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Assignment 7 Question 1 Explain different different groups of Montessori math exercises and how the directress should efficiently present exercises through sequential and parallel work in various groups. E!"#$A% MA&E#$A% $ MA&'MA&$(A% MA&E#$A% The Sensorial Material is mathematical material. It is exact. It is presented with exactness and will be used by the child with exactness. The activities call for precision so that the child can come into contact with the isolated concepts and through repetition, draw from the essence of each and have a clear abstraction. These concepts help the child to order his mind. He is able to classify experience. Clear perception and the ability to classify leads to precise conclusions. The Sensorial wor is a preparation for the study of se!uence and progression. It helps the child build up spatial representations of !uantities and to form images of their magnitudes such as the "in Tower. Tower. The #xercises in arithmetic are grouped. There is some se!uential wor and some parallel wor. )#"*+ 1 ,!*M-E# &'#"*)' &E! The experiences experiences in this group are se!uential. se!uential. $hen the child has a full understandi understanding ng of numbers numbers through through ten%numbers ten%numbers rods&sand rods&sand paper numbers&spi numbers&spindle ndle box&cards box&cards and counter&gold counter&golden en and coloured beds bar'. In this group, the child builds the basic concept of numbers, but also recogni(es the relationships between !uantity and numerals. )#"*+ /,0E($MA% &EM The decimal system can be introduced when the child has a full understanding of numbers through )*. The focus here is on the hierarchy of the decimal system and how the system functi functions ons.. It also also starts starts the child child on the exerci exercises ses of simple simple comput computati ations ons,, which which are the operati operations ons of arithm arithmeti etic. c. It has four four operati operations ons of arithm arithmeti eticc .i.e.a .i.e.addi dditio tion, n, multip multiplic licati ation, on, subtraction and division. They are introduced at this level. )#"*+ 2,("*!&$!) -E"!0 &E! The third group will be started when the decimal system is well underway. underway. +rom then on, these exercises will be given parallel to the continuing of the decimal system. This third group, Counting beyond Ten, includes the teens, the tens, and linear and sip counting.
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Roll#6721
)#"*+ 3,MEM"#$4A&$"! "5 A#$&'ME&$( &A-%E The fourth group is the memori(ation of the arithmetic tables. This wor can begin while the later wor of the decimal system and the counting beyond ten exercises are continued. )#"*+ 6,+AA)E &" A-A(&$"! The fifth group is the passage to abstraction. The #xercises in this group re!uire the child to understand the process of each form of arithmetic and to now the tables of each operation. There is again an overlap. The child who nows the process and tables for addition can begin to do the addition for this group. He may still be woring on learning the tables for the other operations and these will not be taen up until he has the readiness. The exercises in the group for passing to abstraction, allows the child child to drop the use of the material material as he is ready. ready. He can then begin to wor more and more with the symbols on paper, without using the material to find the answers. )#"*+ ,5#A(&$"! The sixth sixth group group of materi materials als,, fracti fractions ons,, can wor wor parall parallel el to the group group of maing maing abstractions and the early wor with the fractions can begin with sensorial wor.
Question / Explain the exercises which ena8le the child to count till 1999: iner exercises helps the child learn to count till till )*** ,along with getting familiar familiar with the decimal system relationships, including the concepts of s!uares and c ubes of numbers. inear counting is presented in two stages. In the first stage, the child learns to count till )**,and in the second stage he masters counting till )***. +urpose To consolidate the child-s child-s nowledge of counting. p until now, now, he wored with tens and hundreds in the decimal system. $ith these exercises, he becomes familiar with the se!uence of numbers from ) through ),***. Counting is a restful activity and tends to become mechanical. Through repetition, the child establishes the mechanism of co unting. $hen the two chains are placed parallel parallel to each other, other, they show in a striing and sensorial way the difference between the s!uare and the cube of ten. In this way wa y, the decimal system relationships are further established by the child.
