operations on integers (danilo s sevillano)
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FUNDAMENTAL OPERATIONS ON INTEGERS
(Using Rathmell Model)
DANILO S. SEVILLANOTAS, CLMD-LRMDS
DepED Regional Office X
Topics for Discussion:
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 Discussion:
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
+ positive chips
- negative chips
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
multiplier1
376
factors
How do we write our solutions?
Division
42
7 42
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 SUBTRACTION MULTIPLICATION 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 SUBTRACTION MULTIPLICATION DIVISION
UNLIKE SIGNSSUBTRACT
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.
(-)(+)=(-)(+)(-)=(-)
(-)÷(+)=(-)(+)÷(-)=(-)
Addition of Integers
LANGUAGE MODEL SYMBOLpositive two plus positive
three++
+++
+ =+++
++ (+
2)(+3)
+ = (+5)
(+2)
(+3)
+ = ?
Combining chips with the same color is the same as adding integers of the same sign.
Rathmell Model
Addition of Integers
LANGUAGE MODEL SYMBOL
positive three plus positive
four
+++
+ = (+3)
(+4)
+ = (+7)
++
++
+++
+++
+
(+3)
(+4)
+ = ?
Addition of Integers
LANGUAGE MODEL SYMBOL
negative two plus negative
two
+ =--
--
--
--
Combining again chips with the same color is the same as adding integers of the same sign.
(-2)
(-2)
+ = ?
(-2)
(-2)
+ = (-4)
Addition of Integers
LANGUAGE MODEL SYMBOL
negative three plus
negative two+ = (-
3)(-2)
+ = (-5)
--
--
---
--
-
(-3)
(-2)
+ = ?
Addition of Integers
LANGUAGE MODEL SYMBOLpositive three plus negative
two
+++
+ =++
+
--
--
Just pair one negative with one positive.Paired chips become zero pair.
(+3)
(-2)
+ = ?
(+3)
(-2)
+ = (+1)
Addition of Integers
LANGUAGE MODEL SYMBOL
negative three plus
positive four
+ = + (-3)
(+4)
+ = (+1)
+-++ +
+---
Again by pairing one negative with one positive,paired chips become zero pair.
(-3)
(+4)
+ = ?
+-+-
Addition of Integers
LANGUAGE MODEL SYMBOL
negative five plus positive
seven
+ =
+
(-5)
(+7)
+ = (+2)
+-++ +
+
+ ++
-----
(-5)
(+7)
+ = ?
+-+-+-+-
+
Addition of Integers
LANGUAGE MODEL SYMBOL
negative four plus positive
four+ = (-
4)(+4)
+ = (0)++
++
Just pairing one negative with one positivecancels out and becomes zero pair.
(-4)
(+4)
+ = ?
---
- +-+-+-+-
Subtraction of Integers
Please keep this in mind…In doing subtraction of integers, the following processes should be followed:1. Manipulation of the signed chips to show
logical explanation on how the operation goes.
2. Using addition instead of subtraction, taking into account the concept of opposites (meaning the reverse operation.)
Subtraction of Integers(+4)
(-3)
- = ?
What is the additive identity for addition? Always remember that any number added to zero equals the number.
LANGUAGE MODEL SYMBOL
positive four minus
negative three
++
++ =
+++
+
++
+
(+7)
(+4)
(-3)=-+- -+
+
++
+
-+-+
+++
--- -+
LANGUAGE MODEL SYMBOL
positive four minus positive six
Subtraction of Integers
++
++
(+4)
(+6)
- = ?
=+++
+++
+
++
+
(-2)
(+4)
(+6)
=-+- -+
+
++
+
-+-+
---
--
-
---
---
Watch out! When the operation was changed, what happened to the chips’ color (subtrahend)? What did we do next?
LANGUAGE MODEL SYMBOL
negative four minus
negative seven
Subtraction of Integers
--
--
(-4)
(-7)- = ?
=
+++
+++
+-
--
-
(+3)
(-4)
(-7)=-+-
-+-
--
-
-+
-+-+
+++
+++
+
---
---
-
LANGUAGE MODEL SYMBOL
negative eight minus
negative five
Subtraction of Integers
--
--
----
(-8)
(-5)- = ?
=+++
++
-
--
---
(-3)
(-8)
(-5)=-
+- -
--
--
--
-
+++
--++
---
--
Multiplication of Integers
What is multiplication?
Multiplication is the shortcut for repeated addition.
Multiplication of IntegersExample:
4 x 3 = ?Sol.
(Addition)3 + 3 + 3 + 3 = 12meaning 3 is being
added to itself 4 times
Multiplication of IntegersExample:
5 x 4 = ?Sol.
(Addition)4 + 4 + 4 + 4 + 4 = 20meaning 4 is being
added to itself 5 times
Multiplication of IntegersExample:
7 x 5 = ?
