Mathematical Exercises Part II Module 7 Andleeb Javed
Andleeb javed
Module 7
Roll # D1081
QUESTION 01 Explain different groups of Montessori Math exercises and how the directress should efficiently present exercises through seuential and parallel wor! in "arious groups# $NS%E&'
!earnin" mathematical concets in a Montessori classroom be"ins concretel$ and ro"resses to%ards the abstract& 'he$ are develoed (rom simle to comlex& Process is tau"ht )rst and (acts come later& *rder+ coordination+ concentration+ and indeendence are exerienced b$ the child usin" these materials& 'he math activities are or"ani,ed into )ve "rous& 1& .& /& 2& 3&
-umbers throu"h ten Decimal s$stem ountin" be$ond ten linear and si countin" Memori,ation o( the arithmetic tables Passa"e to abstractions and 4ractions
'hese exercises are "roued+ some %ors se5uentiall$ and some %ors arallel&
1) Numbers through ten 'his "rou introduces sets o( one throu"h ten %hich reares the child (or countin" and teaches the value o( 5uantit$& hildren be"in to associate numeral and 5uantit$ %ith number rods and number cards& A child %ill "ain a "ro%in" understandin" o( se5uence& 6indle boxes+ cards and counters+ the short bead stair+ and other 1 10 additional countin" activities a teacher ma$ add+ rein(orce the one throu"h ten numeral concets&
2) Decimal system rou t%o involves the decimal s$stem usin" the "olden bead material& 'he child %ill become (amiliar %ith the names o( the decimal cate"ories9 units+ tens+ hundreds+ and thousands& A concrete exerience %ith each cate"or$ is reresented b$ beads& :uantit$ %ill be (ollo%ed b$ s$mbol and association&
3) Counting beyond ten linear and skip counting rou three deals %ith the oerations usin" the "olden bead material& 'he concet and rocess o( addition+ subtraction+ multilication+ and division are resented& hildren %or %ith each other and bene)t (rom these exercises
Andleeb javed
Module 7
Roll # D1081
usin" the ban "ame& Pro"ression then continues usin" oerations %ith the stam "ame&
4) emori!ation o" the arithmetic tables rou (our consists o( linear countin"& :uantit$ is resented usin" the teen and ten boards (ollo%ed b$ s$mbol and association& 'he onehundred board and bead chains develo number concets and reco"nition o( numbers one throu"h onehundred& 'he bead chains also introduce the child to si countin"9 )ve+ ten+ )(teen+ t%ent$+ etc&
#) $assage to abstractions and %ractions rou )ve contains activities such as stri boards+ snae "ame+ and memori,ation o( (acts& 4ractions are also a art o( this "rou& 4raction sittles and insets serve this urose& 'he activities in the Math area are not to be imlemented at a set ace& Providin" the child %ith the materials at recisel$ the ri"ht challen"e level %ill enable the child to demonstrate his develoment to the teacher throu"h his ro"ress& A child that is able to "ras such math concets as addition and subtraction demonstrates the success(ul use o( the math materials& 'he materials are so beauti(ull$ desi"ned and aroriate (or each child durin" his sensitive eriods o( learnin" math& Mathematical aaratus rovides the necessar$ stimulation (or the child to learn math concets&
Andleeb javed
Module 7
Roll # D1081
QUESTION 0( Explain the exercises which ena)le the child to count till 1000# $NS%E&'
&inear Counting !inear Exercises hels the child learn to count till 1000+ alon" %ith "ettin" (amiliar %ith the decimal s$stem relationshis+ includin" the concets o( s5uares and cubes o( numbers+ linear countin" is resented in t%o sta"es& In )rst sta"e the child learns to count till 100 and in second sta"e the child masters countin" till 1000& 'he urose o( these exercises is to consolidate the child;s no%led"e o( countin"& < until no%+ he %ored %ith tens and hundreds in the decimal s$stem& =ith these Exercises+ he becomes (amiliar %ith the se5uence o( numbers (rom 1 throu"h 1+000& ountin" is a rest(ul activit$ and tends to become mechanical& 'hrou"h reetition+ the child establishes the mechanism o( countin"& =hen the t%o chains are laced arallel to each other+ the$ sho% in a striin" and sensorial %a$ the di>erence bet%een the s5uare and the cube o( ten& In this %a$+ the decimal s$stem relationshis are (urther established b$ the child&
