Integernumbers
1st of ESO
index
1.-What are negative numbers?
2.-Absolut value
3.-The opposite numbers
5.-Multiplication and division
6.-Distributing the minus sign
4.-Addition and subtraction
8.- Exponents and radicals.
7.- Nested brackets
9.-Activities
What are negative numbers?
It's time for the next space shuttle launch. The countdown starts: But what happens after lift off, after zero?
Let's look at another example, the weather. On a cold December
night you can watch the thermometer as the temperature drops, as
the numbers go down: 2 degrees, 1 degree, zero degrees
But what happens to the numbers if it gets even colder?
The temperature and the numbers keep going down!
These numbers below zero are called negative numbers
A negative number is less than zero
We write negative numbers like this: negative 2 is the same as -2
The dash is the negative sign.
Sometimes negative numbers are called minus numbers. Be careful,
don't confuse these with subtraction.
Positive numbers: You already use positive numbers all the time! But, unlike negative
number you don't have to put a + sign in front of them. Here are
some examples of positive numbers: 3, 46, 689, 1982
A POSITIVE number is MORE than zero
Why do negative numbers 'get bigger'??
As you extend a number line showing negative numbers, they seem
to get 'bigger'.The numbers seem to increase in value as they go down the number line.
But as the negative number gets bigger, the value gets lower. -10
is a larger number than -5, so it is further below zero. If you look at
the number line you can see that -10 is less than -5.
If it helps you remember, think about the weather. As the
temperature gets lower the negative numbers seem to get bigger.
Compare the temperatures during spring, autumn and winter:
Bank statements
Temperature
examples
Buildings
temperature:
You read about negative numbers in weather reports and on food
packaging. The temperature -5°C is 'negative five degrees' and it
means 5 degrees below zero.
Buildings :
Have you ever been in the lift of a building that goes underground?
Watch the floors as you go down. Starting on the third floor you would see: 3, 2, 1, 0, -1, -2
In the building -2 is the second floor underground.
.
Take a look around to see if you can find other examples where
negative numbers are used to show less than zero.
Bank statements:
Many of us will recognise negative numbers on a bank statement. If you
spend more money than you have in the account it will show up as a
negative number. Sometimes these numbers are written in red, with a
negative sign in front of them.
.
Freezing point of water
Below and above freezing
Negative temperatures are called below freezing and positive
temperatures are above freezing.
Below and above freezing
On the weather map it shows that:
In Newcastle it is 1 which is 1 degree Celsius, 1°C
• 1 degree above freezing
In Glasgow it is -4 which is negative 4 degrees Celsius, -4°C
• 4 degrees below freezing
So on this map it's colder in Glasgow than in Newcastle.
Reading a thermometer
A thermometer is something that is used to measure temperature.
When you've been to the doctors they might have taken your
temperature with a thermometer. Or you might have one at home
showing how hot, or cold, it is inside your house.
• On the first thermometer the scale is between -3°C and 24°C and goes up in lots of 1°C.
• On the second thermometer the scale is between -6°C and 20°C and shows only even numbers.
• On the third thermometer the scale is between -25°C and 40°C and goes up in lots of 5°C.
Temperature increase and decrease
When the weather changes the temperature can go up or down. If a
temperature goes up, gets warmer, it is a temperature increase. If
a temperature goes down, gets colder, it is a temperature
decrease.
Temperature increase and decrease
In the early afternoon the temperature is 15°C. By early evening the temperature goes down to 12°C.
But what happens if we have negative temperatures?
During winter the early morning temperature could be -3°C. By lunchtime it is up to 6°C.
From -3°C to 0°C is 3°C.
From 0°C to 6°C is a further 6°C.
The total of 3°C + 6°C makes 9°C.
In the early afternoon the temperature is 3°C. By early evening the temperature goes down to -5°C.
From 3°C to 0°C is 3°C.
From 0°C to -5°C is a further 5°C.
The total of 3°C + 5°C makes 8°C.
Integer Numbers.
The set of positive and negative numbers, including zero, are called Integer numbers.If a number has no sign, the number is considered positive.
