chapter11 manipulatives

8/8/2019 chapter11 manipulatives

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Page 11.1 1990 Center for Innovation in Education, Saratoga, California

Mathematics Their Way Summary Newsletter

CHAPTER 11:

PLACE VALUE

A firm u nd erstand ing of place value is a pr erequisite for all work

in arithmetic. Stud ents who do n ot und erstand the concept of

place value can not p rogress through the four basic operations

without difficulty. They learn the operations by memorizing an

increasing n um ber of seemingly u nrelated facts and procedures.

Even the most capable students become confused w hen the load

becomes too great, usually at the point w hen they mu st learn

mu ltiplication and division. Mathem atics becomes increasingly

mysterious for these students, wh o have little hope of und erstand-

ing the conten t of second ary-level mathem atics courses. It is

essential that p lace value be given ma jor emp hasis in the primar y

grades and that students have frequent experiences with m anipu-

lative materials that demonstrate place value.

1985 California Mathematics Framework, p.23

In order to effectively implem ent the place value objectives outlined in

the 1985 California Mathem atics Framew ork (above) in the p rimary

grades, the m athematics curriculum mu st include place value experi-ences that:

t use d evelopm entally app ropriate manipu latives.

t use a var iety of bases to teach the place value system.

t explore place value patterns (linear and m atrix).

t provide regroup ing and non -regroup ing multi-digit addition and

subtraction experiences simultaneously.

INTRODUCTION


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Chapter 11: Place Value

1990 Center for Innovation in Education, Saratoga, CaliforniaPage 11.2

PLACE VALUE MANI PULATIV ES

Place value manipulatives fall within two distinct developmental

levels: concrete and repr esentational. Concrete place value experiences

mu st come first. Represen tational activities are appr opr iate later in the

elementary grades after children h ave engaged in a v ariety ofexperiences with concrete mater ials.

Concrete MaterialsConcrete materials beans and cup s and U nifix cubes allow th e

children to construct grou pings. Experiences with th ese materials

stress one-to-one correspondence between the n um ber and the mate-

rial it represents.

Kathy Richardson addresses the issue of one-to-one correspondence at

the p lace value level in her book,Developing Number Concepts Using

Unifix Cubes (Chapter 5, p. 133):

Coun ting grou ps [of objects] requ ires a different kind of thinking

from counting single objects. Childr ens first counting experiences

require an un derstanding of one-to-one correspondence. They

learn that one nu mber w ord goes with one object. But, wh en

dealing with nu mbers above ten, children are required to count

group s as thou gh th ey w ere individual objects. The question

How m any tens in thirty-four? assum es the child can conceive of

ten objects as one entity. The qu estion H ow many hun dred s in

346? assum es the child can conceive of one hun dr ed ob jects as

one group (all the while remembering that each h und red is also

ten group s of ten).

Representational MaterialsBean sticks and mu lti-base blocks are fixed mater ials. The mater ials

impose a ready-mad e structure on the stud ent. Although children

man ipu late materials e.g., chips, m ulti-base blocks and bean sticks

they are requ ired to think abstractly. A child with insu fficient

experiences using concrete materials may be confused w hen a nu mber

is represented by m ulti-base blocks. They must already un derstand

that one block in base ten represents either ten flats, a hu nd red longs,

or a thousand un its. Rather than u nderstand ing that a num ber like 362

is represented by three flats, six longs, and two u nits, a child m ay only

see eleven separate blocks of varying sizes.

Chip trading is also representational but at a m ore abstract level.

Children mu st be able to und erstand the abstraction that one chip

could represent a group of ten or a h und red chips (chips can also

repr esent grou pings in other bases). The necessity for children to be

surroun ded with an abund ance of concrete experiences before they

move on to representational materials cannot be emp hasized strongly

enough.


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Choose AppropriateManipulatives

Because mathematics is made by human beings and exists only in

their minds, it must be made and remade in the mind of each

person w ho learns it . In this sense mat hemat ics can only be

learned by being created.

Author Unknow n

Teachers often overlook the limitations set by representa tional materi-

als. One Math Their Way teacher told how she mad e bean sticks with

her second g rad e class one year. The class had experienced p lace valueactivities earlier in the school year an d in the first grade, using concrete

materials. Once the sticks and rafts were constru cted, the children u sed

them to m ake add ition and subtraction p roblems. The children loved

the bean sticks. They d idnt app ear to have an y p roblems w ith the

fixed arrangement of the sticks and rafts.

The teacher kept the bean sticks and used them w ith classes after that

year. The new second gr aders didn t seem to relate to the materials in

the same m ann er as the first group. In retrospect, the teacher felt the

reason the bean sticks were more su ccessful the first year was because

the children participated in the construction of the sticks and rafts.

They und erstood wh at the beans on the sticks and rafts represented.

Even though they had similar classroom experiences to prepare themfor the activity, the children who used the pre-assembled materials the

following years d idnt experience the creation of the sticks and rafts.

The pre-assembled (representational) materials imposed a structure

that subsequent classes of second grade children w ere not ready to

accept.

Be careful not to move p rematu rely to the representationa l level.

Allow children time to construct their ow n u nd erstanding of place

value by engaging in experiences with an assortm ent of concrete place

value materials.


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USE A VARIETY OF BASESPlace value is merely a system of organ izing large numbers. Teachers

tend to think th at the base ten system is the only base we use in the

real world . For this reason, they often question wh y other bases are

taught in the lower grades. Its true that ou r society frequently uses

base ten, but we also use other bases every day without th inking about

it. Every time w e bake, sew, measu re with a r uler, or tell time, we areworking w ith other bases. The comp uter op erates on a binary system

or base tw o. Yet, textbooks tend to teach isolated p lace value skills

(e.g., borrow ing and carrying) in base ten only. Children w ill develop

a better und erstanding of place value if they are allowed opp ortunities

to compare and contrast relationships (using real materials and real-

life experiences) in other bases.

