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SHAHER BANO
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| PMC Module 5 Mathematical Exercises (Part 1) Solved assignment |
Write answers in your own words. Your answers should be at
least 2 typed pages or 3 handwritten pages. Please don’t forget to write your
name and roll number on each page of the assignment.
Q1:
How would you teach numbers 0 to 10 to a child according to Montessori Method?
Explain all the exercises in this group briefly in your own words.
Answer No 1
Number
Rods: There are 10 wooden rods varying in length
from 1 decimeter (10 centimeters) to 1
meter (100 centimeter). Each decimeter is painted in red and blue colour
section. The shortest rod that is 1 decimeter is red. Second rod which is 2
decimeter long , is colored red and blue, and so on.
·
Exercise 1:
Introduction to rods:
Material:
10 number rods and a floor mat.
Presentation:
The directress starts by reminding
children the use of the long rods. She then informs them that there are rods
similar to long rods save for the fact that they are red and blue in colour.
She then encourages the children to arrange the rods in similar way as the long
rods with red ends on the left and evenly lined.
·
Exercise 2: Learning to count from 1 to 10:
Material:
10 number rods and a floor mat.
Presentation:
The directress takes the first three rods
and points to the first rod and says ”This is the rod of one.” She repeats for emphasis and does the same
for the other two rods as well by calling them by their respective numerical
names. She proceeds by carrying out Period 2 and 3 of the Three Period Lesson
to familiarize children with the numbers, after which she carries on by
processing to 4 and so on until all 10 numbers are attempted.
·
Exercise 3: Sandpaper Numbers:
Material:
Numbers from 0 to 9 , cut out of the
finest grade sandpaper and mounted on wooden or acrylic green cards. (Number
“0” is skipped in this exercise and presented after working with the Spindle Boxes)
Presentation:
The teachers begin by sensitizing fingers
and introducing the children to the material. She then takes out Number 1, traces it with her fingers and pronounce it as
being “One” , asking the child to repeat after her. She repeats this exercise
for number 2 and 3, moving on to doing Three Period Lesson for them before
progressing towards the rest of the numbers.
·
Exercise 4: The Number Rods and the Numerals:
Material:
The number rods, a set of white or acrylic cards with numbers from 1 to 10
and a floor mat.
Presentation:
The directress starts off by placing the Number Rods as well as the
cards onto the mat. She then points at a red rod and ask the child to pronounce
it’s numerical name as well as bring over the relevant number card and place it
next to it. She repeats until the exercise has been repeated for all the rods
and numbers.
·
Exercise 5: The Spindle Boxes:
Material:
A wooden box with ten compartments. At the
back of each compartment is painted a number in black, starting from 0 up to 9, as well as 45 wooden
spindles.
Presentation:
The directress introduces the materials to
the child. She points at the compartments as well as the numbers each and asks
the child to name the numbers. She explain to the child that these numbers will
tell us how many spindles to put into the box. She starts this part of the
exercise by pointing to the number “1” and having the child read it out loud
and then ask him to put “1” spindle in the box. She repeats for all the
numbers, at the end of which, she points at compartment “0” and says “This is
Zero. Zero means nothing, that’s why
there is nothing in this
compartment.”
·
Exercise 6: Number Cards and Counters:
Material:
Number cards with numbers from 1 to 10 and
55 counters of same colour and size.
Presentation:
The directress shows different cards to the children
and has them say the numbers aloud. She places the number 1 card to the left
side and the number 10 card to the right side of the table. She asks the child
to put the other cards in order. She tell the child that he is going to put the
number of counters under the corresponding card. For card 2, she ask to place
the counters next to each other. For
card 3, She ask to put two counters next to each other but place the last
counter under and to the middle of the two counters. She makes sure that the child place the rest of the counters in a similar
way as she has shown. The exercise continues until all of the counters have
been placed. She then runs her finger through the counters that are laid,
places her index finger above the first counter under card 1 and tries to runs it down. When the
finger hits the counter she says ‘odd”. She repeats for the counter 2 and after
running her finger through the two
counters, she says “even” . She repeats it for the rest of the numbers.
After finishing the first period, she then
asks the child to show her the odd and
even number.
These exercises help reinforce the
concept that each number is made up of
different quantities and all this
learning will serve as a strong base for further mathematics in their academia.
Q2;
What do you know about the decimal
system? How would you enable children to count any quantity and identify
numerals till 9999?