The number zero together with the positive integers form the set of whole numbers To add two integers, use a number line to help you. Start on the line at the first number:
To add a positive number count right
To add a negative number count left
Activities
Activities:
1.- Mark the following numbers, smallest to largest, on a number line.
-4, +5, 0, -2, +1, -1, +7, -7
2.-Put these numbers in order, with the smallest first:
-5, -17, -23, +8, -96, +14, -1, +21, 0, +2, -50, +1
4.-Which numbers are exactly halfway between each pair?
a) -2 and -8………………………………………………………………...
b) -3 and +6……………………………………………………………….
3.-Write each pair of numbers with either a < o > sign between them
a) +2 and +4 …………….. b) -8 and +8 ……………..
c) -2 and 0 …………….. d) -1 and +1 ……………..
e) -3 and -2 …………….. f) +2 and -4 ……………..
ABSOLUTE VALUE.
The distance between zero and +1 is the same as the distance between zero and -1; the distance between zero and +2 is the same as the distance between zero and -2; and so on.The way we write absolute value, the mathematical notation, is to use two vertical bars | | . For example, the absolute value of negative seven is the absolute value of seven, which also is seven. |-7| = |7|= 7 Find the absolute value: a) |5|= b) |-5|= c) |-6|-|-3| =
If two integer numbers are positive, the biggest is which has biggest absolute value.Any positive number is bigger than the zero, and the zero is bigger than any negative number. if we have two negative integer numbers, the biggest is which has smallest absolute value
THE OPPOSITE NUMBERS .
The opposite of an integer number is another integer with the same absolute value but with opposite sign Opposite of (+a) = (-a) Opposite of (-a) = (+a) Find the opposite number of: Op ( -8 ) = Op ( +4 ) = Op ( 0 ) = Op ( +1 ) = Op ( -3 ) = Op ( +16 ) =
ADDITION OF INTEGERS :
1.- When signs are alike, add the absolute values. The sign of the sum is the sign of the numbers added.2.- When the signs are different, find the difference between the absolute values. The sign of the answer is the sign of the number with the largest absolute value. Find the sum without using your calculator:
a) ( +4 ) + ( + 8 ) = b) ( - 3 ) + ( + 2 ) =
c) ( - 2 ) + ( + 8 ) = d) ( + 3 ) + ( - 2 ) =
e) ( + 5 ) + ( - 7 ) = f) ( - 3 ) + ( - 2 ) =
Subtraction OF INTEGERS :
1.- Change the subtraction to addition2.- Change the sign of the second number 3.- Add the two numbers. ( Follow the rules for addition ) Solve:
a) ( +4 ) - ( + 8 ) = b) ( - 3 ) - ( + 2 ) =
c) ( - 2 ) - ( + 8 ) = d) ( + 3 ) - ( - 2 ) =
e) ( + 5 ) - ( - 7 ) = f) ( - 3 ) - ( - 2 ) =
Another way :
1.- Eliminate the parenthesis using the sign property:+ + = + - - = + + - = - - + = - We can only use this property when there are two signs together, only separated by a parenthesis. 2.- If the numbers have the same sign, we sum the numbers and put the same sign. 3.- If the numbers have different sign, we subtract the smallest from the biggest and take the sign of the biggest.
addition and subtraction:
MULTIPLICATION AND DIVISION OF INTEGERS .
MULTIPLICATION OF INTEGERS1.- Multiply the absolute values. 2.-a) If the signs are alike, the sign of the product is positive b) If the signs are different, the sign of the product is negative. First determine the sign of the answer. Then calculate the answer.
a) ( +4 ) ·( + 8 ) = b) ( - 3 ) · ( + 2 ) =
c) ( - 2 ) · ( + 8 ) = d) ( + 3 ) · ( - 2 ) =
e) ( + 5 ) · ( - 7 ) = f) ( - 3 ) · ( - 2 ) =
DIVISION OF INTEGERS .
