Aima Ijaz Roll#6721

Aima Ijaz Roll#6721

Assignment 7

Question 1Explain different groups of Montessori math exercises and how the directress should efficiently present exercises through sequential and parallel work in various groups. SENSORIAL MATERIAL IS MATHMATICAL MATERIAL 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 work is a preparation for the study of sequence and progression. It helps the child build up spatial representations of quantities and to form images of their magnitudes such as the Pink Tower.

The Exercises in arithmetic are grouped. There is some sequential work and some parallel work.GROUP 1 (NUMBERS THROUGH TEN)

The experiences in this group are sequential. When the child has a full understanding of numbers through ten(numbers rods/sand paper numbers/spindle box/cards and counter/golden and coloured beds bar). In this group, the child builds the basic concept of numbers, but also recognizes the relationships between quantity and numerals.GROUP 2(DECIMAL SYSTEM)The decimal system can be introduced when the child has a full understanding of numbers through 10. The focus here is on the hierarchy of the decimal system and how the system functions. It also starts the child on the exercises of simple computations, which are the operations of arithmetic. It has four operations of arithmetic .i.e.addition, multiplication, subtraction and division. They are introduced at this level. GROUP 3(COUNTING BEYOND TEN) The third group will be started when the decimal system is well underway.From 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 skip counting.GROUP 4(MEMORIZATION OF ARITHMETIC TABLES) The fourth group is the memorization of the arithmetic tables. This work can begin while the later work of the decimal system and the counting beyond ten exercises are continued.

GROUP 5(PASSAGE TO ABSTRACTION) The fifth group is the passage to abstraction. The Exercises in this group require the child to understand the process of each form of arithmetic and to know the tables of each operation. There is again an overlap.The child who knows the process and tables for addition can begin to do the addition for this group. He may still be working on learning the tables for the other operations and these will not be taken up until he has the readiness. The exercises in the group for passing to abstraction, allows the child to drop the use of the material as he is ready. He can then begin to work more and more with the symbols on paper, without using the material to find the answers.GROUP 6(FRACTIONS) The sixth group of materials, fractions, can work parallel to the group of making abstractions and the early work with the fractions can begin with sensorial work.

Question 2

Explain the exercises which enable the child to count till 1000?

Liner exercises helps the child learn to count till 1000 ,along with getting familiar with the decimal system relationships, including the concepts of squares and cubes of numbers. Linear counting is presented in two stages. In the first stage, the child learns to count till 100,and in the second stage he masters counting till 1000.Purpose To consolidate the childs knowledge of counting. Up until now, he worked with tens and hundreds in the decimal system. With these exercises, he becomes familiar with the sequence of numbers from 1 through 1,000. Counting is a restful activity and tends to become mechanical. Through repetition, the child establishes the mechanism of counting. When the two chains are placed parallel to each other, they show in a striking and sensorial way the difference between the square and the cube of ten. In this way, the decimal system relationships are further established by the child.

Presentation 1:The hundred chain consisting of 10 bars of 10.The hundred square

Containers having arrow labels:- Green labels marked 1 9- Blue labels marked 10 90- A red label marked 100- A large sized mat or runner.

The 100 Chain

Bring the child to the chain cabinet.

Show the child the bars on the shelves and discuss with the child if he has seen bars like these before.

Begin counting with the child starting from the unit to the 10 bar.

Have the child unroll the runner just a little ways.

Show the child how to hold the 100 chain by both ends and have him lay it vertically at the bottom of the mat.

Have him place the tray below the 100 chain.

Slowly fold the chain together to create the hundred square.

Notice that it looks like the hundreds square.

Place the hundreds square on top of the folded ten chain to show that they are the same.

Remove the hundred square and have the child gently re-straighten the ten chain.

Take out the unit tickets (green) and tell the child what they are called. Line them in a vertical line to the left of the ten chain.

Show the child the ten tickets (blue) and place in a vertical line above the unit tickets.

Label the first ten by using the unit tickets and placing them on the left of the chain.

Count with the child 11-20. At the 20 mark, place the ticket that has 20 on it to the right, counting by units; continue placing the ten tickets until you reach 100. Have the child place the red 100 ticket next to the 100. Tell the child: You have just counted to 100.

