Numbers and the Number System 2

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Numbers and the Number System 2
2
Qualifying to Teach 2.1b
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Objectives

• To continue to develop knowledge and
  understanding of numbers and the number system.

• Develop knowledge and understanding of
• Rounding
• The 4 basic operations
• Brackets
• Index notation
• Laws of operations.

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Recap from Last Week
What are the following a) Natural Numbers b)
Integers c) Rational Numbers d) Irrational
Numbers e) Factors f) Multiples g) Prime
Numbers h) Highest Common Factor
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Rounding
Quite often, an approximate answer is acceptable.
Rounding gives approximate answers. Rounding is
very common for numbers in everyday life, for
example - Populations are often expressed to
the nearest million. - The number of people
attending a pop concert may be expressed to the
nearest thousand. - Inflation may be expressed
to the nearest whole number.
If the number we are considering is less than 5
we return to 0. If the number we are
considering is 5 or more we increase the digit to
the left of it by 1.
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Round the Following Numbers to the Nearest 10

• 23
• 25
• 29
• 44
• 79
• 81
• 99
• 126
• 143
• 265

• 20
• 30
• 30
• 40
• 80
• 80
• 100
• 130
• 140
• 270

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Significant Figures
Any number can be rounded to a given number of
significant figures (written s.f.).
Use the following - Count along the digits to
the required number of significant figures. -
Look at the next digit (to the right) in the
number. If its value is less than 5, leave the
digit before it (to the left) as it is. If its
value is 5 or more, increase the digit before it
by 1. - Replace all the digits to the right, but
before the decimal point, by zeros, to keep the
number at its correct size. (Digits to the right
after the decimal point can just be left out)
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Have a go at these
Round 7638.462 to the number of significant
figures shown.
6 s.f. 7638.462 5 s.f. 7638.462 4 s.f.
7638.462 3 s.f. 7638.462 2 s.f. 7638.462 1
s.f. 7638.462
7638.46 7638.5 7638 7640 7600 8000
Fill with 0s to keep the number at its correct
size.
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Decimal Places
Any number can be rounded to a given number of
decimal places (written d.p.).
Use the following - Count along the digits to
the required number of decimal places. - Look at
the next digit (to the right) in the number. If
its value is less than 5, leave the digit before
it (to the left) as it is. If its value is 5 or
more, increase the digit before it by 1. - Leave
out all the digits to the right.
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Have a go at these
Round 6.4275093 to the number of decimals places
shown.
6 d.p. 6.4275093 5 d.p. 6.4275093 4 d.p.
6.4275093 3 d.p. 6.4275093 2 d.p. 6.4275093
1 d.p. 6.4275093
6.427509 6.42751 6.4275 6.428 6.43 6.4
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Operations
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Introduction
Throughout KS 1 and 2 children are not only
introduced to numbers, but also to ways in which
they may be combined or acted upon (which we will
discuss later).
This leads us to the four basic operations or
actions.
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The Four Basic Operations

• Addition

• Subtraction

• Multiplication

• Division

The operations act on 2 or more numbers (or
elements).If we consider addition and add 5 to 3
the result is 8. (This means that the 3 and 5 are
put together and the result is a single number,
in this case 8). There is another way of viewing
3 5.
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Acting Upon
If we continue to consider 3 5.
The first number (3 in this case) is acted upon
or changed by an operator (in this case add 5).
The operator not only defines the type of action
(addition) it also specifies its magnitude (5).
However we see it, one of the most important
things to consider in primary school is the
identification of an appropriate operation to
solve a given problem.
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Brackets
Brackets help complicated sums, expressions or
equations to be written quickly and clearly.
Consider the expression 8 x (4 - 2) primary
form
Which can also be written 8(4 2) First
you take 2 away from 4, then multiply the result
by 8.The number in front of the bracket is to be
multiplied by the result of the calculation
within the bracket.
If there is more than one bracket, all values
within brackets are worked out before the
operations between brackets.
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Have a go at these
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Index Notation (Indices)
When you multiply a number by itself, use the
following shorthand.

• 5 x 5 5² Say, 5 to the power 2 (or 5
  squared)
• 5 x 5 x 5 5³ Say, 5 to the power 3 (or 5
  cubed)
• 5 x 5 x 5 x 5 Say, 5 to the power 4.

Power (or index)
base
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Multiplying/Dividing Numbers with Indices
To multiply two numbers with indices when the
bases are the same you just add their indices.
To divide two numbers with indices when the bases
are the same you just subtract their indices.
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Have a go at these
1. What is the value of
2. Find the value of
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Order of Operations
1. The operations inside the brackets are carried
out first.
2. Next the multiplication and division
operations (including indices) are worked out.
3. Finally the addition and subtraction
calculations are completed.
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Have a go at these

• 10 2 14 3 x 7 6
• 7 4 x 3 - 5 x (6 3) 10
• 26 2 x 7 12 3 4
• 3 5 x 2³ - 4 3 x (7 2)
• (10 2) 2² 5 x 3

• 4
• 14
• 12
• 66
• 17

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Laws of Operations

• Associative Law
• Commutative Law
• Distributive Law

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Associative Law
Does the order matter?
Consider adding three numbers, such as 3 5 2
Starting with the first two numbers (3 5)
2 8 2 10 Starting with the second two
numbers 3 (5 2) 3 7 10
Addition is therefore said to be associative.
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Are the following Associative?

• Subtraction
• Multiplication
• Division

• No
• Yes
• No

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Commutative Law
Can I change places with you?
Lets consider addition again.
4 5 9 and 5 4 9 (It doesnt matter
which way round we write or work out the sum, we
still get the same answer.)
Therefore addition is said to be commutative.
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Are the following Commutative?

• Subtraction
• Multiplication
• Division

• No
• Yes
• No

Note Children usually assume that subtraction
and division are commutative and think that 8
6 6 8 (As you know this is not the case)
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For next week

• Find out what the Distributive Law is and
  whether the four operations follow this law.

• Read the following -
• Chapter 7 from Mathematical Knowledge for
  Primary Teachers by Suggate, Davis and Goulding.
• Chapters 14, 15 and 16 from Mathematics
  Explained for Primary Teachers by Haylock.