Mathematics Class VII Chapter 1 Integers

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

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

Welcome to Module 1

Definition Positive number – a number greater than zero. 1 2 3 4 5 6

Definition Negative number – a number less than zero. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

Definition Opposite Numbers – numbers that are the same distance from zero in the opposite direction -6 -5 -4 -3 -2 -1 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

The absolute value of 9 or of –9 is 9. 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 30 20 10 -10 -20 -30 -40 -50

Negative Numbers Are Used to Show Debt Let’s say your parents bought a car but had 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 11 7 -13 14 -18

Check Your Answers 1. 8 + 13 = 21 2. –22 + -11 = -33 3. 55 + 17 = 72 4. –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 = Larger abs. value Answer = - 4 9 - 5 = 4

Solve These Problems 3 + -5 = -4 + 7 = (+3) + (-4) = -6 + 7 = 5 + -9 = -2 3 + -5 = -4 + 7 = (+3) + (-4) = -6 + 7 = 5 + -9 = -9 + 9 = 5 – 3 = 2 7 – 4 = 3 3 4 – 3 = 1 -1 7 – 6 = 1 1 9 – 5 = 4 -4 9 – 9 = 0

Some more of them… 1. –12 + 22 = 10 2. –20 + 5 = -15 3. 14 + (-7) = 7 4. –70 + 15 = -55

One Way to Add Integers Is With a Number Line When the number is positive, count to the right. When the number is negative, count to the left. - + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

One Way to Add Integers Is With a Number Line +3 + -5 = -2 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

One Way to Add Integers Is With a Number Line +6 + -4 = +2 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

One Way to Add Integers Is With a Number Line +3 + -7 = -4 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

One Way to Add Integers Is With a Number Line -3 + +7 = +4 - 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 +

Integer Subtraction Rule Subtracting 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 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 = c 5 – 2 = 3 To check if your answer is correct, add b and c: a = b + c 5 = 2 + 3

Here are some examples: a – b = c a = b + c 9 – 5 = 4 9 = 5 + 4 20 – 3 = 17 20 = 3 + 17

If the method for checking subtraction works, it should also 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 + c 2 – (-5) = 7 2 = -5 + 7 It works! 2 – (-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 Check: 17 = -12 + 29 Continued on next slide

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 things About adding and subtracting Integers. 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 = Larger abs. value Answer = - 4 9 - 5 = 4

One Way to Add Integers Is With a Number Line When the number is positive, count to the right. When the number is negative, count to the left. - + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

Integer Subtraction Rule Subtracting 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

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Welcome to Module 2

Verbal Problem

Remember

Multiplication and Division Rules When multiplying and dividing integers there are two rules: When multiplying or dividing integers that have the SAME sign, the answer is POSITIVE. Positive x positive = positive Negative x negative = positive +7 x +12 = +84 -6 x –5 = +30

Rules continued… Rule 2: Positive x negative = negative When multiplying or dividing integers whose signs are DIFFERENT, the answer will always be NEGATIVE. Positive x negative = negative Negative 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!!!

What sign will be in front of the answer?? 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

When multiplying and dividing integers: 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?

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Welcome to Module 3

Properties of integers Commutative Property Addition For 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 = 7 Therefore, the integers satisfy commutative property under addition ORDER DOES NOT MATTER

Properties of integers (contd.) Commutative Property 2. Multiplication For 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 = 12 Therefore, the integers satisfy commutative property under multiplication. ORDER DOES NOT MATTER

Properties of integers (contd.) Commutative Property 3. Subtraction For 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 = 1 Therefore, the integers do not satisfy commutative property under subtraction ORDER DOES MATTER

Properties of integers (contd.) Commutative Property 4. Division For 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 = 1/3 Therefore, the integers do not satisfy commutative property under division ORDER DOES MATTER

Properties of integers (contd.) Associative Property 1. Addition For 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 = 9 Therefore, the integers do not satisfy associative property under addition ORDER DOES NOT MATTER

Properties of integers (contd.) Associative Property 2. Multiplication For 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 = 24 Therefore, the integers do not satisfy associative property under multiplication. ORDER DOES NOT MATTER

Properties of integers (contd.) Associative Property 3. Subtraction For 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 = 24 Therefore, the integers do not satisfy associative property under subtraction. ORDER DOES NOT MATTER

Properties of integers (contd.) Associative Property 4. Division For 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 = 2/3 Therefore, the integers do not satisfy associative property under division. ORDER DOES MATTER

Properties of integers (contd.) Additive Identity Zero is the identity element on either side. By adding zero on either side number will not change 5 + 0 = 5 0 + 5 = 5

Properties of integers (contd.) Multiplicative Identity One is the identity element on either side. By multiplying with 1 on either side number will not change 5 x 1 = 5 1 x 5 = 5

Properties of integers (contd.) Distributive Property 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) = 2 Therefore, the integers satisfy distributive property

Verbal Problems

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