+resentation 1; 2
Aima Ijaz
Roll#6721
The hundred chain consisting of )* bars of )*. The hundred s!uare Containers having arrow labels/ 0 1reen labels mared ) 2 3 0 4lue labels mared )* 2 3* 0 5 red label mared )** 0 5 large si(ed mat or runner. The )** Chain 4ring the child to the chain cabinet. Show the child the bars on the shelves and discuss with the child if he has seen bars lie these before. 4egin counting with the child starting from the unit to the )* bar. Have the child unroll the runner 6ust a little ways. Show the child how to hold the )** chain by both ends and have him lay it vertically at the bottom of the mat. Have him place the tray below the )** chain. Slowly fold the chain together to create the hundred s!uare. 7otice that it loos lie the hundreds s!uare. "lace the hundreds s!uare on top of the folded ten chain to show that they are the same. 8emove the hundred s!uare and have the child gently re0straighten the ten chain. Tae out the unit ticets %green' and tell the child what they are called. ine them in a vertical line to the left of the ten chain. Show the child the ten ticets %blue' and place in a vertical line above the unit ticets. abel the first ten by using the unit ticets and placing them on the left of the chain. Count with the child ))09*. 5t the 9* mar, place the ticet that has 9* on it to the right, counting by units: continue placing the ten ticets until you reach )**. Have the child place the red )** ticet next to the )**. Tell the child/ ;= %)**' "oint to the hundred s!uare, ;5nd how many are in this>= %)**' Count with the child all of the ticets/ ), 9, ?, @, A, B, , D, 3, )*, 9*, ?*, @*, A*, B*, *, D*, 3*, and )**. Then count bacwards/ )**, 3*, D*, *, B*, A*, @*, ?*, 9*, )*, 3, D, , B, A, @, ?, 9, ). Have the child replace the ticets into their correct envelop and then replace the rest of the material of the 9* bead. +resentation /; Material 5 thousand chain consisting of )** bars of )* Ten s!uares of hundred The thousand cube containers having arrow labels/ 1reen labels mared ) 2 3 3
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Roll#6721
4lue labels arrowed )* 2 33* 8ed labels from )** 2 3** 5 large green label mared ),*** 5 large si(ed mat or runner Method Tell the child that today we are going to loo at an even longer chain than the )** chain. Have the child unroll the runner all the way. Show the child how to hold the )*** chain. The directress carries the chain to the runner, with all of the strands laid out straight. Have the child bring over the cube and the large box on a tray over to the runner. 5lso bring over the hundred s!uares. Tell the child that you are going to try to fold the chain 6ust lie you did with the )** chain. Mae a hundreds and as the child what you made. "lace a hundred s!uare next to the one you 6ust made. 8epeat until the whole chain has been folded in hundred s!uares. %The child can begin to mae them after a while' "lace each of the hundred s!uares next to the hundred s!uare you have made with the child. Then place the hundred s!uares on top of the hundred s!uares you and the child have made. Count with the child to see how many hundred s!uares there are. Have the child place each hundred s!uare on top of each other. 7otice that it loos 6ust lie the cube. $hen we have )* hundred s!uares, we now that we have )*** beads. "lace the cube next to the ten hundred s!uares %placed on top of one another' to show this to the child. Have the child gently pull the ),*** chain straight. %Have him eep the chain near the left side of the runner'. Have the child lay out all of the ticets. Count each bead and place the correct ticet when needed as in "resentation ). $hen you get to )**, place the ticet as well as a hundred s!uare next to the )**th bead. 8epeat this for every hundred. %#ven at the ),***th bead' 5t the ),***th bead, also place the cube. Stand at the beginning of the runner and wal all the way to the end. Stand at the end and loo at the wor of the child. 1o bac to the beginning and count/ )**, 9**, ?**, @**, A**, B**, **, D**, 3**. 5s the child how many he had at the end/ )***. 1o bac to the beginning and count the tens. )*, 9*, ?*, @*, A*, E )**, ))*, )9*, E @**, @)*, @9*, E 3D*, 33*, )***.Then have the child count by tens bacwards. 5s the labels have to be placed at the end of each bar, the child easily perceives he has made a mistae in counting. 4
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Then child can then put the material away.