Ask, How many beads are in this chain? (100) Point to the hundred square, And how many are in this? (100) Count with the child all of the tickets: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100.

Then count backwards: 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.

Have the child replace the tickets into their correct envelop and then replace the rest of the material of the 20 bead.Presentation 2:

MaterialA thousand chain consisting of 100 bars of 10Ten squares of hundredThe thousand cube containers having arrow labels:Green labels marked 1 9Blue labels arrowed 10 990 Red labels from 100 900 A large green label marked 1,000 A large sized mat or runner

MethodTell the child that today we are going to look at an even longer chain than the 100 chain.

Have the child unroll the runner all the way.

Show the child how to hold the 1000 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. Also bring over the hundred squares.

Tell the child that you are going to try to fold the chain just like you did with the 100 chain.

Make a hundreds and ask the child what you made. Place a hundred square next to the one you just made.

Repeat until the whole chain has been folded in hundred squares. (The child can begin to make them after a while)

Place each of the hundred squares next to the hundred square you have made with the child. Then place the hundred squares on top of the hundred squares you and the child have made.

Count with the child to see how many hundred squares there are.

Have the child place each hundred square on top of each other.

Notice that it looks just like the cube. When we have 10 hundred squares, we know that we have 1000 beads.

Place the cube next to the ten hundred squares (placed on top of one another) to show this to the child.

Have the child gently pull the 1,000 chain straight. (Have him keep the chain near the left side of the runner).

Have the child lay out all of the tickets.

Count each bead and place the correct ticket when needed as in Presentation 1. When you get to 100, place the ticket as well as a hundred square next to the 100th bead. Repeat this for every hundred. (Even at the 1,000th bead)

At the 1,000th bead, also place the cube.

Stand at the beginning of the runner and walk all the way to the end. Stand at the end and look at the work of the child.

Go back to the beginning and count: 100, 200, 300, 400, 500, 600, 700, 800, 900.

Ask the child how many he had at the end: 1000.

Go back to the beginning and count the tens. 10, 20, 30, 40, 50, 100, 110, 120, 400, 410, 420, 980, 990, 1000.Then have the child count by tens backwards.

As the labels have to be placed at the end of each bar, the child easily perceives he has made a mistake in counting.

Then child can then put the material away.Question 3

Print *Dot Game paper (from the link given below) and send three solved problems, each carrying four addends.Dot game

Materials- Squared paper inserted into a frame of ground glass or slate with columns headed 1, 10, 100, 1,000, and 10,000. The columns are divided into small squares so that there are ten in each horizontal row. At the foot of each column are two spaces, the upper one for carrying figures, the lower one for the result. There is a blank column at the right side where the problem to be done is written. - A good lead pencil- A purple or orange pencil- A ruler

Presentation

Stage AInvite a child to come and work 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 10,000.

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 4-digit numbers.

Once all add-ins have been written, draw a line with the ruler and write in a plus sign.

Look at the first number and write a dot in the units column for each unit in the first number.

Repeat for the tens, hundreds and thousands.

Repeat for each add-in until the whole grid is filled with the appropriate amount of dots.

Then count the first row of dots in the units from left to right. When you get to ten dots, cross it out and make an orange dot in the first bottom large square. As you do so, say: This represents one ten.

Continue counting the units in this same way. (Crossing off each ten units and marking with an orange dot.)

Write the number of units left in the second bottom square.

Look at how many orange dots you have in the units column. Mark that amount in a number in the tens column. Tell the child, I am carrying over 2 tens.

Also place two orange dots next to the last pencil dot in the tens column.

Repeat in this way for the tens column, the thousand, and the 10 thousand columns. Always carrying over what needs to be.

Read the answer with the child, emphasizing the ten-thousand number. E.g. Thirty-two 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.

Read the whole problem with the child.Stage B This is to be done in the same way as in Stage A, but this time have the child make 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.

Purpose

Direct- To give the child further understanding of addition in the decimal system and to give him a sense of an ability to work with large numbers.- To emphasize the fact that in each category, there are never combinations that come to more than 9, so that it is just as easy to add tens of thousands together as it is units.- The making of tens focuses on the childs attention on the process of carrying.- To further familiarize the child with the different categories.- A first abstraction in the decimal System.

