Operations on Integers (danilo s sevillano)

FUNDAMENTAL OPERATIONS ON INTEGERS (Using Rathmell Model)

DANILO S. SEVILLANO TAS, CLMD-LRMDS DepED Regional Office X

Topics for Discussion: a. b. c. d. e. f.

What is Rathmell Model?

How does Rathmell Triangle look like?

How do we manipulate signed chips?

How are number items be written?

What does each symbol mean?

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

MODEL

LANGUAG E

SYMBOL

CHIPS as MANIPUL ATIVE

+

positive chips

-

negative chips

How are number items be written?

Hint: • Each number item would be written in either horizontal or vertical manner.

Examples: Vertical way: Horizontal way: 4 +3 7

4 + 3 = 7

What does each symbol mean?

Memorize the following symbols: ( ) + ( ) = addition ( )_ =

( ) subtraction

there is plus or minus sign in between parenthesis

What does each symbol mean?

Memorize the following symbols:

( )( ) = multiplication

there is no sign in between parenthesis

How do we write our solutions?

Addition Example:

Solution:

43 + 32 75 43 + 32 75

addends sum or total

How do we write our solutions?

Subtraction Example:

Solution:

54 _ 24 30 _

54 24 30

minuend subtrahend difference

How do we write our solutions?

Multiplication Example:

43 x 32 1 376

Solution:

43 x 32

86 129 1 376

multiplicand multiplier partial product product

factors

How do we write our solutions?

Division Example:

42 째째 7 = 6

Solution:

6 7 42 42 x divisor

quotient dividend

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 • We always change the sign of the subtrahend from positive to negative or from negative to positive – vice versa.

• Proceed to addition of integers.

Originally (+)_(+) = ( + )_ _( ) =

After changing ( + ) + ( _) = ( + )+ +( ) =

Rules for Multiplication of Integers Like Signs: ( + )( + ) = + ( _ )(_ ) = +

Unlike Signs: ( + )( _ ) = _ ( _ )( + ) =_

• The product of like signs is always positive • The product of unlike signs is always negative

Rules for Division of Integers Like Signs:

Unlike Signs:

( + )° ( + ) = ( + )° _( ) _= ° ° _ _ _ (+ °) ( ) = ( )° ( + ) _= ° ° + • The quotient of like signs is always positive • The quotient of unlike signs is always negative

SUMMARY OF RULES ON INTEGERS ADDITION

SUBTRACTION

MULTIPLICATION

DIVISION

LIKE SIGNS CHANGE THE SIGN OF THE SUBTRAHEND THE PRODUCT OR THE QUOTIENT FROM POSITIVE TO OF TWO NUMBERS WITH SAME ADD THE SIGN IS ALWAYS POSITIVE NUMBERS AND NEGATIVE OR FROM NEGATIVE TO COPY THE POSITIVE COMMON SIGN (-)(-)=(+)

(+)+(+)=(+) (-)+(-)=(-)

from (+)-(+) to (+)+(-)=(+)

THEN PROCEED TO ADDITION OF INTEGERS

(+)(+)=(+) (-)รท(-)=(+) (+)รท(+)=(+)

SUMMARY OF RULES ON INTEGERS ADDITION

SUBTRACTION

MULTIPLICATION

DIVISION

UNLIKE SIGNS CHANGE THE SIGN OF SUBTRACT THE THE SUBTRAHEND THE PRODUCT OR THE QUOTIENT SMALLER FROM POSITIVE TO OF TWO NUMBERS WITH UNLIKE NUMBER FROM NEGATIVE OR FROM SIGNS IS ALWAYS NEGATIVE. THE BIGGER NEGATIVE TO AND COPY THE POSITIVE (-)(+)=(-) SIGN OF THE from (+)-(+) BIGGER (+)(-)=(-) NUMBER AS to (+)+(-)=(+) THE SIGN OF (-)รท(+)=(-) THEN PROCEED TO THE FINAL (+)รท(-)=(-) ADDITION OF ANSWER INTEGERS

Addition of Integers Rathmell Model LANGUAGE positive two plus positive three

(+2 + (+3 = ? ) ) MODEL + + +

+

+ +

+

=

+ +

+ +

SYMBOL (+2 + (+3 = (+5 ) ) )

Combining chips with the same color is the same as adding integers of the same sign.

