1. Module 7
Mathematical exercises (part 2)
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2. Name: rabia shahid Roll no: D6409
Question 1: Explain different groups of Montessori math exercises and how the directress should efficiently
present exercises through sequential and parallel work in various groups.
Numbers through Ten.
Decimal System,
Counting beyond Ten linear and skip counting.
memorization of Arithmetic tables
Passage to Abstractions
Fractions
The Exercises in arithmetic are grouped. There is some sequential work and some parallel work.
The first group is Numbers through Ten. The experiences in this group are sequential. When the
child has a full understanding of numbers through ten,
The second group, The Decimal System, can be introduced. 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.
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.
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.
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.
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.
Q.2: Explain the exercises which enable the child to count till 1000?

3. 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
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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 child’s 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.

4. 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
righCounting 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)
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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:
Material
A thousand chain consisting of 100 bars of 10
Ten squares of hundred
The thousand cube containers having arrow labels:
Green labels marked 1 – 9
Blue labels arrowed 10 – 990
Red labels from 100 – 900
A large green label marked 1,000
A large sized mat or runner
Method
Tell 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.

5. 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
Dot game
Ans.
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 A
Invite 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 and addition problem and write one on the right side of the
grid.

6. 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.)
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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 one 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 catergory, 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 child’s attention on the process of carrying.
- To further familiarize the child with the different categories.
- A first abstraction in the decimal System.

7. Q.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 rabia shahid.
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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.
Purpose
To give practice in multiplication leading to the memorization of the essential
multiplication tables.
Age
5 1/2 - 6 years
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
Chart 1

8. Presentation
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 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 Error
The child checks his work with Chart 1.
Q.5: How is stamp game introduced to the child? Also explain how subtraction problems can be solved with
stamp game.
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

9. Special grid paper
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 1’s 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: Subtraction
Invite 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.
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Tell the child that we are
going to take 3 units from

10. 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 more 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.

11. Special grid paper
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 1’s 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: Subtraction
Invite 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.
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Tell the child that we are
going to take 3 units from