplement the teacher's oral instructions.

The cups, bowls, and tin cans may be simulated on theoverhead by drawing circles on acetate squares. If cups,bowls, and tin cans have been replaced by other containers,the drawings should reflect the shapes and relative sizes ofthe containers actually in use.

If no overhead projector is available, circular cutouts usedto represent the containers should be made for the bulletinboard or flannel board.

Cross-Line Multiplication Flash Cards The flash cardsmay be made on any available paper. Twenty-two cm by26 cm tagboard works well. Each card is made by drawingan appropriate number of lines down and across the card.Circular self-adhesive gummed labels may be placed on theintersection to facilitate counting.

The number of cards to be made depends on the multi-plication facts the teacher wishes the students to study. (Fora description of the use of the flash cards, see Lesson 5-14.)

Spelling Notebooks Students who have difficulty read-ing should not be required to handle this difficulty in theirmath lessons. None of the lessons in this book requirestudents to read; however, some do require them to write.The use of the spelling notebook permits students to writewho cannot yet read.

Spelling notebooks are made by stapling 26 lined pagesinside a construction paper cover. The students then writeone letter of the alphabet on each page.

Anytime a student needs to know how to spell any word,he or she brings the notebook to the teacher opened to thepage on which the student thinks the word belongs. Theteacher then writes the requested word on the page headedby the same beginning letter. Since the teacher will writeany and every word a student needs in the spelling note-book, there is nothing a student cannot write.

At the start of the year, the teacher may have to writemany words for some students, but this is not long-lasting.The number of words the students need diminishes as theyacquire a basic writing vocabulary in their spelling note-books. Some may not be able to read what they have writ-ten, but reading isn't a requirement in math.

No words should be taken from the spelling notebookand used as the basis for spelling tests-this would dis-courage students from asking for more words.

Dice The dice may be anything from cubes cut fromsheets of foam rubber and numbered with marking pens tothe plastic dice sold by mathematics supply houses. Themost versatile dice that can be purchased are those withblank faces whose numbers may be added by attachingself-adhesive labels on each side. Most of the activities inthis book that require the use of dice recommend each diebe numbered from zero to five.

Teacher-made dice may either have changeable numbersor be produced in such quantity that dice whose numbersdo not go as high as five can be made as a separate set.

In schools where the use of dice is discouraged, spinnersmay be used instead. A sample spinner is in this figure.

Materials for Spinner

Cardboard or tagboardsquare

Bent paper clip

, '>Cardboard or tagboard

arrow(draw circle on squareand add numbers)

Polk bent end of paper clipthrough center of square

and arrow

Spinner Assembled

MATERIALS 296

If dice are to be used, the teacher can make a set ofoverhead "dice" by cutting squares of tagboard five by fivecm and punching holes in them. These squares cast dice-likeshadows when placed on the overhead. The zero die is apaper square without holes.

If no overhead projector is available, the same dice cardsmade for the overhead may be placed inside a small bag orbox and drawn out one at a time to simulate dice rolls.

If spinners are used, an overhead projector spinner maybe made from a piece of acetate, a paper clip and a card-board arrow in the same manner as was demonstrated abovefor constructing student spinners. If no overhead projectoris available, a demonstration spinner may be attached to asuction cup which is, in turn, attached to the wall.

Chips and Chip Trading Boards Good materials fromwhich to cut chips are construction paper or drawing paper;a good size is four cm by four cm. Chips made from coloredconstruction paper are ready to use once they are cut.Chips made from drawing paper must be colored in by thestudents first, but permit the students to add new chips totheir sets in any color they need simply by coloring inadditional blank chips.

Chips should be available in sufficient quantity to permitstudents to find an answer to any problem on which theymay be working, even in base ten. A sufficient number ofchips for each student is 20 or 30 of each color, plus asmany as 100 chips of the color used to represent the unitscolumn on their boards. The teacher should also haveseveral hundred chips in reserve in case they are needed byany student or group working on a particularly largeproblem.piece of paper into the necessary number of columns andmarking the top of each column with the appropriate color.

The teacher makes chips for the overhead projector byusing marking pens of the appropriate colors to shade inacetate squares. The trading board can be drawn on theoverhead by lining the necessary number of columns andshading the top of each with the appropriate color. If nooverhead projector is available, strips of colored paper maybe attached to the bulletin board or a fiannel board to formthe necessary number of columns. The paper squares canthen be cut from colored construction paper and placed inthe columns as needed.