Addition of Integers

LANGUAGE positive three plus positive four

(+3 + (+4 = ? ) ) MODEL + + +

+

+ +

+ + +

=

SYMBOL

+

+ + + +

+

(+3 + (+4 = (+7) ) )

Addition of Integers (-2) + (-2) = ?

LANGUAGE negative two plus negative two

MODEL -

+

-

=

SYMBOL

-

-

-

-

(-2) + (-2) = (-4)

Combining again chips with the same color is the same as adding integers of the same sign.

Addition of Integers (-3) + (-2) = ?

LANGUAGE negative three plus negative two

MODEL -

+

-

SYMBOL -

=

-

-

-

-

(-3) + (-2) = (-5)

Addition of Integers

LANGUAGE positive three plus negative two

(+3 + (-2) = ? ) MODEL + + +

+

-

=

+

-

+

+

SYMBOL (+3 + (-2) = (+1) )

Just pair one negative with one positive.

Paired chips become zero pair.

Addition of Integers

LANGUAGE negative three plus positive four

(-3) + (+4 = ? ) MODEL -

+

+ + + +

=

-

+

-

+

-

+

SYMBOL +

(-3) + (+4 = (+1) )

Again by pairing one negative with one positive, paired chips become zero pair.

Addition of Integers (-5) + (+7 = ? ) MODEL

LANGUAGE

negative five plus positive seven

SYMBOL

-

+ +

-

+

+

-

+ +

-

+

+

-

+

-

+

-

+

-

+

+ + +

=

(-5) + (+7 = (+2) )

Addition of Integers (-4) + (+4 = ? ) MODEL

LANGUAGE negative four plus positive four

-

+

-

+

-

+

-

+

-

+

-

+

-

+

+ +

=

SYMBOL (-4) + (+4 = (0) )

Just pairing one negative with one positive cancels out and becomes zero pair.

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 LANGUAG E positive four minus negative three

?

(+4 - (-3) = ) MODEL + + + +

+

-

+ + +

=

SYMBOL

+

+

+

+

+

+

+

(+4 )

- (-3) = +

+

+

+ -

+ +

-

+

What is the additive identity for addition?

Always remember that any number added to zero equals the number.

(+7)

Subtraction of Integers (+4 - (+6)= ? ) MODEL

LANGUAG E positive four minus positive six

+ + + +

+

-

+

+

+

+

+

+

=

+

-

-

+

-

-

+ +

SYMBOL (+4 )

- (+6)= +

+

-

+

+ -

-

+

(-2)

-

+ Watch out! When the operation was changed, what happened to the chips’ color (subtrahend)? What did we do next?

Subtraction of Integers (-4) - (-7) = ?

LANGUAG E negative four minus negative seven

MODEL -

+

-

SYMBOL (-4) - (-7) = (+3)

+

+

-

+

+

+

+

-

+

+

+

-

+

+

-

+

+

-

+ -

-

+

-

+

+

=

-

-

Subtraction of Integers (-8) - (-5) = ?

LANGUAG E negative eight minus negative five

MODEL

-

-

-

-

-

-

-

-

+

-

+

+

+

+

+

=

SYMBOL -

+

-

+

-

+

-

+

-

+

-

-

(-8) - (-5) = (-3) -

-

-

-

-

-

-

-

Multiplication of Integers What is multiplication?

Multiplication shortcut for addition.

is the repeated

Multiplication of Integers Example: 4 x 3 = ?