Sol. (Addition)5 + 5 + 5 + 5 + 5 +
5 + 5 = 35meaning 5 is being added to itself 7
times
Multiplication of Integers
Please keep this in mind…1. The multiplicand will dictate the following:
• The number of groupings to be formed.
• Gives the signal either to keep (when positive) or to take away (when negative).
2. The multiplier will determine what to keep and what to take away.
Definition of Multiplication
please memorize me…
( N)+- ( N)+-
(+) it reminds uson what to keep
(-) it also reminds us on what to take away
use positive chips
use negative chips
multiplicand multiplier
Let N be any real number under the set of Integers
Multiplication of Integers
LANGUAGE MODEL SYMBOL
positive four times
positive three
(+4)
(+3)
= ?
Opposite pair is very important in this operation.Take note of the word keep (+) and take away (-).
+++
(+4)(+3)(+1
2)
x+++
+++
+++
---
---
---
---
Multiplication of Integers
LANGUAGE MODEL SYMBOL
negative two times negative
four
(-2)
(-4)
= ?
In multiplying integers with the same sign, what is the resulting sign of the product?
What conclusion can we make?
++++
(-2)(-4)
(+8)
x++++
----
----
Multiplication of Integers
LANGUAGE MODEL SYMBOL
negative two times positive
three
(-2)
(+3)
= ?
In multiplying integers with the different signs, what is the resulting sign of the product?
What conclusion can we make?
+++
(-2)(+3)(-6)
x+++
---
---
Multiplication of Integers
LANGUAGE MODEL SYMBOL
positive three times
negative four
(+3)
(-4)
= ?
++++
(+3)(-4)(-12)
x++++
++++
----
----
----
In multiplying integers with the different signs, what is the resulting sign of the product?
What conclusion can we make?
Division of Integers
Please keep this in mind…1. The dividend determines what
signed chips to be used.2. The sign of the divisor determines
either to form its opposite or not. When the sign is positive as ease, if the sign is negative use the opposite.
Division of Integers
please memorize me…
( N)+-
it reminds us what color of the chips or
signed chips to be used
it reminds us not to change the chip’s color or signed chips
it reminds us to change the chip’s color or sign
dividend divisor÷( N)+-
Let N be any real number under the set of Integers
Division of Integers
LANGUAGE MODEL SYMBOL
positive fourteen
divided by positive two
(+14)
(+2)= ?÷
++
+
+
+
++
++
+
+
+
++
(+14)(+2)
(+7)=++
+
+
+
++
What’s the sign of the dividend? What color or signed chips do we need to use? What is the sign of the divisor? Do we need to form its opposite or not?
Let’s divide the dividend into groups of _____. How many groupings are formed? _______
Division of Integers
LANGUAGE MODEL SYMBOL
negative twelve
divided by negative
two
(-12)
(-2)= ?÷
--
-
---
--
-
---
(-12)(-2)
(+6)=++
+
+++
++
+
+++
What’s the sign of the dividend? What color or signed chips do we need to use? What is the sign of the divisor? Do we need to form its opposite or not?
Let’s divide the dividend into groups of _____. How many groupings are formed? _______
Division of Integers
LANGUAGE MODEL SYMBOL
negative six divided by
positive two
(-6) (+2)= ?÷
(-6)(+2)
(-3)
=-
- - -
-
- -
- -
What’s the sign of the dividend? What color or signed chips do we need to use? What is the sign of the divisor? Do we need to form its opposite or not?
Let’s divide the dividend into groups of _____. How many groupings are formed? _______
Division of Integers
LANGUAGE MODEL SYMBOL
positive six divided by negative
three
(+6) (-3)= ?÷
(+6)(-3)
(-2)
=++ +
++ +
-- -
-- -
What’s the sign of the dividend? What color or signed chips do we need to use? What is the sign of the divisor? Do we need to form its opposite or not?
Let’s divide the dividend into groups of _____. How many groupings are formed? _______
Division of Integers
LANGUAGE MODEL SYMBOL
negative nine divided by positive
three
(-9) (+3)= ?÷
(-9)(+3)
(-3)
=
-
-
-
-- -
-- -
-- -
What’s the sign of the dividend? What color or signed chips do we need to use? What is the sign of the divisor? Do we need to form its opposite or not?
Let’s divide the dividend into groups of _____. How many groupings are formed? _______
THANK YOU
More Exercises
Perform the following indicated operation:
1. (+2) + (+3) = 2. (+5) + (-7) =3. (-4) + (-8) =4. (-6) + (+7) =5. (+8) + (-9) =
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Perform the following indicated operation:
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Perform the following indicated operation:
16.-5 (-7) =17.(14) ÷ (-2) =18.(-18) ÷ (3) =19.(+15)(-3) =20.(16)(-4) =
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