Question 4

Explain the presentations of Multiplication board and Division board in your own words. Also make illustrations. The multiplication bead board is used for practice with the multiplication tables 1x1 though 10x10. The box consists of a perforated multiplication working with 100 holes in rows of ten arranged in a square, a box with small plastic cards numbering 1-10 which represent the multiplicand, a red disc which marks the multiplier and a box of 100 red beads. At the left side of the board is a window with a slot for the insertion of the cards.PurposeTo give practice in multiplication leading to the memorization of the essential multiplication tables.

Age5 1/2 - 6 yearsMaterialsA 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

Stamp Game

Materials- Large quantities of wooden squares of equal size about 1 inch square like stamps:Each stamp of 1 is green marked with 1.Each stamp of 10 is blue marked with 10.Each stamp of 100 is red marked with 100.Each stamp of 1000 is green marked with 1000.- A pencil and ruler- Special grid paper

NotesThis material is more symbolic, so this work is moving from the concrete to the more abstract.With this material, we will introduce writing the problem and will therefore introduce the symbol for writing the problem. This work will be all individual.

Introduction

Invite the child to come and work 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 Quantity.

Show 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.

Show the child the blue tile and have him read the 10 written on it. Tell the child that this is just like the ten-bar.

Repeat for the tiles of 100 and 1000.

Do a Three Period Lesson with the 1, 10, 100, and 1000 tiles.

Show the child that when we take out the 1 tiles, we place them directly in front of the compartment where the other 1s are.

Tell the child that you are going to take out 5. Take out 5 of the 1 tiles and place them all in front of the 1 compartment.

Put them back and give the child a few numbers to take out. Such as make 3 tens, or 5 hundreds, or 2 thousands.

Then give the child a larger number.

Say, Now we are going to make a larger number. This number will have 3 units, 5 tens, 2 hundreds, 1 thousand.

As you give the child each number, have him take out the appropriate tiles.

Count to check the final product and then have the child put the tiles back into their compartments.

Presentation 1: AdditionTo be done directly following the Introduction.

Static Addition

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.

Write a number, such as 1524 and read it with the child as: 4 units, 2 tens, 5 hundreds, and 1 thousand. Then read it: 1524.

Have the child create the number using the tiles.

Tell the child that we are going to make another number.

Show the child that you will write this new number below the first number on the piece of paper.

Write: 1241 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: 4 + 1 = 5

Write in 5 under the units on the paper.

Have the child count the tens, hundreds, and thousands, each time writing the answer down.

Read the final answer with the child: When we have 1524 and we add 1241 we get 2765!

Allow the child a turn with another example. Guide him with questions.

Dynamic Addition

Have the child construct and write the first add-in, 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 change some of the tiles. Have the child do Chart 1

PresentationShow 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 take a number.

Show the child how to slide the card (4) into the slot on the side of the board.

Tell the child, This tells us we will be doing the table of 4.

Place the little red disc above the 1 at the top of the board.

Say, This tells us we need to take 4 one times.

Using the red beads, place 4 one times in a vertical line.

Have the child count how many beads there are on the board.

Tell the child, 4 x 1 is 4 Have the child write the answer on the paper next to the equation.

Move the disc over above the 2.

Tell the child, We now need 4 two times. But we already have 4 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, 4 x 2 is 8.

Repeat in this manner. When the child reaches 4 x 4, have him say the equation with you.

If the child is making the table with ease, when he reaches 4 x 8 show him that 4 x 7 was 28. Count from 28 up four more. Repeat in this way until he has finished the board.

Have the child read all of the equations and answers written on the piece of paper.

The child can check his work on Multiplication Chart 1.

Control of ErrorThe child checks his work with Chart 1.

Question 5

How is the stamp game introduced to the child? Also explain how subtraction problems can be solved with the stamp game.StampGame

MaterialsLarge quantities of wooden squares of equal size about 1 inch square like stamps:Each stamp of 1 is green marked with 1.

Each stamp of 10 is blue marked with 10.Each stamp of 100 is red marked with 100.Each stamp of 1000 is green marked with 1000.A pencil and rulerSpecial grid paper

IntroductionInvite the child to come and work 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 Quantity.

