PR-6816IRE Essential Study Guides - Number Algebra Strategies

Number facts Algebra facts
Strategies
Maths
Addition table
+ 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10 11
2 3 4 5 6 7 8 9 10 11 12
3 4 5 6 7 8 9 10 11 12 13
4 5 6 7 8 9 10 11 12 13 14
5 6 7 8 9 10 11 12 13 14 15
6 7 8 9 10 11 12 13 14 15 16
7 8 9 10 11 12 13 14 15 16 17
8 9 10 11 12 13 14 15 16 17 18
9 10 11 12 13 14 15 16 17 18 19
10 11 12 13 14 15 16 17 18 19 20
Multiplication table
x 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100
Symbols
+ addition
– subtraction
x
multiplication
÷ division
> greater than
greater than or equal to
< less than
less than or equal to
Currency
£ pound
p pence
€ euro
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= equal to
not equal to
approximately equal to
° degree
% per cent
. decimal point
: ratio
$ dollar
c cent
Number facts
Special numbers
prime numbers
A prime number is a
number that can be
divided evenly by only 1
and itself; e.g. 2, 3, 5, 7,
11, 13 and 17.
prime factors
A prime factor is a
prime number that will
divide evenly into a given
number; e.g. 2, 3 and 5
are prime factors of 30.
even numbers
A whole number exactly
divisible by two.
odd numbers
A number that leaves
a remainder of 1 when
divided by 2.
composite numbers
A composite number is
a number that can be
divided by more than
itself and 1; e.g. 4, 6, 8,
9, 12 (i.e. not a prime
number).
Rounding
factors
A factor of a number is a number
that will divide evenly into that
number; e.g. the factors of 12 are 1,
2, 3, 4, 6 and 12. All numbers except
1 have more than one factor.
factorisation
To represent a counting number as
the product of counting numbers;
e.g. 24 = 4 x 6; 8 x 3; 12 x 2; 24 x 1
To show 24 as a product of its
prime factors, it would look like
this: 24 = 2 x 2 x 2 x 3
multiples
A multiple of a number is that
number multiplied by other whole
numbers; e.g. the multiples of 5 are
5, 10, 15, 20, 25 and so on.
positive numbers
A positive number has a value
greater than zero; e.g. 4, 25, 699
negative numbers
A negative number has a value less
than zero; e.g. -4, -25, -699
square numbers
A square number’s units can be
arranged into a square; e.g. 2 2 = 4 : :
The key to rounding numbers is to understand what is
‘closer to’ and use this to ‘round to the nearest’ so the
answer appears reasonable. Look at the chart below.
Round down to nearest 10 Round up to nearest 10
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
When rounding to the nearest 100:
• amounts ending between 1–49 are rounded down;
e.g. 348 becomes 300
• amounts ending between 50–99 are rounded up;
e.g. 353 becomes 400
Place value
Place value indicates the position of a numeral; e.g. 4032.87
4 thousands, 0 hundreds, 3 tens,
2 units, 8 tenths, 7 hundredths
Th H T U • Tths Hths
4 0 3 2 • 8 7
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Fractions
3
4
Fractions, decimals and percentages
A fraction is a number that describes part of a
group.
numerator
vinculum
denominator
It has a …
denominator
which is the number below the line, indicating how many
equal parts the whole number is divided into; and a …
numerator
which is the number above the line, indicating how many of
these parts are in consideration.
There are different types of fractions …
proper fractions mixed numerals
A proper fraction A mixed numeral is both
is a fraction where a whole number and a
the value of the proper fraction;
numerator is
smaller than the e.g. 1 1 2
denominator;
e.g. 1 equivalent fractions
2
Fractions that name
the same numerical
improper fractions
value even though the
An improper fraction numerals are different;
is a fraction where
the numerator is
1 2 3 4
e.g. 2 , 4 , 6 , 8
larger than the
denominator;
are all equal to each
4
other. They are
e.g. 3
equivalent fractions.
Equivalent fractions
simplest form
A fraction in its simplest form has a numerator and
denominator in their smallest form;
4
e.g.
8 in its simplest form is 1 2
Working with fractions …
adding fractions
Add the numerators
when the denominators
are the same;
e.g. 1 4 + 2 4 = 3 4
Decimal place value
system
The decimal number system is based on the number ten.
The decimal place system is based on multiples of ten,
when a whole number is divided into tenths, hundredths,
thousandths, and so on.