TEACH REGROUPING AND N ON-REGROUPINGADDITION AND SUBTRACTION TOGETHER

The 1985 California Mathematics Framework, p. 22, emph asizes the

importance of teaching m ulti-digit add ition an d subtraction in a w aythat stud ents und erstand the p rocess, rather than viewing each algo-

rithm as a sepa rate, isolated function:

The p ractice of a skill in isolation is seldom effective in developing

the u nd erstanding required to make the skill useful. Instruction in

computational algorithms should emph asize und erstanding the

procedures that are being u sed.

For examp le, it is now com mon to teach tw o-place add ition and

subtraction w ithout regrouping before introducing regrouping.

This app roach leads stud ents to focus on separate procedures and

hinders their un derstanding of the basic operations. Most students

view a problem such as as two one-place problems pu shed

together ( and ) and do not think of add ing 40 and 20.

Because stud ents get a great d eal of practice in this kind of prob-

lem, they assume that can be calculated in the same way and

see nothing w rong w ith the answ er of 612. After learning a new

procedur e in which 1 is carried or p ut on top of the tens colum n

every time, students are confused w hen p resented w ith a mixture

of problems that requ ire regroup ing and problems that do not. The

child w ho asks, Do I regroup on this problem? has almost no

und erstanding of the concept of two-digit ad dition or subtraction.

This d ilemma can be avoided by teaching mu lti-digit ad dition and

subtraction w ith and w ithout regrouping simultaneously, using

manipulative place value materials, and relating the process torealistic situations. When manipu lative materials are used, it is

easy to demonstrate when regroup ing is needed; and th ere is no

need to teach regrou ping as a separate algorithm.

4+2

3+5

43+25

43+29


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EMPHASIZE TH E IMPORTANCE OF PATTERNS

Children should h ave opp ortunities to explore both linear and matrix

place value p atterns in a variety of bases. An und erstanding of pat-

terns encourages children to see relationships betw een the p arts and

the w hole. An und erstanding of the process develops from experi-

encing numerou s patterns in different bases. Bob Baratta-Lorton wrotein Mathematics ...A W ay of Thinking, p. 81:

When stud ents search for patterns in grou pings of three, four, and

five, and they then see these same patterns rep eat for group ings of

ten, they achieve a far greater und erstanding of borrowing and

carrying than is possible from stu dy ing base ten in isolation.


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PLACE VALUE SCHEDULEChoose th e activities that best fit you r classs need s. In general, the

place value focus for each grad e level includ es the following:

KindergartenKindergar tners do not formally explore place value concepts. They are

exposed to the p lace value board and the notion of organizing large

nu mbers of objects into group s of ten throu gh the following types of

activities:

t Estimating and counting g roups of objects in containers (see NL,

Measurem ent pp. 7.12 - 7.15). The place value boar d is used w hen

the objects are counted an d group ed into ten's.

t Opening activities, NL Chap ter 4 (e.g., Straw Cou nt, Tally Cou nt,

Nu mber Line).

First GradePlace value concepts are formally introduced and explored at the

concept and connecting levels late in the school year.

t Estimating and counting g roups of objects in containers (see NL,

pp . 7.12 - 7.15). The place value board is used when the objects are

coun ted and grou ped into ten's.

t Opening activities, (e.g., Straw Cou nt, Tally Cou nt, Num ber Line).

t Place value activities using other bases (concept an d connecting

level).

t Base ten station activities (concept level record ing is not neces-

sary).

Second GradePlace value is the main focus of second gr ade math ematics. Second

grad ers explore all levels of place value th roughou t the school year.

t Estimating and counting g roups of objects in containers. The place

value board is used w hen the objects are counted and group ed into

ten's.

t Opening Activities, (e.g., Straw Coun t, Tally Coun t, Nu mber Line).

t All place value activities using other bases (concept an d conn ecting

and symbolic level).

t All base ten activities (station and indep end ent).


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EXPLORING PLACE VALUE INA VARIETY

OF GROUPINGSThe initial place value activities in this section pr ovide children a

framework for und erstanding place value while adding and subtract-

ing in different group ings. Children acquire a better und erstanding of

the process of regrouping (borrowing and carrying) when they have

experiences with gr oup ings of four , five, and six as well as with

base ten. Independent base ten activities are introduced once the

children d emonstrate a clear u nd erstanding of the p lace value p rocess.

The urge to focus on base ten too qu ickly should be resisted.

Each grou ping u sed (except base ten) is given a nonsense nam e. The

childr en are asked for suggestions. The nonsense name for each

grou ping rem ains the same for the remaind er of the year. A class chart

of the nonsense names is displayed. Each grou ping nam e is added to

the chart after it is introduced throu gh the Cou nting Game.

CONCEPT LEVEL: GROUP ACTIV IT IESSearching forPatterns

Counting Game (MTW, pp. 98-99)Practice counting by the group ings using the nonsense word .

Class CountChoose a nonsense nam e for the group ing to be experienced.

(Zurkles w ill be used instead of four in this description). Ask the

class to stand u p and count off by the grouping chosen. Each p erson

wh o says the nonsense word mu st sit dow n (e.g.,. One, two, three,

zur kle ... one, two, thr ee, zurkle...). The gam e continues u ntil one

person remains standing.