Show 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.

Show the child the blue tile and have him read the 10 written on it. Tell the child that this is just like the ten-bar.

Repeat for the tiles of 100 and 1000.

Do a Three Period Lesson with the 1, 10, 100, and 1000 tiles.

Show the child that when we take out the 1 tiles, we place them directly in front of the compartment where the other 1s are.

Tell the child that you are going to take out 5. Take out 5 of the 1 tiles and place them all in front of the 1 compartment.

Put them back and give the child a few numbers to take out. Such as make 3 tens, or 5 hundreds, or 2 thousands.

Then give the child a larger number.

Say, Now we are going to make a larger number. This number will have 3 units, 5 tens, 2 hundreds, 1 thousand.

As you give the child each number, have him take out the appropriate tiles.

Count to check the final product and then have the child put the tiles back into their compartments.Presentation 2: SubtractionInvite the child to come and work with you.

Write a first number and a second number. Introduce the new subtraction sign.

Have the child construct the first number.

Tell the child that we are going to take 3 units from the four units constructed.

Have the child move 3 units off to the left side of the table.

Count how many units you have left and write the answer.

Have the child take 2 tens away from the 5 and move them off to the side of the table. Count and then write how many tens are left.

Repeat for the hundreds and thousands.

Read the answer with the child.

Subtraction

Write a first large number and a second number under it. Make sure that this will lead to dynamic subtraction

Have the child create the first number.

Ask the child how many units are we going to take away: 3 units. But as the child becomes stuck, say that we are going to have to change one of the tens for units. Take out ten units and replace it with one of the ten tiles.

Then have the child take 3 units away from the now 12 units. Place the unneeded tiles off to the side of the table.

Have the child write how many units he has left.

Repeat for the tens, hundreds, and thousands. Change when needed. Read the final problem with the answer with the child.

Repeat until the child feels comfortable to work alone.

so.

Read the finished problem with the child.

Allow the child a turn with another example. Guide him with questions.ExerciseThe child works alone, creating his own problems.Presentation 2: Subtraction

Static SubtractionInvite the child to come and work with you.

Write a first number and a second number. Introduce the new subtraction sign.

Have the child construct the first number.

Tell the child that we are going to take 3 units from the four units constructed.

Have the child move 3 units off to the left side of the table.

Count how many units you have left and write the answer.

Have the child take 2 tens away from the 5 and move them off to the side of the table. Count and then write how many tens are left.

Repeat for the hundreds and thousands.

Read the answer with the child.

Dynamic SubtractionWrite a first large number and a second number under it. Make sure that this will lead to dynamic subtraction.

Have the child create the first number.

Ask the child how many units we are going to take away: 3 units. But as the child becomes stuck, say that we are going to have to change one of the tens for units. Take out ten units and replace it with one of the ten tiles.

Then have the child take 3 units away from the now 12 units. Place the unneeded tiles off to the side of the table.

Have the child write how many units he has left.

Repeat for the tens, hundreds, and thousands. Change when needed.

Read the final problem with the answer with the child.

Repeat until the child feels comfortable to work alone.

ExerciseThe child works alone, creating his own problems.Presentation 3: Multiplication

Static MultiplicationInvite the child to come and work with you.

Create a problem.

Introduce the new multiplication symbol to the child. This is a new symbol for multiplication. It is called the times symbol.

Read the problem with the child: 2123 times 3.

Have the child create 2123.

Read with the child and say, Yes, this is 2123. But we want 2123 three times. Lets see you make this number a total of three times!

Have the child create 2123 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.

Repeat for the tens, hundreds, and thousands.

Dynamic MultiplicationInvite a child to come and work with you.

Create a problem

Read it: 2635 times 5. Make mention that we are going to take this number 5 times!

Take out the skittles and place them in a vertical line with ample space between them.

Have the child create 2635 to line up with the first skittle.

Have the child create 2635 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.

Once done, have the child count what is left and write the answer on the piece of paper.

ExerciseThe child works alone as shown in the presentationNoteAlthough there is a limitation in the material, the child may multiply any number he wishes to.Presentation 3: Division

Static DivisionInvite a child to come and work 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 3636 using the tiles.