Answer No 2
The
Decimal System:
The decimal system is a numeral system
which organizes and classifies numerical quantities into different hierarchies
of units. And lays a strong foundation for all future math. It is introduced to
the children when they have mastered
counting from 1 to 10, and can recognize the properties of zero as well as the
number 1 to 9. The child is given the total decimal system in a clear and
simple manner with real materials that illustrates the difference between one
unit and one thousand etc. The Montessori approach uses the decimal system materials to introduce addition,
multiplication, subtraction and division as well . The children learn the
operations using numbers in thousands, but
it is easy for them because of the concrete objects and order of the
lessons. They are learning place value from a very early age, but it is in
simple intervals that makes it
approachable.
Geometrical entities are used by
Montessori as Material abstractions for the decimal system of numeration.
Material:
·
A
single golden bead.
·
A
ten bar.
·
A
hundred square.
·
A
thousand cube.
·
Table
mat.
1 golden bead is a ‘unit /point’
10 golden beads make a ‘bar of ten’
10 bars of 10 make ‘hundred square’
10 hundred squares make a ‘thousand cube’
Exercise:
Q3:
Explain addition and multiplication exercises in your own words?
Answer No 3
ADDITION:
Addition is the mathematic operation in
which smaller quantities (addends) are put together to make a larger quantities
(the sum). There are two types of addition.
1.
Addition Without Exchanging:
Materials:
·
Golden
beads Bank
·
1
set of Large Number Cards
·
3
sets of Small Numbers Cards
·
3
trays and small containers for carrying units.
·
A
floor mat.
Exercise:
First the directress invites few group of children to come and work with her,
starting with laying down a mat and
gathering the material on it. One child lays out the large cards while another lays out the beads. The children
place small mats between the two large mats
and set up their set of small cards similar to how they setup the large
cards,
Arrange the bank of
Golden Beads Material in proper sequence on a green felt mat on a table in
correct order. Think of quantities to be added and write each quantity on a separates slip of paper e.g.
1345+2234. Hand over each slip to a
child asking them to build their number with small number cards. When children built their numbers with
small numbers cards take the slip back and
ask them to go to the bank and
bring the quantity of beads according to their numbers. Once they have confirm
that they have brought right number of
beads. Take the small number
cards of first child and placed them out on the mat. Then ask the child to
place the bead material under the
numbers correctly. Repeat the same with the second children. Take the first set
of numbers (1345) and place aside. Take
the second set of numbers (2234) and place
under the first set as if composing an addition equation. Introduce
the addition sign “+” drawn on a square piece of cardboard and place beside the
equation. Place a ruler or a strip
of paper under the number to make a complete addition equation. Put both sets
of material together. Ask one child to count the units and bring the
appropriate number from large number cards. Place the unit card under the
equation. And do so on for the tens, hundreds and thousands also. Finally tell the children ,” We started with
2234. We added 1345 and our answer is 3579.”
Or can say you have added two
small numbers and made one larger number. At the end record show the children
hw to record the answer.
1.
Addition With Exchange:
The presentation begins exactly as in addition without exchanging but have the children take cards for problem
where they will have to carry over. These numbers could be, 3567+ 2787+
1345.
Exercise:
Repeat
all the steps unto the point of
placement of units, tens, hundred and thousands to the side of the
directress tray. When the first child counts the units and reaches 10, point
this out and have him exchange ten units
for a ten bar. Have him count the rest of the units and then go get the card
for that amount. Repeat for the tens, hundreds and thousands changing when
needed. Finish the exercise as for Addition Without Exchanging.
MULTIPLICATION:
Multiplication means adding the same
number again and again. It can be introduced at any time after children have
learned addition.
Exercise:
This exercise should be done in a group format. Gather the children and arrange
all the relevant material just as done
for addition exercise. Select any quantity to multiply and write it on a piece
of paper as many times as we want to
multiply the respective number. The
quantity selected should be such
that the sum of their product does not
exceed 9999 and does not involve exchanging, for example 2121 three times. Pass
one slip to each child and instruct them
to place it on the tray upside down and not
to show their number with small number cards.
Once they have build their numbers with
small cards, retrieve the slip from them and ask the children to go and
bring the quantity of beads that
corresponds to their respective numbers. After making sure the children have brought the correct number
of beads, take the small number cards of the first child and place on the mat.
Then ask that children to take the bead material and place correctly under the
numbers, repeating the same with the other two quantities.