'(ercise 1 *+' +,ND-'D C+A.N 'his exercise re5uires the hundred chain consistin" o( 10 bars o( 10+ the hundred s5uare+ an enveloe containin" arro%ed labels reen labels mared 1 ? @+ lue labels mared 10 ? @0+ A red label mared 100 and a lar"e si,ed mat rin" the child to the chain cabinet& 6ho% the child the bars on the shelves and discuss %ith the child i( he has seen bars lie these be(ore& e"in countin" %ith the child startin" (rom the unit to the 10 bar& Bave the child unroll the runner just a little %a$s& 6ho% the child ho% to hold the 100 chain b$ both ends and have him la$ it verticall$ at the bottom o( the mat& Bave him lace the tra$ belo% the 100 chain& 6lo%l$ (old the chain to"ether to create the hundred s5uare& -otice that it loos lie the hundreds s5uare& Place the hundreds s5uare on to o( the (olded ten chain to sho% that the$ are the same& Remove the hundred s5uare and have the child "entl$ re strai"hten the ten chain& 'ae out the unit ticets C"reen and tell the child %hat the$ are called& !ine them in a vertical line to the le(t o( the ten chain& 6ho% the child the ten ticets Cblue and lace in a vertical line above the
Andleeb javed
Module 7
Roll # D1081
unit ticets& !abel the )rst ten b$ usin" the unit ticets and lacin" them on the le(t o( the chain& ount %ith the child 11.0& At the .0 mar+ lace the ticet that has .0 on it to the ri"ht o( the .0 bead& ountin" b$ units+ continue lacin" the ten ticets until $ou reach 100& Bave the child lace the red 100 ticet next to the 100& 'ell the child FGou have just counted to 100&H As+ FBo% man$ beads are in this chainH C100 Point to the hundred s5uare+ FAnd ho% man$ are in thisH C100& ount %ith the child all o( the ticets 1+ .+ /+ 2+ 3+ + 7+ 8+ @+ 10+ .0+ /0+ 20+ 30+ 0+ 70+ 80+ @0+ 100& 'hen count bac%ards 100+ @0+ 80+ 70+ 0+ 30+ 20+ /0+ .0+ 10+ @+ 8+ 7+ + 3+ 2+ /+ .+ 1& Bave the child relace the ticets into their correct envelos and then relace the rest o( the material&
'(ercise 2 *+' +,ND-'D /0A-D In the activit$+ the child ractices countin" linearl$ (rom one to onehundred& 'his exercises needs board %ith a 10 x 10 "rid+ s5uare tiles %ith numbers 1 100 and bo%l or baset& arr$ the board and container o( number tiles to a mat or table& 6ort the number tiles into columns b$ tens lace& DonJt %orr$ about "ettin" them in the correct order %ithin each column& 6tartin" %ith 1 in the uer le(t corner+ la$ out the number tiles in order& ontinue until all o( the number tiles are in the correct lace&
'(ercise 3 *+' *+0,AND /0A-D 'his exercise re5uires a thousand chain consistin" o( 100 bars o( 10+ 'en s5uares o( ten+ 'he thousand cube+ An envelo containin" arro%ed labels reen labels mared 1 ? @+ lue labels arro%ed 10 ? @@0+ Red labels (rom 100 ? @00+ A lar"e "reen label mared 1+000 and lar"e si,ed mat&
Andleeb javed
Module 7
Roll # D1081
'ell the child that toda$ %e are "oin" to loo at an even lon"er chain than the 100 chain& Bave the child unroll the runner all the %a$& 6ho% the child ho% to hold the 1000 chain& 'he directress carries the chain to the runner+ %ith all o( the strands laid out strai"ht& Bave the child brin" over the cube and the lar"e box on a tra$ over to the runner& Also brin" over the hundred s5uares& 'ell the child that $ou are "oin" to tr$ to (old the chain just lie $ou did %ith the 100 chain& Mae a hundreds and as the child %hat $ou made& Place a hundred s5uare next to the one $ou just made& Reeat until the %hole chain has been (olded in hundred s5uares& C'he child can be"in to mae them a(ter a %hile& Place each o( the hundred s5uares next to the hundred s5uare $ou have made %ith the child& 'hen lace the hundred s5uares on to o( the hundred s5uares $ou and the child have made& ount %ith the child to see ho% man$ hundred s5uares there are& Bave the child lace each hundred s5uare on to o( each other& -otice that it loos just lie the cube& =hen %e have 