When h alf the class remains stand ing, ask the children if they can

pred ict w ho w ill be the last person standing.

Group CountOnce the ru les are established, split the class into small group s. Ask

them to coun t off in their group s. Use the following questions to

challenge the group s:

t Predict w ho w ill be the last person standing.

t Does it make a difference wher e you start counting?

t If you start w ith the same person tw ice, will the last person stand -ing be the same twice? Why? Why not?

t Start with the p erson stand ing to the right (left) of the first starter.Predict w ho w ill be the last person standing.

t Ad d a p erson to the circle. Pred ict wh o will be the last person

standing. Continue add ing a person; is a pattern d eveloping?

t Count in an other group ing. Do you think the outcome w ill be the

same? If not, can you p redict who will be the last person stand ing?


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Introduce Addition and Subtraction on the

Place Value BoardMaterials: Place value board s (MTW, p. 364); man ipu lative materials

(e.g., Unifix cubes or bean s and por tion cups)

Before beginning: Review h ow to use the p lace value board. Thegroup ings (e.g., Zurkles) are placed on th e blue side and the one's are

placed on the w hite side. If the children use Un ifix cubes, they snap

the cubes together into a stick before they place them on th e blue side.

If they use beans, they place the group ing of beans in a cup and move

the cup to the blue side.

Addition: Begin by ad ding one object. Explain to th e childr en that each

time you say p lus one, they add a cube (or bean) to their p lace value

board. Once the m aterial is added to the board , the wh ole class says

the nu mber. The group continues to add to the largest two-digit

number possible, regrouping whenever necessary. For instance, in

zurkles (group ings of four), the largest num ber wou ld be three zu rkles

and three.

Subtraction: Start with the largest two-digit number possible in the

grou ping . Ask the children to su btract one object at a time until they

reach zero regrouping wh enever necessary.

Extension: Add or subtract by two or three objects at a time.

Random Pract ice It is important that children h ave opportu nities to randomly add andsubtract.

Building NumbersMaterials:Beans and cups or Unifix cubes; place value boa rd

Tell the children a n um ber to build on the place value boar d (e.g., 2

zu rkles, three). Then tell them a new nu mber (e.g., 3 zur kles, 2). The

children build the new num ber on the place value board w ith materi-

als. Repeat th e process several times.

Observation: Observe how the children find their answers. Do some

children clear their boards every time they bu ild a new num ber? Do

others add on or subtract the app ropriate objects? Ask the children to

explain how they mad e the new num ber. Ask the children if they think

the new n um ber is larger or smaller than the p revious number. Ask

them to explain their answ er.

Add or SubtractTell the children a nu mber to bu ild on their place value board. Now

ask them to ad d or subtract a certain nu mber of beans.

Observation: Watch to see if the children can regr oup wh en it is

necessary.

Note: Mathematics Their Way, (Ch.12) says to read am oun t of mater i-als on the p lace value board withand (two zu rkles and three). Thisis math ematically in-correct. And signifies a d ecimal p oint. The

correct way to read th e amoun t istwo zurkles, three.


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CONNECTING LEVEL: GROUP ACTIV IT IESThe goal of following activities is to acquaint the children with the

written symbol that represents the num ber on the p lace value board.

The class repeats the addition and subtraction activities introduced at

the concept level. This time the teacher record s the nu merical patterns

either on a long strip or in a matrix frame, as the children add and

subtract the materials on the p lace value board .

Searching forPatterns

Vertical PatternsMaterials: Place value board s (MTW, p. 364); add ing m achine tape;

manipulative materials (Unifix cubes or beans and portion cups)

Procedure: After the n um erical patterns are recorded on the strip, the

class searches for pattern s and th en interpr ets the pattern s in each

column . Ask the children to look for a pattern in the blue colum n.

Then ask them to alternate standing and sitting each time the p attern

sequence changes as they read th e patterns in un ison. Loop th e pat-

terns as the children interpret each p attern.

Ask the children to look for patter ns in the right column (p lain side).

This time, ask the children to stam p th eir foot each time a zero is read.

Loop the repeating sequence as the children interpret the pattern.

Extension:

t Repeat the nu mber strip activities for different coun ting sequences

(by ones, twos, threes,) in one gr oup ing. Continu e to search for

patterns in the on es and cups colum n u ntil the whole counting

sequence begins to repeat. How many n um bers did you have to

count before this occurred ? Is there a pattern ?

tRepeat the activities in a variety of group ings.

Matrix PatternsMaterials: Place value boar ds (MTW, p. 364); class size matrix grid ;

crayons; manipu lative materials (Unifix cubes or beans and p ortion

cups)

Children should have opp ortunities to search for patterns on matrices

as well as num ber strips. Encourage the children to compare th e

matrix patterns and m ake pred ictions from the various p atterns.

Procedure: The teacher records on a matrix grid the sam e size as the

base (e.g., base 4 would be a 4 box by 4 box grid) instead of a nu mber

strip. The class searches for patterns after the sequen ce is recorded .

The teacher loops and shades the p atterns with crayons as the children

discover the patterns.

Extension:

t Repeat the m atrix activity for different coun ting sequences (by

ones, tw os, threes,)

t Chan ge the size of the ma trix grid (e.g., from a 4 box by 4 box

matr ix to a 5 box by 5 box m atrix).