Invite one child to start counting the beads beginning with the units. When the units are added
together if the number is more than 10,
the children are reminded that they can exchange the 10 units beads with the
ten bar at the bank. This ten bar is placed
on the top of “ten “ column . When there are less than 10 units left,
the child is asked to brig the corresponding large number card and placed it
under the equation. Repeat the same for tens, hundreds and thousands also
Tell the children, “2121 three times
is equal to 6363. When we added the same
number over and over again this is called multiplication.” Or can
say that you have multiply a smaller number and made one larger number. At the end, show
the children how to record the answer.
Q4:
Explain how would you give the concept of subtraction and division?
Answer No 4
SUBTRACTION:
Subtraction
means taking away smaller quantities from a larger quantity. Large quantities
are referred to as “minuend” and smaller quantities as “subtrahends”. The subtrahends are smaller
than minuend and finding the difference
between them is called subtraction.
Ø Subtraction
Without Exchanging:
Material:
·
3 sets of small number cards
·
1
set of large number cards
·
3
trays and 3 bowls for units beads
·
A
large tray with an extra bowl
Ø Subtraction
With Exchanging:
Material:
Same as addition.
Presentation:
Material is arranged in the same way as in the exercise above. The directress
writes the minuend and the Subtrahend on two slips, 4755 and 3888. Minuend slip
is given to one child, small number cards are built and beads are placed with
the numbers. Subtrahend number slip is
given to the other child and numbers are built.
The directress asked the second child to take away the beads quantity
equivalent to the subtrahend from the minuend beads. The child will
realize that the subtrahend unit number
is larger than the minuend. The directress will suggest exchanging ten unit
beads with a bar of ten, he will have
twelve beads from which he can take away three. It will go with tens where available beads two as he has already
exchanged while he needs four. The teacher will again suggest exchanging tens
bar of ten with a square of hundred whereby he will be left with twelve from
which he will take away four and so on. Finally, all the beads are counted and
children place the corresponding number card as answers.
DIVISION:
Division is spitting a quantity into equal
parts or groups. There are two values in a division sum:
a.
Quantity
to be divided (dividend)
b.
The
number by which another number is to be divided (divisor)
Ø Division
Without Exchanging:
Material:
·
Golden
Beads Bank
·
1
set of Large Number Cards
·
2
to 3 sets of Small Number Cards
·
3
trays and containers for carrying units
·
A
floor mat
Ø Division
With Exchanging:
Material:
As
for ‘Division without Exchanging’
Presentation:
The directress and two children for the
exercise. She will think of a dividend and a divisor, so that the sum involves
exchanging e.g. 7654/ 2. She will write the dividend on a paper slip, give it
to the child and ask him to build the
number using large number cards and brings the beads quantity. She will then place the material and large number cards on the floor
mat, as she tells the children that she
has 7654 and divide it between both children . She will start the division
from a thousand cubes, giving two cubes
two each child making the each children realize that one thousand is left. She will ask them about what they
should do next and wait for their reply. A child will suggest exchanging it with ten hundred square. By
doing so, she gets 16 hundred squares
which she will equally divide between them. She will repeats it with tens
and units. When the quantity is equally divided, she will ask the children to build their numbers using small number cards.
Each child has 3827. She takes the small cards and says that when she divided
7654 between two children , each got 3827 and nothing is left.
Ø Division
With Remainder:
Material:
As for ‘Division Without Exchanging’
Presentation: The directress
thinks of a division sum that will leave
a reminder, 457/ 3. She writes a
dividend on a paper slip and hands it over to a child, asking him to build the
number using large number cards an bring
the beads quantity. She then arranges it onto the mat. She tells the children
that that she is going to divide 457 equally among three children. She starts
with the hundred squares where 1 hundred is left. She exchange it with ten bars
and then divides 15 tens among them.
Each child gets 5 bars of tens and finally, she starts dividing the 7 beads
unit. Each child gets 2 unit beads where
1 unit is left. She explains that she does not have enough units for everyone ,
and this be called a “remainder” . She
then ask the child to build their amount using small number cards and
each will have 152. She takes the small number cards from the tray, put them
above the dividend and says that she had 457, which she divided equally among
three children so each child got 152, while 1 was a reminder. The directress
can reinforce the terms, dividend, divisor, quotient and remainder as many
times as she deems appropriate.