10 hundred s5uares+ %e no% that %e have 1000 beads& Place the cube next to the ten hundred s5uares Claced on to o( one another to sho% this to the child& Bave the child "entl$ ull the 1+000 chain strai"ht& CBave him ee the chain near the le(t side o( the runner& Bave the child la$ out all o( the ticets& ount each bead and lace the correct ticet %hen needed as in Presentation 1& =hen $ou "et to 100+ lace the ticet as %ell as a hundred s5uare next to the 100th bead& Reeat this (or ever$ hundred& CEven at the 1+000th bead& At the 1+000th bead+ also lace the cube& 6tand at the be"innin" o( the runner and %al all the %a$ to the end& 6tand at the end and loo at the %or o( the child& o bac to the be"innin" and count 100+ .00+ /00+ 200+ 300+ 00+ 700+ 800+ @00& As the child ho% man$ he had at the end 1000& o bac to the be"innin" and count the tens& 10+ .0+ /0+ 20+ 30+ K 100+ 110+ 1.0+ K 200+ 210+ 2.0+ K @80+ @@0+ 1000& 'hen have the child count b$ tens bac%ards& 'hen child can then ut the material a%a$&
Andleeb javed
Module 7
Roll # D1081
QUESTION 0* +rint ,ot -a.e +aper and send three sol"ed pro)le.s/ each carrying four addends $NS%E&'
Andleeb javed
Module 7
Roll # D1081
QUESTION 0 Explain the presentations of Multiplication 2oard and di"ision 2oards in your own words# $lso .a!e illustrations# $NS%E&' Exercise 1' Multiplication 2oard
Purpose of this exercise is to give practice in multiplication leading to the memorization of the essential multiplication tables. Materials •
• • • •
A perforated board with 100 holes in rows of 10 arranged in a square. At the left side of the board is a window with a slot for the insertion of the cards. A red, wooden disc. Tables of multiplication A set of cards from 1 to 10 harts 1 and !
3hart 1
3hart ( +resentation "how the child the material and have him bring it to the table. "how the child the numbers along the top of the board. Tell the child, #These numbers tell us how man$ times to ta% e a number.& "how the child how to slide the card '() into the slot on the side of the board. Tell the child, #This tells us we will be doing the table of (.& Place the little red disc above the 1 at the top of the board. "a$, #This tells us we need to ta%e ( one times. *sing the red beads, place ( one times in a vertical line. +ave the child count how man$ beads there are on the board. Tell the child, #( x 1 is (& +ave the child write the answer on the paper next to the equation. ove the disc over above the !. Tell the child, #-e now need ( two times. ut we alread$ have ( one times.& +ave
Andleeb javed
Module 7
Roll # D1081
the child place the red beads in a vertical line next to the first four. +ave the child count the total number of beads on the board. "a$, #( x ! is /&.
epeat in this manner. -hen the child reaches ( x (, have him sa$ the equation with $ou. f the child is ma%ing the table with ease, when he reaches ( x / show him that ( x 2 was !/. ount from !/ up four more. epeat in this wa$ until he has finished the board. +ave the child read all of the equations and answers written on the piece of paper. The child can chec% his wor% on ultiplication hart 1. Exercise (' Unit ,i"ision 2oard
The purpose of this exercise is to familiarize the child with the various wa$s in which numbers ma$ be divided. Materials
The unit division board 3 green s%ittles • A box with /1 green beads • Printed division slips • Introduction •
"how the child the material and have him bring it to the table. 4oo% at the board with the child. "how him all the places for the beads. "how him the numbers at the top of the boards and tell him that the$ are the dividend. Tell him the numbers on the side will be the answer. -rite !2 5 3 6 on squared paper. egin b$ telling the child the two rules7 the answer cannot be larger than 3 and the remainder cannot be equal to or larger than the divisor. Place !2 green beads into the glass bowl. As% the child how man$ times will we give !2 out. '3) Place 3 green s%ittles along the top of the board. 8ive each s%ittle a bead until there are no beads left.