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Comparing Patterns on the Matrices and

Number StripsAfter the teacher ha s recorded a va riety of sequen ces on m atrix grids

and nu mber strips, the class comp ares, discusses and makes pr edic-

tions using the information.

t Comp are d ifferent coun ting sequ ences (by ones, twos, threes,)

in one grouping on nu mber strips.

t Comp are d ifferent coun ting sequ ences (by ones, twos, threes,)

in one group ing on m atrix grids.

t Compare a coun ting sequence in different groupings on th e num ber

strips.

t Compare p atterns of a counting sequence recorded on a m atrix grid

and a number strip.

t The teacher records the coun ting sequen ce in a d ifferent size matrix

grid. The class searches for patterns in the new grid and th en

comp ares the two grids.

t The teacher records d ifferent group ings in the same size matrix

grid.


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Random Pract ice Place Value N umber FlipsMaterials: Place value board s; place value nu mber flips (MTW, p. 363);

manipu lative m aterials (Unifix cubes or beans and portion cup s)

Show the children how to attach the nu mber flips to th eir p lace value

boards. Only the num bers used in the base being used are on top of the

place value board . All the other nu mbers are tucked into the pocket on

the back of the board.

Building NumbersTell the children a n um ber to make w ith the place value n um ber flips.

Then ask them to build that num ber on the place value boards w ith

materials (either beans and cups or Unifix cubes). Repeat the p rocess

with a d ifferent nu mber.

Add and SubtractAsk the children to make a two-digit numeral w ith their place value

num ber flips. Then ask them to build the nu mber on th eir place value

boards w ith materials. Now ask th em to ad d or subtract a certain

num ber of objects. Once they have ad ded or removed the designated

num ber of materials, ask them to change the p lace value nu mber flips

to represent the new quan tity of objects.

CONNECTING LEVEL: INDEPENDENT ACTIVITIES

Random Pract iceMaterials needed for each child: por tion cups; beans; bowl; place value

board; 1-3 dice

The following activities can be played in any grou ping. Wood en cubes

can be used to make th e dice. Some teachers choose to make a different

set of dice for base 4, 5, and 6. Other teachers m ake a set of base 6 d ice

(numbered zero to five). When they p lay the game in grou pings of 4 or5, they cover the inapprop riate numbers w ith a gu mm ed label. If the

children roll a blank side wh en p laying the games in either groupings of

4 or 5, they repeat the n um ber they previously rolled.

The children w ork in p airs. They take turns rolling the dice and bu ild-

ing the material on the p lace value board.

Dice ArrangementNote: Before the game begins, the players must d ecide how the w inner

will be determined : either the smallest or largest nu mber.

Procedure: The first person rolls two d ice. He or she arran ges the d ice to

make the largest (or smallest) nu mber p ossible. Then th e person bu ildsthe nu mber w ith beans on his or her place value board . The second

child rolls the dice and bu ilds the n um ber with beans and cups on h is or

her place value board .

Once the nu mbers are bu ilt on th e place value board , the children

compare their num bers to determine which nu mber is larger (or

smaller).

Extension: For children w ho m ay need to be challenged, give them

three dice to arrange.


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reaches the bow l colum n.

Race to 0Note: Before the game begins, the players mu st decide w ho the w inner

will be: either th e first person to reach 0 or th e last person to reach 0.

Procedure: Start with the bow l colum n filled (e.g. in Zur kles, the bow l

wou ld contain four cups w ith four beans in each cup). The p layers take

turns rolling the die and subtracting the beans un til one player reaches

zero. Sometimes the children ad d the rule that the exact nu mber m ust

be rolled on th e last roll.

Observation: Watch h ow the children regroup the first time they take

the beans away.

Race to a BowlNote: Before the game begins, the partners d ecide how the w inner will

be determined: either the first person to reach the bowl colum n or the

last person to reach th e bowl colum n.

Procedure: The players start with em pty p lace value board s. Eachplayer will take a turn rolling the d ie and add ing that num ber of beans

to his or her place value board. The players continue rolling the die

and add ing beans to the place value board un til the first person


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Vertical PatternsMaterials needed for each child: Place value strip s (NL Blackline #47);

beans; portion cups; 2-3 small envelopes; a large p lastic zipp ered bag;

clothespin; 3 dice with num erals in the pred etermined group ing

The class works together at th is activity when it is first introd uced.

They decide wh ich grou ping (e.g., Zu rkle) to use. To determ ine how

far to add or w hich n um ber to start subtracting from, three dice are

rolled an d arran ged into a three-digit nu mber.

This process may take more than one day. When the child finishes for

the d ay, he or she rolls up the place value strip an d clips it with a

clothesp in. The place value strip, place value board, po rtion cups, and

envelopes of beans are stored in a large zipp ered p lastic bag w ith the

child's name on it. The next day the child builds the last num ber on the

place value strip and continues: add ing or subtracting m aterials on the

place value board; recording the numerals on the place value strip; and

looping the patterns in each colum n.

Addition Procedure: The class wor ks at this activity together w hen it

is first introdu ced. They ad d materials on their place value board and

record the numerals together to th e end o f the first strip. Then, each

child loops the p atterns in each column on his or h er strip and attaches

a new p lace value strip. The class works together for appr oximately

two an d a half strips. Then they continue w orking at their own p ace

until they reach the n um ber pred etermined by the d ice roll. When the

child has add ed to the third column, the beans in the p ortion cups are

placed in an envelope and sealed.

Subtracti on Procedure: The children begin by building the nu mber

determined by the d ice with m aterials on their place value board. Theobjects for the third column are pu t in envelopes and placed to the left

of the cups colum n. Then the children record the nu mber at the top of

their place value strips. Next, they sub tract one object at a time and

record the nu mber on their place value strips. When they have com-

pleted on e strip, each child loops the patter ns in each colum n of the

place value strip and attaches a new strip. The class works together for

app roximately two an d a half strips. The children continue w orking at

their own pace un til they reach zero. When the child sub tracts all the

objects in the cups and u nits colum ns of the place value board , he or

she opens an envelope of objects in the third colum n an d regroups

them in portion cup s before continuing to subtract.