Answer No 5
Teens and tens boards teach the child the
number names, symbols and sequence from 10 to 99. They are ideal for use with
Montessori beads. Teens boards have two wooden
boards with 9 number slots each
labeled with 10 and the child counts up sliding
the wooden digit cards 1 to 9 into these slots. The numbers 11 to 19 are
particularly difficult for a child to learn as their names are more complicated
than those of the rest of the number system. The teens board helps to develop a
true understanding of how these numbers
are formed from a ten and unit, and thus teaches the foundation of the decimal system. Tens wooden boards have 9
number slots of 10 to 90 and the child
counts up sliding the wooden digit cards 1 to 9 into these slots.
There are three groups of names:
1. Names for a combination of a ten and units 1 to 9, these are ‘teens’
2. Names for a group of ten: 10, 20, 30 etc
3. Names for figures from the tens category
and a unit, these help with linear counting 11-99 teens(beads only).
Ø Coloured
Beads Stair:
Material:
·
A
small felt cloth on a working mat.
·
9
bars of ten Golden Beads representing the units 1-9 and box for each of these.
Each quantity is distinguished by a different colour:
1. Red
2. Green
3. Pink
4. Yellow
5. Light blue
6. Grey or violet
7. White
8. Brown
9. Dark blue
Presentation:
Show the material to the child, removing one bead bar at a time ask the child
to identify the numbers of beads in each bar
at random, make reference to the colour and provide a three period lesson if
necessary. Sort the beads bars into an isosceles triangle, known as a Bead
stair.
Three
Period Lesson:
First
Period: Take
the bar of ten and place the unit to the right of it, adjacent
to the first bead. Count the beads and say, “ One ten and one unit are called
‘eleven’. Repeat the sequence for ‘twelve’ and ‘thirteen’.
Second
Period: Mix all the previously introduced beads
bar and invite the child to make the
numbers, continue mixing to maintain the child’s interest.
Third
Period: Make a quantity and ask the child to name
it . Begin each subsequent three Period Lesson counting up from eleven.
Ø 11-19
Teens Board (cards only)
Material
Description:
·
Two
wooden slated boards with five partitions each, on nine of the partitions a
large 10 is written in black, the last partition is empty.
·
Unit
wooden cards with the digits 1 to 9 which slide into the boards from the right,
covering the “0’.
·
A
mat
Presentation:
Place the board on the working mat and the
cards at random nearby and give the symbols in a Three Period Lesson.
First
period: Slip the ‘1’ over the first ten, saying,
“This is eleven”, do the same with ‘twelve’ and “thirteen”.
Second
period: Ask the child to identify previously
introduced numbers by moving the cards and mixing them , ask the child to make
a number using the cards and boards.
Third
Period: Make a number with the cards and ask the
child to identify it. Continue till 19 on the same day or later, depending on
the child. When complete ask the child to count forwards and backwards.
Direct
Aim: To introduce the child to the symbols for
the numbers 11 to 19 and to continue to associate their names.
Ø Boards
and Beads:
Material:
·
Short
Beads Stair
·
9
bars of tens in a box
·
Teen
Boards
·
A
mat
Presentation:
Lay out the boards on the mat, with the
cards placed at random to the right, and the beads, in a Beads Stair, to the
left , the tens in their box. Place a ‘bar of ten’ and a bead to form eleven to
the left of the top section of the board and slip the card of ‘1’ over the
‘0’ to form the figure ‘11’. Place a
‘bar of ten’ and two beads to form twelve to the left of the top section of the
board and slip the card of ‘2’ over the
‘0’ to form the figure ‘12’. Let the child continue till reaches 19.
When she completes ask her to count
forward and backwards.
Direct
Aim: Continued association of the quantity, name and symbol for 11 to 19, to
rein force the sequence 11 to 19.
Ø 11-99
Tens Boards and Beads:
Material
Description:
·
Two
wooden slated boards with 5 partition each , on nine of the partitions are the
tens numbers 10, 20, etc, the last partition is empty, later use
·
Large
cards.
·
Units
wooden cards with the digits 1 to 9 which slide into the boards from the right,
the ‘0’.
·
45
bars of tens in a box
·
A
mat
Presentation:
Layout the boards on the working mat,
place the boxes with the beads to the left. Place one bar of ten . Indicate
“20” beneath and the child names
however she likes, say, “Twenty also
means two tens”, continue this till reach 90”. Give a Three Period Lesson for
any of the names the child is unfamiliar with.
Direct
Aim:
·
To
learn the conventional names of the tens from 10 to 90.
·
To
realize how the numbers progress from one ten to the next and to see the
pattern in making and counting numbers up to 99.
Ø 11-99
Teens Boards and Beads:
Material
: Same as above mentioned.
Direct
Aim:
·
To
learn the conventional names of the tens from 10 to 90
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