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FUNDAMENTAL OPERATIONS ON INTEGERS (Using Rathmell Model) DANILO S. SEVILLANO TAS, CLMD-LRMDS DepED Regional Office X
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FUNDAMENTAL OPERATIONS ON INTEGERS
(Using Rathmell Model)
DANILO S. SEVILLANOTAS, CLMD-LRMDS
DepED Regional Office X
Topics for Review
a. What is Rathmell Model?b. How does Rathmell Triangle look
like?c. How do we manipulate signed
chips?d. How are number items be
written?e. What does each symbol mean?f. How to show solution on each
item the proper way?
Topics for Review
g. What are the name Parts of each of the Four Basic Operations?
h. What are the rules for every operation given?
i. Do we need drill on rules?
Rathmell Model
Rathmell model was developed by Rathmell Pyne to explain real world situations into language, model and symbol.
Rathmell Triangle
LANGUAGE
SYMBOL
MODEL
CHIPS as MANIPULATIVE
+ means positive
- means negative
How are number items be written?
Hint:
• Each number item would be written in either horizontal or vertical manner.Example
s:
4
Vertical way:
Horizontal way:
3+
7
4 3+ = 7
What does each symbol mean?
Memorize the following symbols:
( ) + ( ) =
addition
subtraction
( ) ( ) =
_
there is plus or
minus sign in
between parenthesi
s
What does each symbol mean?
Memorize the following symbols:
( )( ) =
multiplication
there is no sign in
between parenthesi
s
How do we write our solutions?
Addition
4332
+
4332
+
75
Solution: addend
ssum or total
75
Example:
How do we write our solutions?
Subtraction
5424
_
542430
Solution: minuen
ddifference
30
Example:
subtrahend
_
How do we write our solutions?
Multiplication
4332
x4332
x
86129
1 376
Solution: multiplica
nd
partial product
Example:
product
multiplier
1 376
factors
How do we write our solutions?
Division
42
742
742x
Solution: quotient
Example:
dividend
divisor
=°°
66
Rules for Addition of Integers
Like Signs:
• We just add the numbers and copy the common sign as the sign of the final answer.
( + ) + ( + ) = +( ) ( ) =
+ _ _ _
Rules for Addition of Integers
Unlike Signs:
• We subtract the smaller number from the bigger number and copy the sign of the bigger number as the sign of the final answer.
( + ) + ( ) = +( ) ( + ) =
+ _ _ _
Rules for Subtraction of Integers
Originally
• We always change the sign of the subtrahend from positive to negative or from negative to positive – vice versa.
• Proceed to addition of integers.
( + ) ( + ) = ( + ) ( ) =
_ _ _
After changing
( + ) ( ) = ( + ) ( ) =
+ +
_
+
Rules for Multiplication of Integers
• The product of like signs is always positive
( + )( + ) = +( )( ) = +
_ _ ( + )( ) = ( )( + ) =
_ _
_ _
• The product of unlike signs is always negative
Unlike Signs:Like Signs:
• The quotient of like signs is always positive
Rules for Division of Integers
• The quotient of unlike signs is always negative
°°
( + ) ( + ) = +( ) ( ) = +
_ _ °°
( + ) ( ) = ( ) ( + ) =
_ _
_ _ °
°°°
Unlike Signs:Like Signs:
SUMMARY OF RULES ON INTEGERS
ADDITION SUBTRACTIONMULTIPLICATI
ON DIVISION
LIKE SIGNS
ADD THE NUMBERS AND COPY
THE COMMON
SIGN
(+)+(+)=(+)
(-)+(-)=(-)
CHANGE THE SIGN OF THE SUBTRAHEND
FROM POSITIVE TO NEGATIVE OR FROM NEGATIVE
TO POSITIVE
from (+)-(+) to (+)+(-)=(+)
THEN PROCEED TO ADDITION OF
INTEGERS
THE PRODUCT OR THE QUOTIENT OF TWO
NUMBERS WITH SAME SIGN IS ALWAYS POSITIVE
(-)(-)=(+)(+)(+)=(+)
(-)÷(-)=(+)(+)÷(+)=(+)
SUMMARY OF RULES ON INTEGERS
ADDITION SUBTRACTIONMULTIPLICATI
ON DIVISION
UNLIKE SIGNS
SUBTRACT THE
SMALLER NUMBER
FROM THE BIGGER AND
COPY THE SIGN OF THE
BIGGER NUMBER AS THE SIGN OF THE FINAL ANSWER
CHANGE THE SIGN OF THE SUBTRAHEND
FROM POSITIVE TO NEGATIVE OR FROM NEGATIVE
TO POSITIVE
from (+)-(+) to (+)+(-)=(+)
THEN PROCEED TO ADDITION OF
INTEGERS
THE PRODUCT OR THE QUOTIENT OF TWO
NUMBERS WITH UNLIKE SIGNS IS ALWAYS
NEGATIVE.
