Mathematics Class VII Chapter 1 Integers. Module Objectives Learn the basic concepts of Integers ...

Mathematics

Class VIIChapter 1Integers

Module Objectives

Learn the basic concepts of Integers Learn integer addition and subtraction Multiply positive integer by positive integer as well as

negative integer Multiply negative integer by positive integer as well as

negative integer Compare integers and identify smaller and greater

among them Follow proper method in multiplication as well as relate

multiplication and division of numbers

By the end of this chapter, you will be able to

Module Objectives

Divide an integer by another integer Understand why an integer cannot be divisible by 0 Understand the basic properties of integers with respect

to fundamental operations.

By the end of this chapter, you will be able to

Welcome to Module 1Welcome to Module 1

Definition

Positive number – a number greater than zero.

0 1 2 3 4 5 6

Definition

Negative number – a number less than zero.

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition

Opposite Numbers – numbers that are the same distance from zero in the opposite direction

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition

Integers – Integers are all the whole numbers and all of their opposites on the negative number line including zero.

7 opposite -7

Definition

Absolute Value – The size of a number with or without the negative sign.

The absolute value of 9 or of –9 is 9.

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level

0102030

-10-20-30-40-50

Negative Numbers Are Used to Show Debt

Let’s say your parents bought a car buthad to get a loan from the bank for $5,000.When counting all their money they add in -$5,000 to show they still owe the bank.

Hint

If you don’t see a negative or positive sign in front of a number it is positive.

9+

Integer Addition Rules

Rule #1 – If the signs are the same, pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.

9 + 5 = 14-9 + -5 = -14

Solve the Problems

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

-8

-18

14-13

7

11

Check Your Answers

1. 8 + 13 = 212. –22 + -11 = -333. 55 + 17 = 724. –14 + -35 = -49

Integer Addition Rules

Rule #2 – If the signs are different pretend the signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer.

-9 + +5 =9 - 5 = 4

Larger abs. value

Answer = - 4

Solve These Problems

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

-25 – 3 = 2

0 -4

1-1

3

9 – 9 = 0

9 – 5 = 4

7 – 6 = 14 – 3 = 1

7 – 4 = 3

Some more of them…

1. –12 + 22 = 102. –20 + 5 = -153. 14 + (-7) = 74. –70 + 15 = -55

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

When the number is positive, countto the right.

When the number is negative, countto the left.

+-

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -5 = -2

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+6 + -4 = +2

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -7 = -4

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

+

-3 + +7 = +4

Integer Subtraction RuleSubtracting a negative number is the same as adding a positive. Change the signs and add.

2 – (-7) is the same as

2 + (+7)

2 + 7 = 9!

Here are some more examples.

12 – (-8)

12 + (+8)

12 + 8 = 20

-3 – (-11)

-3 + (+11)

-3 + 11 = 8

Try these out

1. 8 – (-12) = 8 + 12 = 20

2. 22 – (-30) = 22 + 30 = 52

3. – 17 – (-3) = -17 + 3 = -14

4. –52 – 5 = -52 + (-5) = -57

How do we know that “Subtracting a negative number is the same as adding a positive” is true?

We can use the same method we use to check our answers when we subtract.

Suppose you subtract a – b and it equals c: a – b = c5 – 2 = 3

To check if your answer is correct, add b and c:a = b + c5 = 2 + 3

Here are some examples:

a – b = c a = b + c9 – 5 = 4 9 = 5 + 4

a – b = c a = b + c20 – 3 = 17 20 = 3 + 17

If the method for checking subtraction works, it shouldalso work for subtracting negative numbers.

If a – b = c, and….

2 – (-5) is the same as

2 + (+5), which equals 7,

Then let’s check with the negative numbers to see if it’s true…

a – b = c a = b + c2 – (-5) = 7 2 = -5 + 7

It works!

a – b = c a = b + c-11 – (-3) = -8 -11 = -3 + -8

YES!

Check Your Answers

1. Solve: 3 – 10 = 7 Check: 3 = 10 + (-7)

2. Solve: 17 – ( 12) = 29

Continued on next slide

Check: 17 = -12 + 29

Check Your Answers

1. Solve: 20 – ( 5) = 25

Check: 20 = -5 + 25

1. Solve: -7 – ( 2) = -5

Check: -7 = -2 + -5

You have learned lots of thingsAbout adding and subtractingIntegers. Let’s review!

Integer Addition Rules

Rule #1 – If the signs are the same, pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.

9 + 5 = 14-9 + -5 = -14

Integer Addition Rules

Rule #2 – If the signs are different pretend the signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer.

-9 + +5 =9 - 5 = 4

Larger abs. value

Answer = - 4

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

When the number is positive, countto the right.

When the number is negative, count

to the left.

+-

Integer Subtraction RuleSubtracting a negative number is the same as adding a positive. Change the signs and add.

2 – (-7) is the same as

2 + (+7)

2 + 7 = 9!