This diagram shows how we use decimal fractions or the
decimal place system.
1 whole
divided into ten equal parts
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
decimal notation
The expression of a numeral
in a form which uses the
decimal place value system;
e.g. 13.5
Percentages
A percentage is a
number or quantity
represented in
hundredths.
To convert a number or
fraction to a percentage, it
is necessary to multiply the
number by 100;
e.g.
8
10 x 100
1 = 800
10 = 80%
subtracting fractions
Subtract the numerators
when the denominators are
the same;
e.g. 3
4 – 1 4 = 2 4
each tenth can be divided
into a further ten equal
parts to make hundredths
decimal numeral
A base ten numeral;
e.g. 36, 96.1
common conversions
3
4 75% 0.75
2
3 66.66% 0.66
Viewing Sample
1
2 50% 0.5
1
3 33.33% 0.33
1
4 25% 0.25
1
5 20% 0.2
1
10 10% 0.1
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addend
Any number which is to be added;
e.g. 2 + 5 = 7
(2 and 5 are the addends).
addition
A mathematical operation that involves
combining; e.g. 3 + 4
ascending order
The arrangement of numbers from
smallest to largest.
commutative law
The order in which two numbers are
added or multiplied does not affect the
result;
e.g. 3 + 7 = 7 + 3 and 4 x 6 = 6 x 4
This is not the same for subtraction
and division.
complementary addition
The complement is the amount needed
to complete a set.
‘Counting on’ to a higher total;
e.g. the way change is paid after a
purchase.
The method of ‘subtracting’ which
changes the subtraction to an addition;
e.g. 7 – 3 = gives the same result
as 3 + = 7
consecutive
Consecutive numbers follow in order
without interruption; e.g. 11, 12, 13.
descending order
The arrangement of numbers from
largest to smallest.
difference
By how much a number is bigger or
smaller than another.
digit
Any one of the ten symbols 0 to 9
(inclusive) used to write numbers.
distributive law
To multiply a sum by a number is the
same as multiplying each addend by the
number and then adding the products;
e.g. 3 x (4 + 2) = (3 x 4) + (3 x 2)
3 x 6 = 12 + 6
18 = 18
dividend
A number which is to be divided by
another number;
e.g. 21 ÷ 3 (the dividend is 21).
Mathematical terms
divisible
A number is divisible by another
number if the second number is a
factor of the fi rst;
e.g. 6 is divisible by 2 because 2 is a
factor of 6.
division
The inverse operation of multiplication;
e.g. 21 ÷ 7 = 3 is the inverse of
7 x 3 = 21
Repeated subtraction can also be used
to achieve the same result.
21 – 7 – 7 – 7
e.g.
3
divisor
A number which is to be divided into
another number;
e.g. 21 ÷ 3 (the divisor is 3).
index notation
A shortened way of writing large
numbers as products of repeated
factors;
e.g. 1 000 000
= 10 x 10 x 10 x 10 x 10 x 10
= 10 6 where 6 is the index or
exponent and 10 is the base.
e.g. 6200
= 6.2 x 10 x 10 x 10
= 6.2 x 10 3
inverse operation
Opposite operations; addition and
subtraction are inverse operations;
multiplication and division are inverse
operations; halving and doubling are
also inverse operations.
multibase arithmetic
blocks (Base Ten
blocks)
Blocks used to give a concrete
representation of numbers, showing
the aspect of place value in base ten.
multiplication
A mathematical operation;
e.g. 7 x 2 = 14
Repeated addition can also be used to
achieve the same result;
e.g. 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14
number
An indication of quantity.
number line
A line on which equally spaced points
are marked. Points correspond, in
order, to the integers.
number sentence
A mathematical sentence that uses
numbers and operation symbols;
e.g. 6 + 7 = 13; 6 + 7 > 10
numeral
A symbol used to represent a number.
operation
The four operations of arithmetic:
addition, subtraction, multiplication and
division.
ordinal number
A number which indicates position in
an ordered sequence;
e.g. fi rst, second, third.
partitioning
A method of simplifying a problem in
order to calculate the solution;
e.g. 47 + 54 = (40 + 50) + (7 + 4)
= 90 + 11 = 101
product
The result when two or more numbers
are multiplied;
e.g. the product of 2, 3 and 4 is 24,
2 x 3 x 4 = 24
sequence
A set of numbers or objects arranged
in some order.
seriate
To put in order.
subtraction
A mathematical operation used in
three types of situations:
1. Take away
2. Difference (fi nding a difference)
3. Complementary addition (fi nding a
complement).
sum
The result when two or more numbers
are added.
total
The result when two or more numbers
are added.
whole number
The numbers 0, 1, 2, 3, 4 … are called
whole numbers.