Note: Pattern is a self-correcting tool and w henever a child find s anerror, rather th an erasing, the child m erely cuts off the place value

strip w here the error occurred and attaches a new strip.

Do not be surp rised abou t how many children forget to write in the

nu mbers at th e regrouping stage. This error corrects itself as soon as

the children realize the value of pa ttern. They begin to find their

mistakes by looking for the recorded evolving patterns.

Extension: Ask the children to search for patterns w hile adding or

subtracting by two (or three) objects each time.

SYMBOLIC LEVEL: IN DEPENDENT ACTI VI TI ES



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Matrix PatternsM aterials needed for each child: matr ix fram es of various sizes (NL

Blackline #46); mater ials to count; po rtion cup s; place value boar d;

colored p encils, crayons, or markers to loop pattern s

Same Grouping Different Size FramesThe children can w ork in p artners or alone. They begin by ad ding

objects to the place value board on e at a time and recording the

counting sequ ence for the chosen grou ping in the matrix fram e. They

record the nu merals in sequence un til the matr ix frame is filled.

Extensions:

t When the children are experienced w ith recording in a matrix

frame, they can record on e nu meral at a time u ntil they begin to see

patterns d eveloping in the colum ns and rows (vertically, horizon-

tally and d iagonally). They complete the matrix by recording the

pa tterns as they d iscover them . If theyre un sure of a pattern, they

can check it by using concrete mater ials.

t Once one matr ix has been comp leted, the children can try a d iffer-

ent size matrix frame. Some of the pa tterns in the first matrix may

be useful when th e children p redict patterns in the new frame.

t The children can record p atterns by adding by tw o or three num -

bers at a time. Concrete materials can be u sed to confirm the m atrix

patterns.

Same Size Frame Different Groupings

The pa ir chooses a m atrix frame (e.g., 5 boxes by 5 boxes) and a

group ing. They record u ntil they begin to see patterns d eveloping in

the columns and r ows (vertically, horizontally and d iagonally). Theyrecord the matrix patterns as they are d iscovered. When the frame is

completed, the p air searches for add itional patterns. They loop and / or

shade p atterns with the side of a crayon.

When th e pair is finished searching for pattern s in the first group ing,

they choose a different grou ping an d record it in the same size fram e.

They comp are the patterns in the two group ings in the same matrix

grids.


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BASE TEN WORK

GROUP ACTIVITIES

Searching for

Patterns

Materials: class-size (10 by 10) matrix fram e or over head transp arency

of NL Blackline #45; class-size hu nd red 's chart or an overhead trans-parency of NL Blackline #44

Searching for Patterns on the Hundreds ChartProcedure: Record the coun ting sequ ence on a class-size 10 by 10

matrix frame as the children count by one's. Complete approximately

one and one-half rows in sequence before asking the children if they

see any patterns d eveloping (vertically, horizontally or diagonally).

Complete the m atrix as the children d iscover the p atterns.

Extensions:

t Using the same p rocedu re, the class searches for pa tterns in other

counting sequ ences (by tw o's, three's, five's) as you r ecord thepatter n sequ ences on a large class-size (10 by 10) frame.

t The class searches for p atterns w hile subtracting. Begin the activity

by record ing 100 in the first box. The class coun ts backwar d, one

nu mber at a time, as you record the nu merals on the matrix. Com-

plete appr oximately one and one-half rows. Ask the children to

search for patterns. Record th e patterns as the children d iscover

them.

t A comp leted class-size hun dr eds chart is used for this activity. Ask

the children to search for pa ttern sequ ences. Ask the children if they

see the same sequence anywhere else on the chart. Loop and/ or

shade th e patterns w ith a colored m arker or crayon. Ask the chil-dr en to search for a different pattern sequ ence. Continu e to search

for patterns in this fashion.

Random Pract iceVisualizationAfter the children have had extensive manipulative experiences

regrouping w hile add ing and su btracting large num bers in base ten,

allow some time for the children to visualize the regrou ping p rocess.

Visualization enables children to transition from their concrete experi-

ences to a m ore abstract level. This is necessary for them to be su ccess-

ful in school testing situations.

Procedure: Write a problem on th e board. Ask the children to close

their eyes and imagine a certain typ e of manipu lative on th e place

value board as the teacher records the action num erically. The childr en

should experience visualizing both ad dition and subtraction equ ations.


16/24



Sample Tea chingStrate gy Visualization


17/24



INDEPENDENT ACTIVI TIES


Before You Begin: Run off NL blackline #43 for each child. The master

has two r ows w ith five stars in each row.

Place a copy of the star blackline on the ones side of each childs placevalue boa rd . It is difficult to tell at a glance (withou t counting ) how

many counters are on the board in base ten. The stars give the child an

organ izing tool for work in base 10. By covering each star w ith the

mater ial, children w ill think in term s of nu mber combina tions. With-

out the ad dition of the star working area, they may count by ones and

miss the opp ortunity to reinforce the ad dition and subtraction combi-

nations.

Vertical PatternsM aterials needed for each child: Place valu e strip s (NL Blackline #47); a

small cylinder m ade of tagboard; a clothesp in; beans; 10 portion cup s;

2-3 envelopes; large zipp ered bag (to store the m aterials)

Procedure: The childr en pr actice add ing to or subtracting from a large

three-digit nu mber in ba se ten. Its helpful if the children w ork in

pairs. Each p air chooses a small container from the comp aring station

and fills it with beans. The beans ar e placed in a zipp ered bag for

storage.

