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TOPIC OUTLINE
• Directed Numbers
• Mental and Written Arithmetic Strategies
• Operations with Directed Numbers
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DIRECTED NUMBERS
• Understands what a directed number is and gives examples of the +
and – signs in context
• Places directed numbers on a number line
• Compares and orders directed numbers
DIRECTED NUMBERS
An integer is a _____ _________. It has no fractional or decimal part.
eg. 5 is an integer. 5½ is not an integer.
Directed numbers have ______ and ________. They can be positive or
negative.
• +5 is a ________ integer
• ‐5 is a ________ integer
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In real life, negative numbers can be
represented in many ways:
• 1m below mean sea level
• $13 200 in debt
• ‐18oC in temperature
• 1 floor below ground
Assets Liabilities
Car 4000 Car Loan 2500
Savings 300 Student Loan 15000
TOTAL 4300 TOTAL 17500
Net Worth (13200)
DIRECTED NUMBERS IN REAL LIFE
DIRECTED NUMBERS
A number line is used to represent positive and negative numbers.
Negative numbers are written to the _____, zero in the ________ and positive numbers are written on the ______.
negative positive
‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5
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NUMBER LINE
To plot numbers on a number line, draw coloured in circles where the
numbers are positioned.
eg. Plot {‐2, 3, 4½} on a number line.
Note: When writing a set of plotted numbers, write them in ascending
order, separated by commas, in curly brackets.
‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5
DIRECTED NUMBERS
The more to the left a number is, the smaller it is. So, to order
numbers, put them in order as per the number line.
eg. Write 4, ‐2, ‐7, 0 and 3½ in ascending order.
‐7 ‐6 ‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5
• • •• •
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COMPARING DIRECTED NUMBERS
To compare any numbers we use:
• Less than
• Greater than 5 10
4 2
COMPARING DIRECTED NUMBERS
Comparing directed numbers is exactly the same. Choose the sign that
‘opens’ to the larger number.
eg. Put < or > in the box to make the statement true.
7 2
2 1
4 6
‐7 ‐6 ‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5
• •• •••
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MENTAL AND WRITTEN ARITHMETIC STRATEGIES
• Uses the commutativity, associativity and distributive laws to aid
mental computation
• Uses knowledge of factors and multiples to aid mental computation
• Performs long division with two digit divisors
COMMUTATITIVY LAW
The Commutativity Law states that we can swap numbers around in
addition or multiplication and we get the same answer.
For example: 5 8
5 8
We cannot do this with subtraction or division:
4 2
4 2
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WHY IS IT IMPORTANT?
We can use our knowledge of the Commutativity Law to help us with
mental arithmetic.
Calculate 60 71 40.
We can do this by adding from left to right, but it is easier if we swap
the 71 and the 40.
60 40 71 100 71 171
WHY IS IT IMPORTANT?
We can use our knowledge of the Commutativity Law to help us with
mental arithmetic.
Calculate 5 9 4.
We can do this by multiplying from left to right, but it is easier if we
swap the 9 and the 4.
5 4 9 20 9 180
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ASSOCIATIVITY LAW
The Associativity Law states that it doesn’t matter how we group
numbers for addition or multiplication.
For example: 3 6 2
3 6 2
It DOES matter how we group numbers for subtraction and division.
6 3 2
12 4 2
WHY IS THIS IMPORTANT?
We can use our knowledge of the Associativity Law to help us with
mental arithmetic.
Calculate 5 16 4.
It is easier if we add the 16 and the 4 first.
5 16 4 5 20 25
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WHY IS THIS IMPORTANT?
We can use our knowledge of the Associativity Law to help us with
mental arithmetic.
Calculate 15 5 4.
It is easier if we multiply the 5 and the 4 first.
15 5 4 15 20 300
DISTRIBUTIVE LAW
The Distributive Law states that multiplying a group of numbers is the
same as doing the multiplications separately then adding them
together. In other words:
4 2 6
Is it true?
4 2 6 4 8 32
4 2 4 6 8 24 32
This is called ___________ the brackets.
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WHY IS IT IMPORTANT?
Find 99 15
It is easiest to find 100 15 and then subtract 1 15.
99 15
FACTORS AND MULTIPLES
We can use factors and multiples to aid mental arithmetic as well.
Remember, a factor is a number that divides into another number
without ___________.
What are the factors of 24?
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FACTORS AND MULTIPLES
We can use factors and multiples to aid mental arithmetic as well.
A multiple is a multiplication of a number by a counting number.
State the first six multiples of 7.
FACTORS AND MULTIPLES
How can we use a knowledge of factors and multiples to help us with
mental arithmetic?
Suppose we want to calculate 288 24 without a calculator.
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LONG DIVISION
Long division is when we divide by a number with more than one _____.
For example, 632 21.
Sometimes, there will be a remainder. Sometimes, there won’t.
One of the easiest ways to do long division, is to guess, by using
multiplication.
LONG DIVISION
Consider the problem 632 21.