Andleeb javed
Module 7
Roll # D1081
Table of division division remainder
"a$, #!2 5 3 is 9&. Point to the 9 along the vertical side of the board. +ave the child write in the answer to the equation. +ave him replace the beads into the bowl. As% the child how man$ beads we have in the bowl. '!2) Tell him as $ou write the next part of the equation on the second line of paper, #!2 5 / 6& +ave him do this equation as he had :ust done the last. +ave him write the answer with the remainder. ead the whole equation with the child. Tell the child that there is something special about the remainder7 #-e can have a remainder as long as it is smaller than the dividend.& epeat as above for !2 5 ; 6 After the child has written the answer and the remainder chec% to see if the remainder is smaller than the dividend. epeat in this wa$ for the entire sheet of paper. -hen $ou reach !2 5 !, $our remainder will be bigger than the dividend. Point this out to the child and then cross out the entire equation because7 #-e can never have a remainder that is the same or bigger than the dividend.&
Andleeb javed
Module 7
Roll # D1081
+resentation
-rite /1 5 3 6. +ave the child count /1 beads into the glass bowl and place nine s%ittles at the top of the board. +ave the child do this equation as written above. ead the finished equation out loud with the child. Then write7 /1 5 / 6. +ave the child do this equation and write in the remainder. 4oo% with the child at the remainder and notice that it is bigger than the dividend. ross off the whole equation. epeat for /1 5 2 6. epeat for /1 5 ; 6 4oo% at the last three crossed off equations and wonder out loud7 # wonder if the others will also have a remainder larger than the dividend<& The child should notice that the remainder gets bigger and bigger as the dividend gets smaller and smaller. =ecide together that because of this pattern, onl$ /1 5 3 wor%s. >ou can finish this presentation here.
Andleeb javed
Module 7
Roll # D1081
QUESTION 04 5ow is Sta.p -a.e introduced to the 3hild $lso explain how su)traction pro)le.s can )e sol"ed with the sta.p ga.e $NS%E&' Sta.p -a.e Materials
4arge quantities of wooden squares of equal size about 1 inch square li%e stamps7 o ?ach stamp of 1 is green mar%ed with @1. o ?ach stamp of 10 is blue mar%ed with @10. o ?ach stamp of 100 is red mar%ed with @100. o ?ach stamp of 1000 is green mar%ed with @1000. A pencil and ruler • "pecial grid paper • Introduction •
nvite the child to come and wor% with $ou. "how the child the material and have him first bring over the paper needed. Then show the child the material and have him bring over the box of wooden tiles as well as the tra$ from ntroduction to Buantit$. "how the child the 1 green tile and show the 1 unit to the child. Tell the child that it is the same as the unit bead. "how the child the blue tile and have him read the @10 written on it. Tell the child that this is :ust li%e the tenC bar. epeat for the tiles of 100 and 1000. =o a Three Period 4esson with the 1, 10, 100, and 1000 tiles. "how the child that when we ta%e out the 1 tiles, we place them directl$ in front of the compartment where the other 1s are. Tell the child that $ou are going to ta%e out D. Ta%e out D of the 1 tiles and place them all in front of the 1 compartment. Put them bac% and give the child a few numbers to ta%e out. "uch as ma%e 9 tens, or D hundreds, or ! thousands. Then give the child a larger number. "a$, #Eow we are going to ma%e a larger number. This number will have 9 units, D tens, ! hundreds, 1 thousand. As $ou give the child each number, have him ta%e out the appropriate tiles. ount to chec% the final product and then have the child put the tiles bac% into their compartments. Exercise 1' Static Su)straction
nvite the child to come and wor% with $ou. -rite a first number and a second number. ntroduce the new subtraction sign.
Andleeb javed
Module 7
Roll # D1081
+ave the child construct the first number. Tell the child that we are going to ta%e 9 units from the four units constructed. +ave the child move 9 units off to the left side of the table. ount how man$ units $ou have left and write the answer. +ave the child ta%e ! tens awa$ from the D and more them off to the side of the table. ount and then write how man$ tens are left. epeat for the hundreds and thousands. ead the answer with the child.
Exercise (' ,yna.ic Su)straction
-rite a first large number and a second number under it. a%e sure that this will lead to d$namic subtraction. +ave the child create the first number. As% the child how man$ units we are going to ta%e awa$7 9 units. ut as the child becomes stuc%, sa$ that we are going to have to change one of the tens for units. Ta%e out ten units and replace it with one of the ten tiles. Then have the child ta%e 9 units awa$ from the now 1! units. Place the unneeded tiles off to the side of the table.
Andleeb javed
Module 7
Roll # D1081
+ave the child write how man$ units he has left. epeat for the tens, hundreds, and thousands. hange when needed. ead the final problem with the answer with the child. epeat until the child feels comfortable to wor% alone.