Addition: The children count by ones one person records the

sequence as the other person ad ds th e material to the place value

board. They search for and loop patterns every time an individual

place value strip is completed. On ce the patterns are looped and any

mistakes are snipped off, a new place value strip is add ed an d th e pair

continu es add ing beans. When they reach 100, they emp ty the ten cup s

of beans into an envelope and seal it. The envelope is placed off the

place value board (to the left of the 10s column).

Subtraction: This time, the container full of beans is grouped into

hun dred s, tens and ones. The hu nd red's are sealed in an en velope.

The total is recorded at the top of the place value strip. The pair

subtracts (one bean at a time), record s, and searches for patterns on the

num ber strip u ntil they are out of beans.

Extension: Children should h ave experiences add ing and subtracting

by two, three, four objects at a time. (see MTW p. 314).

Matrix PatternsMaterials needed for each child: hu nd red's char t (NL blackline #44);

blank (10 by 10) matr ix fram e (NL blackline #45)

Procedure: The children ind ependen tly search for and record p atterns

on a matr ix. See NL, page 11.15 for a detailed d escription.


18/24



Dice GamesPlay Dice Arran gem ent (N L, p. 11.11), Race to a Bowl (p. 11.12), Race to

Zero (N L. p. 11.12) in base t en.

Bobs DittoAdditionMaterials: small objects; portion cups; ad dition board * (NL Blackline

#39); record ing sheet (NL Blackline #38)

Step One: Take a hand ful of junk and

count it into counting cups. Place the

jun k on the top place value boxes of the

add ition board. Record the n um ber on

the recording sheet.

Step Two: Take another handful and

count it into counting cups. Place thejun k in the middle place value boxes.

Record the nu mber on th e recording

sheets.

Step Three: Move the jun k dow n to the

bottom p lace value boxes.

Step Four: Regroup to find the total.

Record th e total on the record ing

sheet.

SubtractionMaterials: place value boar d; small objects; portion cups; a w ooden

cube n umbered 4, 5, 6, 7, 8, 9; record ing sheet (NL Blackline #38)

Step One: Take a handfu l of objects and

regrou p into tens if possible. Place the

objects on the place value board .

Step Two: Record the am oun t of objects on

the recording sheet.

Step Three: Roll the die. Record the number

on the p aper. Subtract that man y objectsfrom the p lace value board.

Step Four: Record th e answer.

Random Pract ice

(*) Note: Large place valueadd ition boards can be made with9" x 11" file folder s. (see d irectionsp. 11.23)


19/24



PLACE VALUE STATION ACTIVITIES

The p lace value station activities fall into tw o categories:

Counting Act iv it ies

Addition and Subtraction Activities

Place Value Counting ActivitiesThe counting activities provide children opportunities to rebuild inbase ten. The counting stations should be introduced before the

add ition an d subtraction stations.

Measuring MTW, p. 307

Estimating and Checking MTW, p. 308

Counting Jars of Objects MTW, p. 310

Perimeter MTW, p. 315

Unifix Stack MTW, p. 320

Magazine Pictures NL, p. 11.19

Place Value Addition and Subtraction ActivitiesThe following activities provide children opportunities to regroup

wh ile add ing and subtracting. Add these stations when the children

demon strate a clear und erstanding of regrouping w hile counting in

base ten.

Determining Prices MTW, p. 312

The Store MTW, p. 317

Inside-Out NL, p. 11.20

Bobs Ditto NL, p. 11.18

Place ValueAct ivities Not Foundin Mat hematicsTheir Way

Magazine PicturesMaterials: Magazine pictures, old wrap ping pap er and/ or stickers

scattered on a page of pap er; water-soluble pen; wet r ags; NL Blackline

#42

Note: Find either magazine p ictures, wrap ping p aper, or create

pictures by sticking stickers on a page. Each picture shou ld hav e

between 12-100 objects that are easily coun table. Label each p ictur e

with a different letter. Cover the pictures w ith acetate.

Procedure: The child chooses a picture and records the letter on the

recording sheet. He or she circles as man y grou ps of ten as possible.

Next the child counts the total nu mber of objects and records theanswer on the recording sheet.

Extension:

Procedure: Some teachers p lace several different types of rubber

stamps, a stamp pad , and blank pap er in this station. The child chooses

a stamp and covers the pap er with the stamp 's image. He or she circles

the stamped images in groups of ten. Next the child coun ts the total

and records the n um ber on the p age. The recording can be taken

home.


20/24



Inside-OutsideThis activity is d escribed inMathematics Their

Way (see Geoboard Designs, MTW, p. 316).

The description (and p ictur e illustration) uses

only one geoboard. In order to provide chil-

dren with regroup ing experiences, there must

be a m inimum of two boards side-by-side.

PLACE VALUE MATERIALSPlace Value BoardsM aterials for each place value board: 12" x 9" tagboard ; a 5-1/ 2" x 8-1/ 2"

piece of blue mim eo or constru ction p aper ; a 12" x 9" piece of white

construction paper; glue stick or ru bber cement; 2 library p ockets

(Use protective covering after the p lace value boar d is completed.)

Procedure: Glue the white paper to the tagboard. Glue the blue paper

on the left side on top of the white paper. Draw a h app y face in the

up per left corner to p rovide a "right side up " signal to the children .

Place Value Flip Pockets: Glue two library pockets on the back side of

the place value board . (One for the blue side and one for the wh ite

side.) Be sure to p lace the p ockets as close to the mid dle of the board as

possible.