Verbal Problem

Thank you

Welcome to Module 2Welcome to Module 2

Verbal Problem

Remember

Multiplication and Division Rules

When multiplying and dividing integers there are two rules:

1. When multiplying or dividing integers that have the SAME sign, the answer is POSITIVE.

Positive x positive = positiveNegative x negative = positive

+7 x +12 = +84-6 x –5 = +30

Rules continued…

Rule 2:When multiplying or dividing integers whose signs are

DIFFERENT, the answer will always be NEGATIVE.Positive x negative = negativeNegative x positive = negative

+8 x –4 = -32-5 x +7 = -35

Activity: Let’s Sing a Song!Multiplying Integers Song

(sing to “Dead Bones” tune)

Verse 1

A negative times a negative is a, (um) positive

A negative times a positive is a, (um) negative

A negative times a negative is a, (um) positive

These are the rules for signs.

Verse 2

If you have the same signs, you get a positive,

But if they’re different, you get a negative,

If you have the same signs, you get a positive,

Yes, these are the rules for signs.

Let’s Practice!

What sign would the answer have in front of the integer???

-5 x -7 = ???

POSITIVE!!!

How about…

-6 x 9 = ???What sign will be in front of the answer??

NEGATIVE!!!

Try this one!

What sign will be in front of the answer???

-81 ÷ +9 = ???

NEGATIVE!!!

Let’s solve some problems!

-8 x –9 =

+72

+4 x +12 =

+48

-45 ÷ +5 =

-9

-63 ÷ +7 =

-9

-8 x –11 =

+88

Remember…

When multiplying and dividing integers:

Same signs = a positive

Different signs = a negative

Verbal Problem

Activity

Activity

Why is it not possible to divide by zero?

Thank you

Welcome to Module 3Welcome to Module 3

Properties of integers

1. AdditionFor any two integers a and b , if a + b = b+a , then integer addition is

said to be commutative. e.g. 3 + 4= 7 4 + 3 = 7Therefore, the integers satisfy

commutative property under additionORDER DOES NOT MATTER

Commutative Property

Properties of integers (contd.)

2. MultiplicationFor any two integers a and b , if a x b = b x a , then integer multiplication is said to be commutative. e.g. 3 x 4= 12 4 x 3 = 12Therefore, the integers satisfy commutative property under multiplication. ORDER DOES NOT MATTER

Commutative Property

Properties of integers (contd.)

3. SubtractionFor any two integers a and b , if a - b = b - a , then integer subtraction is said to be commutative. e.g. 3 - 4= -1 4 - 3 = 1Therefore, the integers do not satisfy commutative property under subtraction ORDER DOES MATTER

Commutative Property

Properties of integers (contd.)

4. DivisionFor any two integers a and b , if a / b = b / a , then integer subtraction is said to be commutative. e.g. 12/ 4= 3 4 /12 = 1/3Therefore, the integers do not satisfy commutative property under division ORDER DOES MATTER

Commutative Property

Properties of integers (contd.)

1. AdditionFor any two integers a and b , if a + (b + c) = (a + b) + c , then integer subtraction is said to be associative. e.g. 2+ (4 + 3) = 9 (2 + 4) + 3 = 9Therefore, the integers do not satisfy associative property under addition ORDER DOES NOT MATTER

Associative Property

Properties of integers (contd.)

2. MultiplicationFor any two integers a and b , if a x (b x c) = (a x b) x c , then integer subtraction is said to be associative. e.g. 2x (4 x 3) = 24 (2 x 4) x 3 = 24Therefore, the integers do not satisfy associative property under multiplication. ORDER DOES NOT MATTER

Associative Property

Properties of integers (contd.)

3. SubtractionFor any two integers a and b , if a - (b x c) = (a x b) x c , then integer subtraction is said to be associative. e.g. 2x (4 x 3) = 24 (2 x 4) x 3 = 24Therefore, the integers do not satisfy associative property under subtraction. ORDER DOES NOT MATTER

Associative Property

Properties of integers (contd.)

4. DivisionFor any two integers a and b , if a / (b / c) = (a / b) / c , then integer subtraction is said to be associative. e.g. 12/ (6 / 3) = 6 (12 / 6) / 3 = 2/3Therefore, the integers do not satisfy associative property under division. ORDER DOES MATTER

Associative Property

Properties of integers (contd.)

Zero is the identity element on either side. By adding zero on either side number will not change

5 + 0 = 50 + 5 = 5

Additive Identity

Properties of integers (contd.)

One is the identity element on either side. By multiplying with 1 on either side number will not change

5 x 1 = 51 x 5 = 5

Multiplicative Identity

Properties of integers (contd.)

For any three integers a, b, c, a x (b + c) = (a x b) + (a x c) . This is called Distributive property e.g. 2x (4 + 3) = (2 x 4 ) + (2 x 3) = 14 2 X (4 – 3) = (2 x 4) – (2 x 3) = 2Therefore, the integers satisfy distributive property

Distributive Property

Verbal Problems

Thank you !