operations on integers (danilo s sevillano)

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


LANGUAGE MODEL SYMBOL

positive three plus positive

four

+++

+ = (+3)

(+4)

+ = (+7)

++

++

+++

+++

+

(+3)

(+4)

+ = ?



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)



negative three plus

negative two+ = (-

3)(-2)

+ = (-5)

--

--

---

--

-

(-3)

(-2)

+ = ?


LANGUAGE MODEL SYMBOLpositive three plus negative

two

+++

+ =++

+

--

--

Just pair one negative with one positive.Paired chips become zero pair.

(+3)

(-2)

+ = ?

(+3)

(-2)

+ = (+1)



negative three plus

positive four

+ = + (-3)

(+4)

+ = (+1)

+-++ +

+---

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

(-3)

(+4)

+ = ?

+-+-



negative five plus positive

seven

+ =

+

(-5)

(+7)

+ = (+2)

+-++ +

+

+ ++

-----

(-5)

(+7)

+ = ?

+-+-+-+-

+



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.


positive four minus

negative three

++

++ =

+++

+

++

+

(+7)

(+4)

(-3)=-+- -+

+

++

+

-+-+

+++

--- -+


positive four minus positive six


++

++

(+4)

(+6)

- = ?

=+++

+++

+

++

+

(-2)

(+4)

(+6)

=-+- -+

+

++

+

-+-+

---

--

-

---

---

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


negative four minus

negative seven


--

--

(-4)

(-7)- = ?

=

+++

+++

+-

--

-

(+3)

(-4)

(-7)=-+-

-+-

--

-

-+

-+-+

+++

+++

+

---

---

-


negative eight minus

negative five


--

--

----

(-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


5 x 4 = ?Sol.

(Addition)4 + 4 + 4 + 4 + 4 = 20meaning 4 is being

added to itself 5 times


7 x 5 = ?

Sol. (Addition)5 + 5 + 5 + 5 + 5 +

5 + 5 = 35meaning 5 is being added to itself 7

times


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



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+++

+++

+++

---

---

---

---



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++++

----

----



negative two times positive

three

(-2)

(+3)

= ?

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


+++

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

x+++

---

---



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?


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.


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



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? _______



negative twelve

divided by negative

two

(-12)

(-2)= ?÷

--

-

---

--

-

---

(-12)(-2)

(+6)=++

+

+++

++

+

+++





negative six divided by

positive two

(-6) (+2)= ?÷

(-6)(+2)

(-3)

=-

- - -

-

- -

- -





positive six divided by negative

three

(+6) (-3)= ?÷

(+6)(-3)

(-2)

=++ +

++ +

-- -

-- -





negative nine divided by positive

three

(-9) (+3)= ?÷

(-9)(+3)

(-3)

=

-

-

-

-- -

-- -

-- -



THANK YOU

More Exercises

Perform the following indicated operation:

1. (+2) + (+3) = 2. (+5) + (-7) =3. (-4) + (-8) =4. (-6) + (+7) =5. (+8) + (-9) =

More Exercises


6. (+5) - (-7) =7. (4) - (-5) =8. (-3) - (-10) =9. (+9) - (-11) =10.(-5) - (+7) =

More Exercises


11.-(-7) =12.(+5)(+7) =13.(+15) ÷ (-3) =14.(-20) ÷ (-2) =15.(+12) ÷ (-3) =

More Exercises


16.-5 (-7) =17.(14) ÷ (-2) =18.(-18) ÷ (3) =19.(+15)(-3) =20.(16)(-4) =

More Exercises - Addition

More Exercises - Subtraction

More Exercises - Multiplication

More Exercises - Division

operations on integers (danilo s sevillano)

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