Question 2 +rint <0ot )ame paper ,from the link given 8elow and send three solved pro8lems= each carrying four addends. 0ot game
Materials 0 S!uared paper inserted into a frame of ground glass or slate with columns headed ), )*, )**, 5
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Roll#6721
),***, and )*,***. The columns are divided into small s!uares so that there are ten in each hori(ontal row. 5t the foot of each column are two spaces, the upper one for carrying figures, the lower one for the result. There is a blan column at the right side where the problem to be done is written. 0 5 good lead pencil 0 5 purple or orange pencil 0 5 ruler
+resentation tage A Invite a child to come and wor with you. Introduce him to the new paper and have him bring it over to the table. Show the child the different columns on the paper and introduce the child to the new number of )*,***. Tell the child you are going to write an addition problem and write one on the right side of the grid. Have the child choose at least three more @0digit numbers. Fnce all add0ins have been written, draw a line with the ruler and write in a plus sign. oo at the first number and write a dot in the units column for each unit in the first number. 8epeat for the tens, hundreds and thousands. 8epeat for each add0in until the whole grid is filled with the appropriate amount of do ts. Then count the first row of dots in the units from left to right. $hen you get to ten dots, cross it out and mae an orange dot in the first bottom large s!uare. 5s you do so, say/ ;This represents one ten.= Continue counting the units in this same way. %Crossing off each ten units and maring with an orange dot.' $rite the number of units left in the second bottom s!uare. oo at how many orange dots you have in the units column. Mar that amount in a number in the tens column. Tell the child, ;I am carrying over 9 tens.= 5lso place two orange dots next to the last pencil dot in the tens column. 8epeat in this way for the tens column, the thousand, and the )* thousand columns. 5lways carrying over what needs to be. 8ead the answer with the child, emphasi(ing the ten0thousand number. #.g. Thirty0two thousand, one hundred and fifty two. Have the child write the answer under the problem on the right side of the paper and show the child where we place the comma to separate the thousands. 8ead the whole problem with the child.
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tage This is to be done in the same way as in Stage 5, but this time have the child mae the dots for all of the units, then all of the tens, then all of the hundreds, and then all of the thousands. This is to be done from the top unit to the bottom unit.
+urpose 0irect 0 To give the child further understanding of addition in the decimal system and to give him a sense of an ability to wor with large numbers. 0 To emphasi(e the fact that in each category, there are never combinations that come to more than 3, so that it is 6ust as easy to add tens of thousands together as it is units. 0 The maing of tens focuses on the child-s attention on the process of carrying. 0 To further familiari(e the child with the different categories. 0 5 first abstraction in the decimal System.
Question 3 Explain the presentations of Multiplication 8oard and 0ivision 8oard in your own words. Also make illustrations.
The multiplication bead board is used for practice with the multiplication tables )x) though )*x)*. The box consists of a perforated multiplication woring with )** holes in rows of ten arranged in a s!uare, a box with small plastic cards numbering )0)* which represent the multiplicand, a red disc which mars the multiplier and a box of )** red beads. 5t the left side of the board is a window with a slot for the insertion of the cards. +urpose To give practice in multiplication leading to the memori(ation of the essential multiplication tables. 5ge A )&9 0 B years Materials 5 perforated board with )** holes in rows of )* arranged in a s!uare. 5t the left side of the board is a window with a slot for the insertion of the cards. 5 red, wooden disc. 7
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Tables of multiplication 5 set of cards from ) to )* tamp )ame
Materials 0 arge !uantities of wooden s!uares of e!ual si(e about ) inch s!uare lie stamps/ #ach stamp of ) is green mared with G)-. #ach stamp of )* is blue mared with G)*-. #ach stamp of )** is red mared with G)**-. #ach stamp of )*** is green mared with G)***-. 0 5 pencil and ruler 0 Special grid paper !otes This material is more symbolic, so this wor is moving from the concrete to the more abstract. $ith this material, we will introduce writing the problem and will therefore introduce the symbol for writing the problem. This wor will be all individual. Introduction Invite the child to come and wor with you. Show 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 tray from Introduction to uantity. Show the child the ) green tile and show the ) unit to the child. Tell the child that it is the same as the unit bead. Show the child the blue tile and have him read the G)*- written on it. Tell the child that this is 6ust lie the ten0bar. 8epeat for the tiles of )** and )***. o a Three "eriod esson with the ), )*, )**, and )*** tiles. Show the child that when we tae out the ) tiles, we place them directly in front of the compartment where the other )-s are. Tell the child that you are going to tae out A. Tae out A of the ) tiles and place them all in front of the ) compartment. "ut them bac and give the child a few numbers to tae out. Such as mae ? tens, or A hundreds, or 9 thousands. Then give the child a larger number. Say, ;7ow we are going to mae a larger number. This number will have ? units, A tens, 9 hundreds, ) thousand. 5s you give the child each number, have him tae out the appropriate tiles. 8
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Count to chec the final product and then have the child put the tiles bac into their compartments. +resentation 1; Addition To be done directly following the Introduction. Static 5ddition Show the child the paper on which we write our problems. Tell the child that the first column is where we write the units. The second column is where we write the tens, the third column is where we write the hundreds, and the fourth column is where we write the thousands. $rite a number, such as )A9@ and read it with the child as/ @ units, 9 tens, A hundreds, and ) thousand. Then read it/ )A9@. Have the child create the number using the tiles. Tell the child that we are going to mae another number. Show the child that you will write this new number below the first number on the piece of paper. $rite/ )9@) and read it with the child as before. Show the child that we will place the tiles for this number a little below the other tiles. Have the child create this number using the tiles.