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LONG DIVISION
For the proper method, we set the problem out like this:
21 632
LONG DIVISION
Divide 32 into 5240.
32 5240
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OPERATIONS WITH DIRECTED NUMBERS
• Adds and subtracts directed numbers
• Multiplies and divides directed numbers
• Uses a calculator to perform operations with directed numbers
• Uses strategies to perform mental operations with directed numbers
including grouping symbols
ADDING AND SUBTRACTING NUMBERS
To add or subtract numbers, use a number line.
To add, go to the right. To subtract, go to the left.
eg. 5 7
‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5
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ADDING AND SUBTRACTING NUMBERS
To add or subtract numbers, use a number line.
To add, go to the right. To subtract, go to the left.
eg. 5 – 8
‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5
DIRECTED NUMBERS
Directed numbers have size and direction.
We can write positive 5 as: 5, +5 or +5
Negative 5 can be written: – 5 or –5
If we are calculating with directed numbers, they can be placed in
brackets.
eg. Find 3 4
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ADDING AND SUBTRACTING DIRECTED NUMBERS
Find 3 4
The easiest way to handle questions with and – signs together is to
remember the following rules:
+ +
+
+
IT’S JUST LIKE TAKING A SHOWER!
When you take a shower:
• If you want the shower to be colder (MINUS), you can:
• Turn down the hot tap (
• Turn up the cold tap
• If you want the shower to be hotter (PLUS), you can:
• Turn up the hot tap
• Turn down the cold tap
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EXAMPLES
Let’s try these:
4 3 4 3 1
1 2 1 2 3
2 1 2 1 3
4 5 4 5 9
MULTIPLYING AND DIVIDING
Multiplying and dividing directed numbers uses the same rules as
before:
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MULTIPLYING AND DIVIDING
3 3 9 3 3 9
3 2 6 3 2 6
3 1 3 3 1 3
3 0 0 3 0 0
3 1 3 3 1 3
3 2 6 3 2 6
3 3 9 3 3 9
MULTIPLYING AND DIVIDING
Examples:
3 2 6 4 3 12
4 2 8 30 2 15
6 3 2 10 5 2
10 2 5 4
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USING THE CALCULATOR
There is only one extra key that we should know about for directed
numbers. It is the negative key.
We use this key for questions such as:
6 3
3 5
USING THE CALCULATOR
Brackets are also useful for directed number questions.
For example:
6 3 5
2 3 4
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STRATEGIES FOR MENTAL ARITHMETIC
We have used a number of strategies to aid mental arithmetic:
• Commutativity law
• Associativity law
• Distributive law
• Factors and multiples
COMMUTATIVITY LAW
Calculate 3 9
To make things easier, we can turn this around to become 9 3.
The only trick is to pick up the plus or minus sign in front of the number
when you move it.
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ASSOCIATIVITY LAW
Calculate 3 8 2
If we calculate the second two numbers first, we get an easier problem
to solve.
3 8 2 3 6 6 3 3
3 8 2 8 3 2 3
Likewise: 3 2 5 is easier if we calculate the last two digits first.
3 2 5 3 10 30
HANG ON A MINUTE!
Commutativity and Associativity Laws only work with addition and
multiplication!!!!
3 9 is really 3 9
So, it is an addition and providing we take the sign with the number it
will work.
3 9 9 3 6 6
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DISTRIBUTIVE LAW
Sometimes it is easier to calculate brackets first as BODMAS tells us to.
Sometimes it is easier to use the distributive law to ‘expand’ the
brackets.
Calculate: 2 3 5
Doing brackets first gives us:
2 3 5 2 5 3 2 2 4
DISTRIBUTIVE LAW
Sometimes it is easier to calculate brackets first as BODMAS tells us to.
Sometimes it is easier to use the distributive law to ‘expand’ the
brackets.
Calculate: 2 3 5
Expanding the brackets gives us:
2 3 5 2 3 2 5 6 10 4
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FACTORS AND MULTIPLES
As for positive integers, we can use our knowledge of factors and
multiples to help mental arithmetic.
Calculate 4 75 2
Firstly I know that multiplying the last two numbers first is going to be
easiest.
4 75 2
FACTORS AND MULTIPLES
As for positive integers, we can use our knowledge of factors and
multiples to help mental arithmetic.
Calculate 4 75 2
4 75 2
I know that 4 is equal to 2 2 so I am going to double 150, then
double it again.
4 75 2
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GLOSSARY
Integer A whole number with no fractional or decimal part
Directed Numbers Numbers with size and direction
Commutativity Law The order of the numbers in addition and multiplication is unimportant
Associativity Law With addition and multiplication, operations can be performed in any order
Distributive Law A method for removing brackets by multiplying the outside number by all numbers inside the brackets
Expanding brackets To remove the brackets using the distributive law
Factor A number that divides into another number without remainder
Multiple The number obtained when a number is multiplied by a counting number
Counting numbers The numbers 1, 2, 3, 4, …
Remainder The leftover number after a division
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