Sol. (Addition) 3 + 3 + 3 + 3 = 12 meaning 3 is being added to itself 4 times

Multiplication of Integers Example: 5 x 4 = ?

Sol. (Addition) 4 + 4 + 4 + 4 + 4 = 20 meaning 4 is being added to itself 5 times

Multiplication of Integers Example: 7 x 5 = ?

Sol. (Addition) 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35 meaning 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 (+) it reminds us on what to keep

use positive chips

+ + ( N)( N)multiplier

multiplicand

(-) it also reminds us on what to take away

-

-

use negative chips

Let N be any real number under the set of Integers

please memorize me‌

Multiplication of Integers LANGUAG E positive four times positive three

(+4 (+3) = ? ) MODEL

SYMBOL

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

(+4) x (+3) (+12 )

Opposite pair is very important in this operation.

Take note of the word keep (+) and take away (-).

Multiplication of Integers (-2)(-4) = LANGUAG E negative two times negative four

?

MODEL

SYMBOL

-

-

+

+

-

-

+

+

-

-

+

+

-

-

+

+

(-2) x (-4) (+8)

In multiplying integers with the same sign, what is the resulting sign of the product?

What conclusion can we make?

Multiplication of Integers ?

(-2)(+3) = LANGUAG E negative two times positive three

MODEL

SYMBOL

+

+

-

-

+

+

-

-

+

+

-

-

(-2) x (+3) (-6)

In multiplying integers with the different signs, what is the resulting sign of the product?

What conclusion can we make?

Multiplication of Integers LANGUAG E positive three times negative four

(+3 (-4) = )

?

MODEL

SYMBOL

-

-

-

+

+

+

-

-

-

+

+

+

-

-

-

+

+

+

-

-

-

+

+

+

(+3) x (-4) (-12)

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 it reminds us not to change the chip’s color or signed chips

it reminds us what color of the chips or signed chips to be used

+ + ÷ ( N) ( N)divisor

dividend

-

-

it reminds us to change the chip’s color or sign Let N be any real number under the set of Integers

please memorize me…

Division of Integers (+14)÷ (+2)= LANGUAGE positive fourteen divided by positive two

?

MODEL

SYMBOL

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

(+14) = (+7) (+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 negative twelve divided by negative two

?

(-12) ÷ (-2) = MODEL

SYMBOL

-

-

+

+

-

-

+

+

-

-

+

+

-

-

+

+

-

-

+

+

-

-

+

+

(-12) = (+6) (-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 (-6) ÷ (+2)= MODEL

LANGUAGE negative six divided by positive two

?

SYMBOL

-

-

-

-

-

-

-

-

-

(-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 positive six divided by negative three

?

(+6)÷ (-3) = MODEL

SYMBOL

+

+

+

-

-

-

+

+

+

-

-

-

(+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 (-9) ÷ (+3)= LANGUAGE negative nine divided by positive three

?

MODEL

SYMBOL

-

-

-

-

-

-

-

-

-

-

-

-

(-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.

4.

5.

(+2) + (+3) = (+5) + (-7) = (-4) + (-8) = (-6) + (+7) = (+8) + (-9) =

More Exercises Perform the following indicated operation: 6. (+5) - (-7) = 7. (4) - (-5) = 8. (-3) - (-10) = 9. (+9) - (-11) = 10. (-5) - (+7) =

More Exercises Perform the following indicated operation: 11. -(-7) = 12. (+5)(+7) = 13. (+15) รท (-3) = 14. (-20) รท (-2) = 15. (+12) รท (-3) =

More Exercises Perform the following indicated operation: 16. -5 (-7) = 17. (14) รท (-2) = 18. (-18) รท (3) = 19. (+15)(-3) = 20. (16)(-4) =

More Exercises - Addition

More Exercises - Addition

More Exercises - Subtraction

More Exercises - Subtraction

More Exercises Multiplication

More Exercises - Multiplication

More Exercises - Division

More Exercises - Division