Tell him that we will see how much we have all together. Tell the child that we show this by using the addition sign. Show the child the sign and where to place it on the paper. Then draw a line under the last number using the ruler. Have the child count all of the units/ @ J ) K A $rite in A under the units on the paper. Have the child count the tens, hundreds, and thousands, each time writing the answer down. 9
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8ead the final answer with the child/ $hen we have )A9@ and we add )9@) we get 9BAL
5llow the child a turn with another example. 1uide him with !uestions. ynamic 5ddition Have the child construct and write the first add0in, first the units, tens, hundreds, and then thousands. Have the child write another add in, but guide the child so that there will be a need to change the numbers. Have the child construct the two numbers using the tiles. Count all of the tiles and notice that you are going to need to chang e some of the tiles. Have the child do (hart 1
+resentation 10
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Show the child the material and have him bring it to the table. Show the child the numbers along the top of the board. Tell the child, ;These numbers tell us how many times to tae a number.= Show the child how to slide the card %@' into the slot on the side of the board. Tell the child, ;This tells us we will be doing the table of @.= "lace the little red disc above the ) at the top of the board. Say, ;This tells us we need to tae @ one times. sing the red beads, place @ one times in a vertical line. Have the child count how many beads there are on the board. Tell the child, ;@ x ) is @= Have the child write the answer on the paper next to the e!uation. Move the disc over above the 9. Tell the child, ;$e now need @ two times. 4ut we already have @ one times.= Have the child place the red beads in a vertical line next to the first four. Have the child count the total number of beads on the board. Say, ;@ x 9 is D=. 8epeat in this manner. $hen the child reaches @ x @, have him say the e!uation with you. If the child is maing the table with ease, when he reaches @ x D show him that @ x was 9D. Count from 9D up four more. 8epeat in this way until he has finished the board. Have the child read all of the e!uations and answers written on the piece of paper. The child can chec his wor on Multiplication Chart ). Control of #rror The child checs his wor with Chart ).
Question 6 'ow is the stamp game introduced to the child: Also explain how su8traction pro8lems can 8e solved with the stamp game. tamp)ame
Materials arge !uantities of wooden s!uares of e!ual si(e about ) inch s!uare lie stamps/ #ach stamp of ) is green mared with G)-.
#ach stamp of )* is blue mared with G)*-. #ach stamp of )** is red mared with G)**-. #ach stamp of )*** is green mared with G)***-. 5 pencil and ruler 11
Aima Ijaz
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Special grid paper $ntroduction Invite the child to come and wor with you. Show 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 tray from Introduction to uantity. Show the child the ) green tile and show the ) unit to the child. Tell the child that it is the same as the unit bead. Show the child the blue tile and have him read the G)*- written on it. Tell the child that this is 6ust lie the ten0bar. 8epeat for the tiles of )** and )***. o a Three "eriod esson with the ), )*, )**, and )*** tiles. Show the child that when we tae out the ) tiles, we place them directly in front of the compartment where the other )-s are. Tell the child that you are going to tae out A. Tae out A of the ) tiles and place them all in front of the ) compartment. "ut them bac and give the child a few numbers to tae out. Such as mae ? tens, or A hundreds, or 9 thousands. Then give the child a larger number. Say, ;7ow we are going to mae a larger number. This number will have ? units, A tens, 9 hundreds, ) thousand. 5s you give the child each number, have him tae out the appropriate tiles. Count to chec the final product and then have the child put the tiles bac into their compartments. +resentation /; u8traction Invite the child to come and wor with you. $rite a first number and a second number. Introduce the new subtraction sign.