DIVISION OF INTEGERS1.- Divide the absolute values of the two numbers. 2.- a) If the signs are alike, the sign of the quotient is positive b) If the signs are different, the sign of the quotient is negative Calculate:First determine the sign of the answer. Then calculate the answer.
a) ( +48 ) : ( + 8 ) = b) ( - 32 ) : ( + 2 ) =
c) ( - 24 ) : ( + 8 ) = d) ( + 32 ) : ( - 2 ) =
e) ( + 56 ) : ( - 7 ) = f) ( - 30 ) : ( - 3 ) =
Multiplication and division:
MULTIPLICATION AND DIVISION WITH MORE THAN TWO NUMBERS
MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
An even number of negative signs gives a positive answer
An odd number of negative signs gives a negative answer
distributing the minus sign
One of the trickier situations is when a pair of brackets is preceded by a minus sign. In such instances, regard the minus sign as meaning “multiply by –1.”For example, -(5 – 3 + 6) = (-1) · (5 – 3 + 6) = =(-1) · (5) + (-1) · (-3) + (-1) · (6) = -5 + 3 – 6 = -8 When we remove brackets which are preceded by a minus sign, we need to reverse the signs of every term that was inside the brackets.
Nested brackets:
When the expression within a pair of brackets itself contains brackets, we say that the brackets are nested. In such situations, the removal of brackets must progress from the innermost pair of brackets to the outermost pair. For example, 5 + 3[ 4 + 6(7 – 2) ] = 5 + 3[ 4 + 6 · 5 ] = 5 + 3[ 4 + 30 ] = 5 + 3 · 34 = 5 + 102 =107. Now you do this exercise: (-8) + 2[ 9 - 2(8 – 6) ]
Exponents and radicals:
Powers of Negative Numbers
If a negative number is raised to an even-numbered power, the result is a positive number.
If it is raised to an odd-numbered power, the result is a negative number
There are not the square root of a negative number because
( - a ) · ( - a ) = + a
COMMON MISTAKES:
Powers of Negative Numbers
If a negative number is raised to an even-numbered power, the result is a positive number.
If it is raised to an odd-numbered power, the result is a negative number
There are not the square root of a negative number because ( - a ) · ( - a ) = + a
Activities:
Solve the next activities: 1)a) + ( - 2 ) ( - 6 ) – 8 : ( - 4 ) = b) ( -14 ) : ( - 2 ) + 4 ( - 5 ) = 2) Calculate:
ThankS!
by Iván Fagundo
1st of ESO integer numbers
ivanfag74
Created on January 27, 2021
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Transcript
Integernumbers
1st of ESO
index
1.-What are negative numbers?
2.-Absolut value
3.-The opposite numbers
5.-Multiplication and division
6.-Distributing the minus sign
4.-Addition and subtraction
8.- Exponents and radicals.
7.- Nested brackets
9.-Activities
What are negative numbers?
It's time for the next space shuttle launch. The countdown starts: But what happens after lift off, after zero?
Let's look at another example, the weather. On a cold December night you can watch the thermometer as the temperature drops, as the numbers go down: 2 degrees, 1 degree, zero degrees But what happens to the numbers if it gets even colder? The temperature and the numbers keep going down! These numbers below zero are called negative numbers
A negative number is less than zero
We write negative numbers like this: negative 2 is the same as -2 The dash is the negative sign. Sometimes negative numbers are called minus numbers. Be careful, don't confuse these with subtraction. Positive numbers: You already use positive numbers all the time! But, unlike negative number you don't have to put a + sign in front of them. Here are some examples of positive numbers: 3, 46, 689, 1982
A POSITIVE number is MORE than zero
Why do negative numbers 'get bigger'??
As you extend a number line showing negative numbers, they seem to get 'bigger'.The numbers seem to increase in value as they go down the number line. But as the negative number gets bigger, the value gets lower. -10 is a larger number than -5, so it is further below zero. If you look at the number line you can see that -10 is less than -5. If it helps you remember, think about the weather. As the temperature gets lower the negative numbers seem to get bigger.
Compare the temperatures during spring, autumn and winter:
Bank statements
Temperature
examples
Buildings
temperature:
You read about negative numbers in weather reports and on food packaging. The temperature -5°C is 'negative five degrees' and it means 5 degrees below zero.
Buildings :
Have you ever been in the lift of a building that goes underground? Watch the floors as you go down. Starting on the third floor you would see: 3, 2, 1, 0, -1, -2 In the building -2 is the second floor underground. .
Take a look around to see if you can find other examples where negative numbers are used to show less than zero.