Geoboards Geoboards are readily available commer-cially, and they can also be made easily. Given the necessarytools, intermediate-grade students can make their own.

The actual size of the geoboard is not as important asits relative dimension. The board's 25 nails should beplaced in 5 rows of 5, forming 16 small squares. Thedistance from the edge of the geoboard to the first row orcolumn of nails should be one-half the distance betweentwo adjacent nails.

5

o 0

o 0

o o

o oo o o

o o ooo

Popular sizes of geoboards range from between 12 and13 cm square to 25 cm on a side.

Each geoboard should be equipped with about fourdifferent-colored bands, so students can distinguish thevarious parts of the shapes they make. A good size rubberband is one that forms a square around four nails withoutfitting loosely or having to be stretched.

Too ,....0 0Too "gh<

oGood fit

A simulated geoboard may be made by outlining aboard on a piece of acetate and representing the head ofeach nail by an X. If no overhead projector is available, alarger outline may be drawn and used in the giant pocketchart. In either situation, the. rubber bands may be simu-lated by drawing on the acetate with marking pens.

Objects To Be Sorted Although the first objects sortedin the chapter on sorting and classifying are buttons, anyother material that silhouettes a variety of attributes whenplaced on the overhead would do as well: nuts and bolts,cut-up pieces of paper, puzzle pieces, et cetera.

If no overhead projector is available, the teacher shouldmake enlarged cutouts for the bulletin board or flannelboard of whatever material the students will be using first·for sorting. If buttons are to be used, then button cutoutsshould be made.

The sorting objects may include any items to which theteacher or the students have ready access. The students

297 MATERIALS

may sort such things as Christmas cards, wrapping paper,ribbon and string, assorted jars and cans, food boxes, mar-bles, pot holders, pencils, paint samples, linoleum squares,pattern samples, puzzle pieces, corks, stamps, spools, metalor wood scraps, or anythirig else that may be found byrummaging through basements, attics, garages, old toychests, or kitchen cupboards.

Name Boxes The name boxes the students use in theirearly work on graphing are cut-down half-gallon milk car-tons with acetate pockets taped on what was formerlythe bottom. The steps in constructing a name box may beseen in the figure below. Other boxes will do as well, butthey must all be of the same size and stackable.

Roll ofmystic tape

Toplessbox

"Name" box with"name" card in it

"Name" card

The acetate pockets are used to house the students'names written on cards. These name cards should be avail-able in sufficient quantity to permit students to replace themeach time they are used. If acetate is not available, smallpockets may be made. by taping a strip of tagboard to thebottom edge of the surface of the box.

Large Cardboard Box (screen for graphs) Any large boxor piece of cardboard may be made into a shield for graphs,

or the teacher may string a rope between two chairs anddrape a sheet over it.

Metric Measuring Devices If funds are not available forthe purchase of rulers and tape measures, they may bemade from cut-up strips of tagboard and adding-machinepaper. Weights may be made from washers and pieces ofclay. Metric measuring devices must be useful in obtainingmeasurements in standard metric units. Handmade meas-uring tools calibrated against commercial implements are ascapable of producing accurate measurements as are thoseimplements.

Cardboard "Coins" Probability coins may be made bycoloring or spray painting one side of a piece of tagboard orcardboard, producing a marked and a plain side. The card-board is cut into squares approximately two cm on a side.

Students could also make coins by marking their ownpieces of cardboard, but using the already uniformly markedcoins makes it more likely no student will see by looking atthe material alone there are more ways for some events tooccur than others.

Number Machine Instructions for making a number ma-chine from a half-gallon milk carton are included in thefigure below. The machine turns cards upside down, sowhen a card is put in the machine top side up it comes outbottom side up.

c7 cm---->i Taped-----.-- -,/1 I

I"-I

T "-2cm

,,\I

E I,2.5cm

1(.)I

M ,I... ",

2.5cm 11./....

Sidewith tagboard

"slide" inserted

Frontwithout tagboard"slide" inserted

Tagboardslide

The teacher-made cards are constructed out of five cmby eight cm pieces of tagboard. Cards are marked on oneside and left plain on the other so the teacher can tell the

MATERIALS 298