Show the child the green skittle and place them in a row to the right of the tiles. Explain to the child that we are going to give each skittle the same amount of tiles.

Tell the child that when we divide, we always start with the thousands.

Give each skittle a thousand tile.

Notice that there are no more 1000 tiles to give.

Give each skittle a 100 tile. Notice that there are more 100 tiles. Give each skittle another 100 tile.

Give each skittle a 10 tile and notice that there are no more left to give.

Give each skittle a 1 tile until there are none left.

Tell the child: In division, we always look at what one gets, so lets see how many one skittle got.

Count what one skittle got, writing in the units, the tens, the hundreds, and then the thousands.

Dynamic Division: no remainderInvite a child to come and work with you and have him bring the material to the table.

Create a problem: 6525/5

Have the child create 6525 using the tiles.

remind the child that when we divide, we start with the thousand.

Give each skittle a thousand tile.

Notice that we have 1 left over. Change it for ten hundred tiles.

Give each skittle a hundred until there are no more hundred tiles. Notice that there are none left after we gave each skittle 3 hundreds.

Notice that there are not enough ten tiles to give to each skittle. Change one ten tile into 10 unit tiles and then change the second ten tile into 10 unit tiles.

Give each skittle a unit tile until there are none left.

Count how many tiles one skittle has and write down the number each number at a time.Dynamic Division: remainderCreate a problem for the child: 6738/5

The procedure for this exercise is the same as in the above procedure for Division with no remainder except for the end.

Notice with the child that there are some unit tiles left and not enough to give to each skittle. Tell the child that these tiles are called a remainder. Show the child how to write the remainder.

Dynamic Division: 2 level divisorInvite a child to come and work with you and have himbring the material to the table.Create a problem:Have the child create 4583 using the tiles.

Discuss how many units there are in 12 = 2 units.

Place two green skittles at the top of the table.

Discuss how many tens there are in 12 = 1 ten.

Place one blue skittle at the top of the table to the left of the green skittles.

Begin by giving the blue skittle a tile of a thousand. Discuss with the child that if the blue skittle 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.

Give each green skittle a hundred tile for every thousand tile you give to the blue skittle.

When there are two thousand tiles left but only a hundred tile, change one of the thousand tiles for ten hundreds. Give the last thousand to the blue skittle and two hundred tiles to the two green skittles.

Continue in this manner, giving the green skittles ten times less that what you give the blue skittle.

Once all of the tiles that can be shared are shared, remind the child that in division, we always look at what one person gets. Have the child count the tiles for one of the green skittles, writing in how many units, tens, hundreds, and thousands one skittle gets and have the child write in the remainder.

Dynamic Division: 3 level divisorThis is to be done in the same way as Division with a 2 level divisor. The hundred should be represented by a red skittle.

Dynamic Division: 3 level divisor with a zero in the tensAs above but since there are no tens, keep the place of the ten by placing a blue circle where the skittle would have gone. Tell the child that this holds the tens spot. From time to time, ask the child what the ten would get if it was not a zero.

Move the circle down before beginning to give out a new grouping of tiles. (The child should answer that he would get 100 times less than the hundred skittle.) See diagram for finished product:

Dynamic Division: 3 level divisor with a zero in the unitsAs above but since there are no units, keep the place of the unit by placing a green circle where the skittle would have gone. Tell the child that this holds the units spot. From time to time, ask the child what the unit would get if it was not a zero.

Move the circle down before beginning to give out a new grouping of tiles. (The child should answer that the unit would get 10 times less than the ten skittle.) Once the operation is complete, remind the child that in division we always look at what one gets so you will need to divide what a ten skittle got into ten unit skittles. See diagram for finished product:

ExerciseThe child can work alone, creating his own division problems to solve as he was shown in the presentations.PurposeDirectTo give the child the opportunity of carrying out individual exercises in the four operations. Previously, he needed the collaboration of other children to do these operations with the bead material of the decimal system.

Long divisionAs 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.Control of Error

The childs growing knowledge.AgeTo 5 1/2 years for addition, subtraction, multiplication and short division.To 6 years for long division.3