Bank statements:
Many of us will recognise negative numbers on a bank statement. If you spend more money than you have in the account it will show up as a negative number. Sometimes these numbers are written in red, with a negative sign in front of them. .
Freezing point of water
Below and above freezing Negative temperatures are called below freezing and positive temperatures are above freezing.
Below and above freezing
On the weather map it shows that: In Newcastle it is 1 which is 1 degree Celsius, 1°C • 1 degree above freezing In Glasgow it is -4 which is negative 4 degrees Celsius, -4°C • 4 degrees below freezing So on this map it's colder in Glasgow than in Newcastle.
Reading a thermometer
A thermometer is something that is used to measure temperature. When you've been to the doctors they might have taken your temperature with a thermometer. Or you might have one at home showing how hot, or cold, it is inside your house. • On the first thermometer the scale is between -3°C and 24°C and goes up in lots of 1°C. • On the second thermometer the scale is between -6°C and 20°C and shows only even numbers. • On the third thermometer the scale is between -25°C and 40°C and goes up in lots of 5°C.
Temperature increase and decrease
When the weather changes the temperature can go up or down. If a temperature goes up, gets warmer, it is a temperature increase. If a temperature goes down, gets colder, it is a temperature decrease.
Temperature increase and decrease
In the early afternoon the temperature is 15°C. By early evening the temperature goes down to 12°C.
But what happens if we have negative temperatures?
During winter the early morning temperature could be -3°C. By lunchtime it is up to 6°C. From -3°C to 0°C is 3°C. From 0°C to 6°C is a further 6°C. The total of 3°C + 6°C makes 9°C.
In the early afternoon the temperature is 3°C. By early evening the temperature goes down to -5°C. From 3°C to 0°C is 3°C. From 0°C to -5°C is a further 5°C. The total of 3°C + 5°C makes 8°C.
Integer Numbers.
The set of positive and negative numbers, including zero, are called Integer numbers.If a number has no sign, the number is considered positive. The number zero together with the positive integers form the set of whole numbers To add two integers, use a number line to help you. Start on the line at the first number: To add a positive number count right To add a negative number count left
Activities
Activities:
1.- Mark the following numbers, smallest to largest, on a number line. -4, +5, 0, -2, +1, -1, +7, -7
2.-Put these numbers in order, with the smallest first: -5, -17, -23, +8, -96, +14, -1, +21, 0, +2, -50, +1
4.-Which numbers are exactly halfway between each pair? a) -2 and -8………………………………………………………………... b) -3 and +6……………………………………………………………….
3.-Write each pair of numbers with either a < o > sign between them a) +2 and +4 …………….. b) -8 and +8 …………….. c) -2 and 0 …………….. d) -1 and +1 …………….. e) -3 and -2 …………….. f) +2 and -4 ……………..
ABSOLUTE VALUE.
The distance between zero and +1 is the same as the distance between zero and -1; the distance between zero and +2 is the same as the distance between zero and -2; and so on.The way we write absolute value, the mathematical notation, is to use two vertical bars | | . For example, the absolute value of negative seven is the absolute value of seven, which also is seven. |-7| = |7|= 7 Find the absolute value: a) |5|= b) |-5|= c) |-6|-|-3| =
If two integer numbers are positive, the biggest is which has biggest absolute value.Any positive number is bigger than the zero, and the zero is bigger than any negative number. if we have two negative integer numbers, the biggest is which has smallest absolute value
THE OPPOSITE NUMBERS .