Some teachers cut the place value board s in half and hinge them with

tape on the back side. The folded p lace value board s are easy to store.

Note: If the class is going to be working a lot with 3 colum n nu mbers,

you m ay w ant to attach an add itional (5-1/ 2" x 8-1/ 2") piece of

tagboard (with a d ifferent color of construction pap er glued on top of

it) to the left of the blue side. The childr en can fold this piece und erthe blue side of the place value board wh en they are not u sing it.

Star WorksheetNL, Blackline #43; tape; place value boards

Note: The star paper s are to be used w ith the base ten activities only.

(see NL, p. 11.17 for an explanation of its pu rpose)

Procedure: Dup licate the star blackline. Cut the pap er on the do tted

line. Tape the star paper on to the wh ite side of the place value board.

Place Value FlipsMaterials: Number Flips*; chicken rings**, Chinese Jacks or small metal

rings; scissors

* Making Num ber Flips from scatch can be time consum ing. We

recommen d that you pu rchase this item, predrilled and ready to

assemble from the Center for Innova tion in Education. If you w ant to

assemble you r ow n, please consu lt MTW, p. 363 for the d irections.

** Chicken rings can be pu rchased in a feed store.


21/24

Activity

Description

Recording

*

Description

* Recording materials are placed in the tubbing stations only after the children have had sufficient time to work at the concrete (concept) level with the manipulatives. Depending on the

class's experience and the material in the tubbing station, the time allowed may span from several weeks to several months. Recording should be introduced as an optional activity.

Additional materials needed:

Materials needed: Materials needed: Materials needed: Materials needed: Materials needed:

Additional materials needed:Additional materials needed:

Additional materials needed: Additional materials needed:

Page #'s

11.21 1990 Center for Innovation in Education, Saratoga

UNIFIX STACKS

MTW, pp . 320, 396

3 more/ less spinners

3 wooden cubes with numbers

0,1,2,3,4, and 5 written on them

Unif ix cubes

3 Unifix stack gameboards

(MTW, Blackline #29)

4-5 place value boards

Procedure: Two children take turns

rolling their dice and p lacing a

stack of cubes on their row of the

gameboard. If a child rolls a zero,

he or she leaves the box empty.

When the row s are completed, they

snap their cubes together in groups

of ten's.

MTW Blackline #61 in a small

plastic zippered bag

Recording: Each child w rites his or

her name on the recording sheet.

They play the gam e. Then each

child adds his or h er Unifix cubes,

and records the number in the

appr opriate box. One child spins

the more/ less spinner to determine

the winner.

PERIMETERS

MTW, p. 315

old geoboard records (MTW, p.

177)

Unifix cubes in one color (about 80

cubes)


Procedure: The children p redict the

num ber of Unifix cubes it will take to

measure the perim eter of a geoboard

design. (One unit equals one side of

the colored square on the recording

sheet.) Then they m ark each unit

with a Un ifix cube, snap the cubes

together into a train, and d ivide the

train into groups of ten's and one's.

The Unifix cubes are then placed on

the place value board.

MTW Blackline #40 in a small


Recording: The children write their

estimates on the recording sheets.

Then they check their estimates by

placing the Unifix cubes around the

perimeter of the d esign. The children

then group the Unifix cubes used to

measure the perimeter into ten's and

one's. They record th e actual

number.

MAGAZINE PICTURES

NL, p . 11.19

Magazine pictures or wrapping

paper sh owing between 12-100

objects easily countable (a flock o

birds, a parking lot of cars, a

picket fence, a row of toys, a

crowd of p eople, a box of

chocolates, etc.) covered with

acetate

6 water color marking pens

Procedure: The children choose a

picture, loop the objects into group

of tens and then coun t the total.

NL Blackline #42 in a small


Recording:The children r ecord the

picture's letter onto the recording

sheet, loop the objects in the pictur

in groups of tens, and record the

total on the recording sheet next to

the app ropriate letter. (Three

pictures can be recorded on one

recording sheet)

Extension: The children can m ake

recordings to take home by stampi

blank paper w ith a picture on a

rubber stam p. Next, they circle the

stamped images in grou ps of ten's

and record the total on the paper.

ESTIMATING ANDCHECKING

MTW, p. 308, 360

7-10 empty jars or containers

3-4 large margarine tubs of beans

(each w ith a different type of bean,

e.g., large kidney, lima beans, pinto

beans) 1 or 2 oz. portion cups


5-6 empty margarine tubs

Procedure: The children select an

empty container and fill it with beans.Then they estimate how many beansare in the container. Then they p laceeach group of ten beans in a p ortioncup. When there are enough to makea hundred, the ten portion cups ofbeans are stacked and placed inside amargarine tub. The children arrangethe beans on the place value boardand count to find the total num ber ofbeans.

PLACE VALUE STATIONS

COUNTING CONTAINERS

OF OBJECTS

MTW, p. 310

6-10 small containers with small

objects inside (bread tabs,

macaroni, buttons, rocks, dried

pinto beans, etc.)

1 or 2 oz. portion cups 5-6 empty margarine tubs


Procedure: The children select one

container and estimate how many

objects are inside. Then they place

each group of ten objects in a

portion cup. When there are

enough to make a hundred, the ten

portion cup s of objects are stacked

and p laced inside a margarine tub.

The children place objects on th e

place value board and read the

number.

NL Blackline #14 stored in asmall plastic zippered bag

Recording: The children record theirestimates in the first box, count theobjects into portion cups and record theactual number in the second box on therecording sheet.

Extension: The children count theobjects in the con tainers by 1s, 2s, 3s,5s, 10s etc. They record the countingsequence on either a 2 column or 3column place value strips.