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Aima Ijaz
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Have the child construct the first number.
Tell the child that we are going to tae ? units from the four units constructed. Have the child move ? units off to the left side of the table. Count how many units you have left and write the answer. Have the child tae 9 tens away from the A and move them off to the side of the table. Count and then write how many tens are left. 8epeat for the hundreds and thousands. 8ead the answer with the child.
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u8traction $rite a first large number and a second number under it. Mae sure that this will lead to dynamic subtraction
Have the child create the first number. 5s the child how many units are we going to tae away/ ? units. 4ut as the child becomes stuc, say that we are going to have to change one of the tens for units. Tae out ten units and replace it with one of the ten tiles. Then have the child tae ? units away from the now )9 units. "lace the unneeded tiles off to the side of the table. 14
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Have the child write how many units he has left. 8epeat for the tens, hundreds, and thousands. Change when needed. 8ead the final problem with the answer with the child.
8epeat until the child feels comfortable to wor alone. so.
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8ead the finished problem with the child.
5llow the child a turn with another example. 1uide him with !uestions. Exercise The child wors alone, creating his own problems. +resentation /; u8traction tatic u8traction Invite the child to come and wor with you. $rite a first number and a second number. Introduce the new subtraction sign. 16
Aima Ijaz
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Have the child construct the first number.
Tell the child that we are going to tae ? units from the four units constructed. Have the child move ? units off to the left side of the table. Count how many units you have left and write the answer. Have the child tae 9 tens away from the A and move them off to the side of the table. Count and then write how many tens are left. 8epeat for the hundreds and thousands. 8ead the answer with the child.
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0ynamic u8traction $rite a first large number and a second number under it. Mae sure that this will lead to dynamic subtraction.
Have the child create the first number. 5s the child how many units we are going to tae away/ ? units. 4ut as the child becomes stuc, say that we are going to have to change one of the tens for units. Tae out ten units and replace it with one of the ten tiles. Then have the child tae ? units away from the now )9 units. "lace the unneeded tiles off to the side of the table.
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Have the child write how many units he has left. 8epeat for the tens, hundreds, and thousands. Change when needed.
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8ead the final problem with the answer with the child.
8epeat until the child feels comfortable to wor alone. Exercise The child wors alone, creating his own problems. +resentation 2; Multiplication tatic Multiplication Invite the child to come and wor with you. Create a problem.
Introduce the new multiplication symbol to the child. ; This is a new symbol for multiplication. It is called the Gtimes- symbol.= 8ead the problem with the child/ 9)9? times ?. Have the child create 9)9?. 8ead with the child and say, ;
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Roll#6721
Have the child create 9)9? three times.
Slide all of the tiles from the same category up together to create only four rows. Have the child count all the units and then fill in the answer on the paper. 8epeat for the tens, hundreds, and thousands.
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0ynamic Multiplication
Invite a child to come and wor with you. Create a problem
8ead it/ 9B?A times A. Mae mention that we are going to tae this number A timesL Tae out the sittles and place them in a vertical line with ample space between them. Have the child create 9B?A to line up with the first sittle. Have the child create 9B?A four more times. Have the child count the total amount of units, changing when needed. Count the total amount of tens, changing when needed. Count the total amount of hundreds, changing when needed. Fnce done, have the child count what is left and write the answer on the piece of paper.
Exercise The child wors alone as shown in the presentation 22
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!ote 5lthough there is a limitation in the material, the child may multiply any number he wishes to. +resentation 2; 0ivision tatic 0ivision Invite a child to come and wor with you and have him bring the material to the table. Create a problem.
Introduce the two new symbols to the child. Have the child create ?B?B using the tiles. Show the child the green sittle and place them in a row to the right of the tiles. #xplain to the child that we are going to give each sittle the same amount of tiles. Tell the child that when we divide, we always start with the thousands. ;1ive= each sittle a thousand tile. 7otice that there are no more )*** tiles to give. ;1ive= each sittle a )** tile. 7otice that there are more )** tiles. 1ive each sittle another )** tile.