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Order of operation
This is a rule for performing
operations in expressions
which have more than
one operation, to ensure
calculations are handled in
the same way.
Some calculators use an
‘algebraic operating system’
(AOS). This is used to follow
the Rule of Order.
Note:
Multiplication and division are equally powerful operations,
completed left to right in order as they appear, as are
addition and subtraction.
Patterns
Patterns are repeated designs. Patterns occur in
mathematics in many different ways and always follow a rule.
Simple number patterns such as odd and even numbers are
easily identifi ed. Here are some more complex patterns.
Triangular numbers
A triangular number is a number that can form the shape of a
triangle.
1 3 6 10
May also be shown as
Square numbers
A square number is a number that can form the shape of a
square.
1 4
9
16
Word problems as equations
Sequences
A sequence is a list of items. Any item in the list can be
named by its position: first, second, third, fourth and so on.
Some lists have patterns which define the position of each
item. There are two kinds of sequences:
Mathematical terms
A word problem needs to be changed into an equation before it can be solved. ‘Key words’ and phrases are used in word
problems to tell what type of operation (addition, subtraction, multiplication, division) should be used to solve the problem.
Look at the
table. It shows
some common
key words and
phrases, together
with the correct
operation needed
to solve the
problem.
Brackets ( )
Index notation 2 3
Multiplication x
Division ÷
Addition +
Subtraction –
Operation Key word Word problem Equation
addition
subtraction
multiplication
division
Algebra facts
sum The sum of my age and 15 equals 32. a + 15 = 32
total
The total of my age and my brother’s age, who is 11
years old, is 24.
a + 11 = 24
more than
Fifteen more than my age equals 32.
(Can also be subtraction.)
a + 15 = 32
difference
The difference between my age and my younger
sister’s age, who is 9 years old, is 3 years.
a – 9 = 3
less than Twelve less than my age equals 49. a – 12 = 49
product The product of my age and 21 is 252. a x 21 = 252
times Five times my age is 60. 5 x a = 60
group A number grouped into lots of 6 is 5. a ÷ 6 = 5
shared
equally
A ‘finite’ sequence is a list made up of
a limited number of items.
For example,
9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0 is the sequence of 6
alternating 9s and 0s.
1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the
fi rst 10 odd numbers.
An ‘infinite’ sequence is a list that
continues without end.
For example,
d, e, f, d, e, f, d, e, f, d, e ... is the sequence of the
letters d, e, f, repeating in this pattern forever.
2, 4, 6, 8, 10, 12, 14, 16 ... is the sequence of even
whole numbers. The 50th number in this sequence
is 100.
constant
Adding, subtracting,
multiplying or dividing by the
same amount each time.
equivalent
Having the same value.
equation
A statement of equality
between two expressions;
e.g. 3 x 4 = 6 + 6
132 lollies shared equally among a number of
children is 11 lollies each.
growing
A pattern that is becoming
larger.
inequality
Not having the same value.
repeating
The pattern uses the same
symbols or pictures over
and over.
e.g. 3, 4, 3, 4, 3 …
132 ÷ a = 11
To make mental maths easier, use your known number facts first. Here are some
ideas to help you …
Counting on
3 + 5 =
(change the order) 5 + 3 =
1 2 3
5 6 7 8
6 + 11 =
(change the order) 11 + 6 =
1 2 3 4 5 6
11 12 13
Doubling
3 + 3 = 6
5 + 5 = 10
13 + 13 = 26
Ask
clarifying
questions
What is the
question
asking?
Do I know
anything about
this topic?
What does …
really mean?
What else
could I find out
about to help
me answer the
question?
14 15 16 17
Make
assumptions
Using my
knowledge
of … I can
assume …
I think …
because I
know …
If I know …
then it could
be …
Mental strategies
25 + 25 = 50
33 + 33 = 66
70 + 70 = 140
Estimate
Can I use my
judgment
to make
a suitable
guess?
What
strategies
could I use
to estimate
a solution to
the problem?
Near doubles
5 + 6 =
(5 + 5) + 1 =
10 + 1 = 11
Survey
Can I find out
information
by asking a
sample of
people?