A blank 10 x 10 matrix (NL Blackline#45), 0-99 matrix (MTW Blackline #56)or hundreds chart (NL blackline #44)can also be used to record the countingsequence.

NL Blackline #14 stored in a small


Recording: The children record theirestimates in the first box, count theobjects into portion cups and recordthe actual number in the second boxon the recording sheet.

Extension: The children count thebeans in the cont ainers b y 1s, 2s, 3s,5s, 10s etc. They record the countingsequence on either a 2 column or 3column place value strips.

A blank 10 x 10 matrix (NL Blackline#45), 0-99 matr ix (MTW Blackline#56) or hun dreds chart (NLblackline #44) can also be used torecord the counting sequence.


22/24

Activity

Description

Recording

*

Descriptio

n

* Recording materials are placed in the tubbing stations only after the children have had sufficient time to work at the concrete (concept) level with the manipulatives. Depending on the

class's experience and the material in the tubbing station, the time allowed may span from several weeks to several months. Recording should be introduced as an optional activity.

Additional materials needed:

Materials needed: Materials needed: Materials needed: Materials needed: Materials needed:

Additional materials needed: Additional materials needed:Additional materials needed: Additional materials needed:

Page #'s

1990 Center for Innovation in Education, Saratoga, CA 1

DETERMINING PRICES

MTW, pp . 312, 396

pennies and dimes in empty

bullion tubes or coin cylinders

place value addition boards

(NL Blackline # 39) or large

place value add ition boards

(see p . 11.23)

cans of food (real food inside)labeled and priced less than

50.

Procedure: The children select two

cans of food to buy. They put

the needed coins on their addition

board and compute the total.

MTW Blackline #59

Recording: The children select two

cans of food to buy and record

the names of the items and p rices.

Then they put the needed coins

on their addition board and

compute the total. They record the

process on the recording paper.

PLACE VALUE STATIONS

MEASURING

MTW, p. 307

Unifix cubes

word and picture cards (door,

chair, bookcase, table, desk,

wind ow, trash can, coat, book,

paper, easel, box, record

player, etc.) for the children to

use to spell these word s

Procedure: The children select a

card. (A simple picture of the

object can be drawn on the card to

help the children.) The children

measure th e object described on

the card w ith Unifix cubes in

whatever d irection they choose.

NL Blackline #41

Recording: The children w rite

dow n the nam e of the object

measured and the number of

Unifix cubes it took to m easure it.

(The children can measure an d

record thr ee different objects on

each recording sh eet.)

THE STORE

MTW, pp . 317, 396

10-12 objects in plastic zippered

bags and pr iced less than 50

real coins (dimes and pennies

only) in coin cylinders (from a

coin shop) with the level marked

with a marking pen.

place value addition boards(NL Blackline # 39) or large p lace

value addition board s (see p.

11.23)

Procedure: The children select two

items from the store to bu y. Then

they put the need ed coins on their

addition board and compute the

total.

MTW Blackline #59

Recording: The children choose two

things to buy and write down the

names and p rices. They put the

money needed on the place value

addition board and compute the

total. Then they r ecord the p rocess

on the recording paper.

GEOBOARD DESIGNS(INSIDE-OUTSIDE)

MTW, pp. 316

approximately 100 Unifix cubes in

two colors


6-8 geoboards

Procedure: The children m ake a

design on two side-by-side geoboards

with one geoband . They cover the

inside nails that are not touched by

the geoband w ith one color Unifix

cube and the ou tside nails that are

not touched by the geoband with the

second color. They take both colors

off the geoboard, snap them together

into groups of tens and p lace them

on the place value board.

NL Blackline #42

Note: At least two geoboards must beused so there w ill be enough nails

inside and outside the geoband for

the children to experience addition

with regrouping.

Recording: The children count and

record the num ber of Unifix cubes on

the inside of the geoband. Then they

count and record the number of

Unifix cubes on the ou tside on the

geoband. They add two groups of

Unifix cubes together and record the

total number of Unifix cubes inside

and outside the geoband design.

BOB'S DITTO

NL, p . 11.18

small object (e.g., elbow macaro

or kidney beans)

place value addition boards

(NL Blackline # 39) or large

place value add ition boards

(see p. 11.23)

place value boards

port ion cups

6-8 dice numbered 4, 5, 6, 7, 8, 9

Procedure: Consult NL, p. 11.18 for

a detailed d escription of Bob's Ditt

NL Blackline #38

Cut the recording sheets on the

dotted line. Place the additionand su btraction sheets in separa

small zippered bags.

Recording: Consult NL, p. 11.18 for

a detailed description.


23/24



LARGE PLACE VALUE ADDIT ION BOARD

Materials for each board: 1 file folder (9" x 11"); 1 med ium black perm a-

nent mar ker, 3 star w ork sheets (NL, Blackline # 43)

Trim the file folder to 8-1/ 2" x 11-3/ 4".

Open the file. Draw tw o horizon tal lines across the file folder. Draw

the first horizontal line 3-3/ 4" from the top of the file folder. Draw the

second horizontal line 7-1/ 2" from the top of the file folder . Make the

second line heavier than the first line.

Draw a vertical line 5" from the righ t side of the file folder.

Trim the 3 star pap ers to app roximately 3" x 4". Glue one star paper in

each of the boxes in the right colum n.

Draw a small smiley face on top of the place value add ition board.

Note: Large place value add ition boards p rovide more room for

materials on th e 10s side of the board .

7.5"

3.75 5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5"


24/24


chapter11 manipulatives

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chapter11 manipulatives