The opposite of an integer number is another integer with the same absolute value but with opposite sign Opposite of (+a) = (-a) Opposite of (-a) = (+a) Find the opposite number of: Op ( -8 ) = Op ( +4 ) = Op ( 0 ) = Op ( +1 ) = Op ( -3 ) = Op ( +16 ) =
ADDITION OF INTEGERS :
1.- When signs are alike, add the absolute values. The sign of the sum is the sign of the numbers added.2.- When the signs are different, find the difference between the absolute values. The sign of the answer is the sign of the number with the largest absolute value. Find the sum without using your calculator: a) ( +4 ) + ( + 8 ) = b) ( - 3 ) + ( + 2 ) = c) ( - 2 ) + ( + 8 ) = d) ( + 3 ) + ( - 2 ) = e) ( + 5 ) + ( - 7 ) = f) ( - 3 ) + ( - 2 ) =
Subtraction OF INTEGERS :
1.- Change the subtraction to addition2.- Change the sign of the second number 3.- Add the two numbers. ( Follow the rules for addition ) Solve: a) ( +4 ) - ( + 8 ) = b) ( - 3 ) - ( + 2 ) = c) ( - 2 ) - ( + 8 ) = d) ( + 3 ) - ( - 2 ) = e) ( + 5 ) - ( - 7 ) = f) ( - 3 ) - ( - 2 ) =
Another way :
1.- Eliminate the parenthesis using the sign property:+ + = + - - = + + - = - - + = - We can only use this property when there are two signs together, only separated by a parenthesis. 2.- If the numbers have the same sign, we sum the numbers and put the same sign. 3.- If the numbers have different sign, we subtract the smallest from the biggest and take the sign of the biggest.
addition and subtraction:
MULTIPLICATION AND DIVISION OF INTEGERS .
MULTIPLICATION OF INTEGERS1.- Multiply the absolute values. 2.-a) If the signs are alike, the sign of the product is positive b) If the signs are different, the sign of the product is negative. First determine the sign of the answer. Then calculate the answer. a) ( +4 ) ·( + 8 ) = b) ( - 3 ) · ( + 2 ) = c) ( - 2 ) · ( + 8 ) = d) ( + 3 ) · ( - 2 ) = e) ( + 5 ) · ( - 7 ) = f) ( - 3 ) · ( - 2 ) =
DIVISION OF INTEGERS .
DIVISION OF INTEGERS1.- Divide the absolute values of the two numbers. 2.- a) If the signs are alike, the sign of the quotient is positive b) If the signs are different, the sign of the quotient is negative Calculate:First determine the sign of the answer. Then calculate the answer. a) ( +48 ) : ( + 8 ) = b) ( - 32 ) : ( + 2 ) = c) ( - 24 ) : ( + 8 ) = d) ( + 32 ) : ( - 2 ) = e) ( + 56 ) : ( - 7 ) = f) ( - 30 ) : ( - 3 ) =
Multiplication and division:
MULTIPLICATION AND DIVISION WITH MORE THAN TWO NUMBERS
MULTIPLICATION AND DIVISION OF SIGNED NUMBERS An even number of negative signs gives a positive answer An odd number of negative signs gives a negative answer
distributing the minus sign
One of the trickier situations is when a pair of brackets is preceded by a minus sign. In such instances, regard the minus sign as meaning “multiply by –1.”For example, -(5 – 3 + 6) = (-1) · (5 – 3 + 6) = =(-1) · (5) + (-1) · (-3) + (-1) · (6) = -5 + 3 – 6 = -8 When we remove brackets which are preceded by a minus sign, we need to reverse the signs of every term that was inside the brackets.
Nested brackets:
When the expression within a pair of brackets itself contains brackets, we say that the brackets are nested. In such situations, the removal of brackets must progress from the innermost pair of brackets to the outermost pair. For example, 5 + 3[ 4 + 6(7 – 2) ] = 5 + 3[ 4 + 6 · 5 ] = 5 + 3[ 4 + 30 ] = 5 + 3 · 34 = 5 + 102 =107. Now you do this exercise: (-8) + 2[ 9 - 2(8 – 6) ]
Exponents and radicals:
Powers of Negative Numbers If a negative number is raised to an even-numbered power, the result is a positive number. If it is raised to an odd-numbered power, the result is a negative number There are not the square root of a negative number because ( - a ) · ( - a ) = + a
COMMON MISTAKES:
Powers of Negative Numbers If a negative number is raised to an even-numbered power, the result is a positive number. If it is raised to an odd-numbered power, the result is a negative number There are not the square root of a negative number because ( - a ) · ( - a ) = + a
Activities:
Solve the next activities: 1)a) + ( - 2 ) ( - 6 ) – 8 : ( - 4 ) = b) ( -14 ) : ( - 2 ) + 4 ( - 5 ) = 2) Calculate:
ThankS!
by Iván Fagundo