1ive each sittle a )* tile and notice that there are no more left to give. 1ive each sittle a ) tile until there are none left. 23
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Tell the child/ ;In division, we always loo at what one gets, so lets see how many one sittle got.= Count what one sittle got, writing in the units, the tens, the hundreds, and then the thousands.
0ynamic 0ivision; no remainder Invite a child to come and wor with you and have him bring the material to the table. Create a problem/ BA9A&A Have the child create BA9A using the tiles. remind the child that when we divide, we start with the thousand. ;1ive= each sittle a thousand tile. 7otice that we have ) left over. Change it for ten hundred tiles. ;1ive= each sittle a hundred until there are no more hundred tiles. 7otice that there are none left after we gave each sittle ? hundreds. 7otice that there are not enough ten tiles to give to each sittle. Change one ten tile into )* unit tiles and then change the second ten tile into )* unit tiles. 1ive each sittle a unit tile until there are none left. Count how many tiles one sittle has and write down the number each number at a time. 0ynamic 0ivision; remainder Create a problem for the child/ B?D&A The procedure for this exercise is the same as in the above procedure for ivision with no remainder except for the end. 7otice with the child that there are some unit tiles left and not enough to give to each sittle. Tell the child that these tiles are called a ;remainder=. Show the child how to write the remainder.
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0ynamic 0ivision; / level divisor Invite a child to come and wor with you and have him bring the material to the table. (reate a pro8lem;
Have the child create @AD? using the tiles. iscuss how many units there are in )9 K 9 units. "lace two green sittles at the top of the table. iscuss how many tens there are in )9 K ) ten. "lace one blue sittle at the top of the table to the left of the green sittles. 4egin by giving the blue sittle a tile of a thousand. iscuss with the child that if the blue sittle gets a thousand tile, the green tiles must each get a hundred tile because the green tile is ten times less than the blue tile. 1ive each green sittle a hundred tile for every thousand tile you give to the blue sittle. $hen there are two thousand tiles left but on ly a hundred tile, change one of the thousand tiles for ten hundreds. 1ive the last thousand to the blue sittle and two hundred tiles to the two green sittles.
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Continue in this manner, giving the green sittles ten times less that what you give the blue sittle. Fnce all of the tiles that can be shared are shared, remind the child that in division, we always loo at what one person gets. Have the child count the tiles for one of the green sittles, writing in how many units, tens, hundreds, and thousands one sittle gets and have the child write in the remainder.
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Aima Ijaz
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0ynamic 0ivision; 2 level divisor This is to be done in the same way as ivision with a 9 level divisor. The hundred should be represented by a red sittle. 0ynamic 0ivision; 2 level divisor with a >ero in the tens 5s above but since there are no tens, ;eep= the place of the ten by placing a blue circle where the sittle would have gone. Tell the child that this ;holds= the tens spot. +rom time to time, as the child what the ten would get if it was not a (ero.
Move the circle down before beginning to give out a new grouping of tiles. %The child should answer that he would get )** times less than the hundred sittle.' See diagram for finished product/
0ynamic 0ivision; 2 level divisor with a >ero in the units 5s above but since there are no units, ;eep= the place of the unit by placing a green circle where the sittle would have gone. Tell the child that this ;holds= the unit-s spot. +rom time to time, as the child what the unit would get if it was not a (ero.
Move the circle down before beginning to give out a new grouping of tiles. %The child should answer that the unit would get )* times less than the ten sittle.' Fnce the operation is complete, remind the child that in division we always loo at what one gets so you will need to divide what a ten sittle got into ten unit sittles. See diagram for finished product/
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Exercise The child can wor alone, creating his own division problems to solve as he was shown in the presentations. +urpose irect To give the child the opportunity of carrying out individual exercises in the four operations. "reviously, he needed the collaboration of other children to do these operations with the bead material of the decimal system. ong division 5s above, and so become more familiar with all the steps involved in long division. The use of this more symbolic material helps the child to move closer towards abstraction. (ontrol of Error The child-s growing nowledge. Age To A )&9 years for addition, subtraction, multiplication and short division. To B years for long division.
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