Do I need to
investigate
similar data
and compare
it to my
information?
OR
Bridging a ten
Example 1
9 + 6 =
(9 + 1) + 5 =
10 + 5 = 15
Partitioning
5 + 8 =
(5 + 5) + 3 =
10 + 3 = 13
Problem-solving strategies
Locate
information
Where
could I find
information to
help me solve
the problem?
Internet
Library
Things I
already know
Ask an expert
Friends
Adults
5 + 6 =
(6 + 6) – 1 =
12 – 1 = 11
Example 2
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7 + 5 =
(7 + 3) + 2 =
10 + 2 = 12
Present
findings
What is the
best way to
show what I
have found?
Diagrams
Tables
Graphs
Calculations
Explanations
It is important to check your work to make sure answers to problems are
correct and sensible. Checking your work can be done in many ways; some
are shown below.
Question your
answer
Ask yourself if the answer sounds
right. A question you might ask
yourself is …
‘Is the answer way too big or way
too small?’
If you think the answer does
not seem right, try some of the
following checking strategies.
Addition Subtraction Multiplication
odd + odd = even
odd + even = odd
even + even = even
Consider the size of the answer …
When you read a question, do you ever consider how big the answer will be? Think
about if the answer will be in tens, hundreds, thousands … or bigger! There is a
pattern that can help alert you to potential errors.
Addition
Multiplication
Number of digits
Answer likely to be in the …
1 digit + 1 digit units or tens
2 digits + 1 digit tens or hundreds
2 digits + 2 digits tens or hundreds
3 digits + 1 digit hundreds or thousands
3 digits + 2 digits hundreds or thousands
3 digits + 3 digits hundreds or thousands
1 digit x 1 digit units or tens
2 digits x 1 digit tens or hundreds
2 digits x 2 digits hundreds or thousands
3 digits x 1 digit hundreds or thousands
3 digits x 2 digits thousands or ten thousands
3 digits x 3 digits ten thousands or hundred thousands
Estimating
Estimating gives you an answer that is
close to the exact answer. It is usually
found by rounding or by using judgment
to make a ‘best guess’.
Front-end rounding
1. Look at the left-most digit in the
number.
2. Consider the place value of the digit.
For example,
3 2 1 5
6 9 1 0
+ 4 3 4 2
3 + 6 + 4 = 13
Viewing Sample
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So the estimate would be 13 000.
Checking strategies
Odd and even numbers
Odd and even numbers follow a pattern. Once you are aware of the pattern, all
you need to do is look at the units digits of the numbers in the problem, and the
answer to determine whether the answer is defi nitely wrong or possibly correct.
Think about the context of the
numbers before rounding.
For example,
When calculating the cost of groceries,
it is probably better to overestimate so
you do not run short of money.
When calculating, you
should:
1. Estimate
2. Calculate
3. Evaluate (How close was your
estimate? Could you improve on
your technique?)
odd – odd = even
odd – even = odd
even – odd = odd
even – even = even
odd x odd = odd
odd x even = even
even x even = even
Divisibility
rules
It is often helpful to know when one
number may be divided into another
without leaving a remainder.
2
3
4
5
6
9
10
11
12
25
100
Any even number;
e.g. 2, 4, 6, 8 or 0
Sum of all digits = multiple of 3;
e.g. 8652 = 8 + 6 + 5 + 2 = 21;
2 + 1 = 3
Last two digits divisible by 4;
e.g. 23 632, 45 656, 13 988
Any number ending in 0 or 5;
e.g. 20, 45, 670, 9845
Any number divisible by 2 and 3;
e.g. 54, 972, 2196
Sum of all digits = multiple of 9;
e.g. 72 567 = 7 + 2 + 5 + 6 + 7
= 27; 2 + 7 = 9
Any number ending in 0;
e.g. 10, 200, 7680, 98 450
Alternate digit sums differ by a
multiple of 11 or 0;
e.g. 76 285 = 7 + 2 + 5 = 14
6 + 8 = 14; 14 – 14 = 0
Any number divisible by 3 and 4;
e.g. 180, 720, 3600
Last two digits divisible by 25;
e.g. 6475, 8950
Last two digits are 00;
e.g. 67 400, 589 780 300
Repeat the calculation:
•
carefully, in exactly the same
way.
• using the inverse operation.
• using a different method.
Maths
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PR–6816
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ISBN 1-84654-025-9
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