(-)(+)=(-)(+)(-)=(-)
(-)÷(+)=(-)(+)÷(-)=(-)
Cooperative Learning
Round Robin Brainstorming:a. Group yourselves equally.b. Same number should group
together.c. Make yourselves in circular form.
Do it now.d. Choose a leader who is
knowledgeable in all fundamental operations.
e. At the same time, choose a recorder to record all that has been discussed by the group.
Cooperative Learning
In step-by-step manner, thoroughly discuss among yourselves how the following operations on integers are done (30 minutes):
• Addition• Subtraction• Multiplication• Division
Cooperative Learning
f. Reporting will be done by group.g. Follow this group assignment:
• Group 1 – addition • Group 2 – subtraction• Group 3 – multiplication • Group 4 – division
EVERY MEMBER OF THE GROUP SHOULD TELL WHERE HIS DIFFICULTY LIES. THIS WOULD BECOME THE FOCUS OF THE DISCUSSION.
Addition of Like Signs
LANGUAGE MODEL SYMBOL
positive two plus positive
three
+
+
+
+
+
+ =
+
+
+
+
+(+2)
(+3)
+ = (+5)
Like Sign: (+2)
(+3)
+ = (+5)
Combining the chips of same color is the same as adding integers of same sign.
Rathmell Model
Addition of Like Signs
LANGUAGE MODEL SYMBOL
positive three plus positive
four
+
+
+
+ = (+3)
(+4)
+ = (+7)
+
++
+
+
+
++
+
+
+
(+3)
(+4)
+ = (+7)
Addition of Like Signs
LANGUAGE
MODEL SYMBOL
negative two plus negative
two
+ =-
-
-
-
-
-
-
-
Combining again the chips of same color is the same as adding integers of same sign.
(-2)
(-2)
+ = (-4)
(-2)
(-2)
+ = (-4)
Addition of Like Signs
LANGUAGE
MODEL SYMBOL
negative three plus
negative two
+ = (-3)
(-2)
+ = (-5)
-
--
-
-
-
-
-
-
-
(-3)
(-2)
+ = (-5)
Addition of Unlike Signs
LANGUAGE MODEL SYMBOL
positive three plus negative
two
+
+
+
+ =
+
+
+
-
-
-
-
Just pair one negative with one positive.Paired chips become zero.
(+3)
(-2)
+ = (+1)
(+3)
(-2)
+ = (+1)
Addition of Unlike Signs
LANGUAGE
MODEL SYMBOL
negative three plus
positive four
+ = + (-3)
(+4)
+ = (+1)
+-+
+ +
+-
-
-
Again by pairing one negative with one positive,paired chips become zero.
(-3)
(+4)
+ = (+1)
+-
+-
Addition of Unlike Signs
LANGUAGE
MODEL SYMBOL
negative four plus positive
four+ = (-
4)(+4)
+ = (0)
+
+
+
+
Just pairing one negative with one positivecancels out and becomes zero.
(-4)
(+4)
+ = ( 0 )
-
-
-
- +-
+-
+-
+-
Subtraction of Integers
LANGUAGE
MODEL SYMBOL
positive four minus
negative three
- =
+
+
+
+-
-
-
?
Originally(+4)
(-3)
- = ?
Always examine the subtrahend. Always use the opposite chips in the subtrahend. Then proceed to addition.
(+4)
(-3)
- = ?
Subtraction of Integers
LANGUAGE
MODEL SYMBOL
positive four minus
negative three
+ = (+4)
(+3)
+ = (+7)
+
+
+
+
Instead of white chips, we use red chips. Proceed to addition.
(+4)
(+3)
+ = (+7)
+
+
+
+
+
+
+
+
+
+
AFTER
Subtraction of Integers
LANGUAGE
MODEL SYMBOL
positive four minus
positive six- =
+
+
+
+ ?
Originally(+4)
(+6)
- = ?
Always examine the subtrahend. Always use the opposite chips in the subtrahend. Then proceed to addition.
(+4)
(+6)
- = ?+
+
+
+
+
+
Subtraction of Integers
LANGUAGE
MODEL SYMBOL
positive four plus
negative six+ = (+
4)(-6)+ = (-2)
+
+
+
+
Instead of red chips, we use white chips. Proceed to addition.
AFTER
-
-
-
-
-
-+
+
+
+
-
-
-
-
-
-
(+4)
(-6)+ = (-2)
Multiplication of Integers
What is multiplication?
Multiplication is the shortcut for repeated addition.
Multiplication of Integers
Example:
4 x 3 = ?
Sol. (Addition)4 + 4 + 4
= 12meaning 4 is being added by itself 3
times
Multiplication of Integers
Example:
5 x 4 = ?
Sol. (Addition)5 + 5 + 5 +
5 = 20meaning 5 is being added by itself 4
times
Multiplication of Integers
Example:
7 x 5 = ?
Sol. (Addition)7 + 7 + 7 + 7 + 7
= 35meaning 7 is being added by itself 5
times
Multiplication of Integers
IMPORTANT REMINDERS:
1. The multiplier will dictate the following:• How many groupings will be formed • What color of signed chips to be used.
2. When the factors have the same sign (both positive or both negative), always use the red chips (positive chips).
3. When the factors are of different signs, always use the white chips or the negative chips.
Multiplication of Integers
LANGUAGE
MODEL SYMBOL
positive four times
positive three
+ =
+
+
+
+
(+4)
(+3)
= ?
Notice that both factors have the same sign (+), therefore, we need to use positive chips.
How many times did we add groups of four?
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
(+4)(+3)(+1
2)
x
Multiplication of Integers
LANGUAGE
MODELSYMB
OLNegative five times negative
four+ =+
+
+
+
+
Notice that both factors have the same sign (-), therefore, we still need to use positive chips.
How many times did we add groups of five?
+
+
+
+
+
+ +
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
(-5)(-4)
(+20)
x
(-5)
(-4)
= ?
Multiplication of Integers
LANGUAGE
MODELSYMB
OLpositive
four times negative
three
+ =
Notice that both factors have different signs,therefore, we must always use negative chips.
How many times did we add groups of negative four?
-
-
-
-
+-
-
-
-
(+4)(-3)(-
12)
x-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
(+4)
(-3)
= ?
ALWAYS REMEMBER THAT WHEN
PERFORMING DIVISION OF INTEGERS, THE
RULES FOR MULTIPLICATION OF
INTEGERS APPLY.
Division of Integers
Division of Integers
LANGUAGE
MODEL SYMBOL
positive fourteen
divided by positive two
(+14)
(+2)= ?÷
+
+
+
+
+
+
+
+
+
+
+
+
+
+
(+14)(+2)
(+7)=
+
+
+
+
+
+
+
Let’s divide the minuend into groups of _____. How many groupings were made? _______
The number of groupings become our quotient.
Division of Integers
LANGUAGE
MODEL SYMBOL
negative twelve
divided by negative
two
(-12)
(-2)= ?÷
+
+
+
+
+
+
+
+
+
+
+
+
(-12)(-2)
(+6)=
+
+
+
+
+
+
Let’s divide the minuend into groups of _____. How many groupings were made? _______
The number of groupings become our quotient.
Division of Integers
LANGUAGE
MODEL SYMBOL
negative six divided by
positive two
(-6) (+2)= ?÷
(-6)(+2)
(-3)
=-
- - -
-
- -
- -
Let’s divide the minuend into groups of _____. How many groupings were made? _______
The number of groupings become our quotient.
Division of Integers
LANGUAGE
MODEL SYMBOL
positive six divided by negative
three
(+6) (-3)= ?÷
(+6)(-3)
(-2)
=-- - -
--- -
Let’s divide the minuend into groups of _____. How many groupings were made? _______
The number of groupings become our quotient.
Division of Integers
LANGUAGE
MODEL SYMBOL
negative nine divided by positive
three
(-9) (+3)= ?÷
(-9)(+3)
(-3)
=
-
-
-
-- -
-- -
-- -
Let’s divide the minuend into groups of _____. How many groupings were made? _______
The number of groupings become our quotient.
THANK YOU
