RIC-6831 Maths Essentials - Number Algebra and Strategies 1 (Ages 11-15)

Mathematical terms
Number
Checking strategies
Strategies
Mental strategies
Strategies
Algebra Number
Strategies
Maths
addend
Any number which is to be added;
e.g. 2 + 5 (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; e.g. the way change is
paid after a purchase.
The method of ‘adding on’ 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
Multiplying the sum of two or more
numbers is the same as multiplying each
one by the number and then adding
their 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 (21).
divisible
A number is divisible by another
number if the second number is a factor
of the first; 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 (3).
equality
Having the same value.
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.
inequality
Not having the same value.
integers (directed numbers)
Numbers which are positive (+8) or
negative (–6).
multiplication
A mathematical operation;
e.g. 7 x 2 = 14
Repeated addition can also be used to
achieve the same result;
i.e. 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14.
negative number
A number smaller than zero. A negative
number is always written with a minus
sign; e.g. –6.
number
An indication of quantity.
number line
A line on which equally spaced points
are marked.
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. first, 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.
positive number
A number bigger than zero; e.g. +8.
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
e.g. How many eggs are left when
three are taken from a box of
six? 6 – 3 =
2. Difference (finding a difference)
e.g. What is the difference
between 18 and 13?
18 – 13 =
3. Complementary addition (finding a
complement)
e.g. How much change is given
from $5 for an article costing
$3.50? 3.50 + = 5.00
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.
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
Estimating
Number of digits
Answer likely to be in the …
1 digit + 1 digit ones 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 ones 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 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
So the estimate would be 13 000.
NOTE: you will always end up with an
underestimate.
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 ones digits of the numbers in the problem, and the answer to
determine whether the answer is definitely 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
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
©R.I.C. Publications
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10
11
12
25
100
Any number ending in 0;
e.g. 10, 200, 7680, 98 450
Alternative 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.
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 =
6 + 11 =
(change the order) 11 + 6 =
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?
1 2 3
5 6 7 8
2 3
12 13
Doubling
3 + 3 = 6
5 + 5 = 10
13 + 13 = 26
4 5 6
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 …
25 + 25 = 50
33 + 33 = 66
70 + 70 = 140
Problem solving strategies
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
Bridging a ten
9 + 6 =
(9 + 1) + 5 =
10 + 5 = 15
Partitioning
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
5 + 8 =
(5 + 5) + 3 =
10 + 3 = 13
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 1 Example 2
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
6831RE maths 1 y8.indd 1
3/02/11 3:35 PM

Fractions
A fraction is a
number that
describes part
of a group.
3
4
There are
different types
of fractions …
proper fractions
1
e.g. 2
The value of the numerator
is smaller than the
denominator.
improper fractions
4
3
e.g.
The numerator is larger
than the denominator.
Percentages
1
10 10% 0.1
So the estimate would be 13 000.
numerator
vinculum
denominator
mixed numerals
e.g. 1 1 2
Both a whole number and a
proper fraction.
equivalent fractions
Fractions that name the
same numerical value even
though the numerals are
different;
1
2 ,
2
4 ,
3
6 ,
4
8
e.g.
are all equal to each
other. They are equivalent
fractions.
simplest form
A fraction in its simplest
form has a numerator
and denominator in their
smallest form.
For example:
4
8 in its simplest form is 1 2
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;
8
e.g.
10 x 100
1 = 800
common conversions
3
4 75% 0.75
2
3 66.66% 0.66
1
2 50% 0.5
1
3 33.33% 0.33
1
4 25% 0.25
1
5 20% 0.2
Working with fractions …
adding and subtracting fractions
Add or subtract the numerators when the denominators are
the same:
e.g.
1
4 + 2 4 = 3 4
This system is based on multiples of ten, when a whole number
is divided into tenths, hundredths, thousandths … and so on.
or
3
4 – 1 4 = 2 4
If the denominators are different the fractions have to be
changed to ‘equivalent’ fractions before completing the sum.
For example:
multiplying fractions
Fractions do not need to
have the same denominator
to multiply them.
For example:
dividing fractions
Think of inverse operations
when dividing fractions.
The inverse of division is
multiplication. So perform the
inverse operation on the second
fraction and change the ÷ to x
For example:
Decimal place value system
finding percentages
of whole numbers
10 = 80% 1. Write the percentage as a
fraction.
2. Multiply the fraction,
with the whole number,
simplifying where possible.
For example:
15% of 75
15
100 x 75 1
15
=
100 x 75 4 1
15
= 4
x 3 1
= 45 4
= 11 1 4
Number
3
5
6 + 1 4
1. Multiples of:
6 = 6, 12, 18, 24
4 = 4, 8, 12, 16
LCD = 12
5
2. 6 x 2 2 + 1 4 x 3 3
3. 10
12 + 3 12 = 13
12 or 1 1 12
6
7 x 3 12
1. =
1
6
7 x 12
3 2
2. =
1
7 x 3 2
3. = 3 14
decimal notation
The expression of a numeral in
a form which uses the decimal
place value system; e.g. 13.5
11
12 – 5 6
1. Multiples of:
12 = 12, 24, 36
6 = 6, 12, 18, 24
LCD = 12
11
2. 12 – 5 6 x 2 2
3. 11
12 – 10
12 = 1
12
6
7 ÷ 1 3
1.
6
7 ÷ 1 3
2. =
6
7 x 3 1
3. =
18
7
4. = 2 4 7
inverse
LCD: Lowest Common Denominator – the lowest multiple
common to each denominator.
decimal numeral
A base ten numeral;
e.g. 36, 96.1
rounding off with decimals
It is easier to work with numbers which have been rounded
off to two or three decimal places. The rules for rounding
decimals are the same as for whole numbers.
0, 1, 2, 3, 4 – round down
6, 7, 8, 9 – round up
5 – may round up or down depending on context
For example,
1.834529 (round to 2 d.p.) = 1.83 (the third-place digit
is in the round down category, so round down)
1.837529 (round to 2 d.p.) = 1.84 (the third-place digit
is in the round up category, so round up)
Symbols
+ addition
– subtraction
x
multiplication
÷ division
> greater than
greater than or equal to
< less than
less than or equal to
= equal to
Multiplication table
Ratios
not equal to
approximately equal to
° degree
% per cent
. decimal point
: ratio
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
Rounding
c
cent
$ dollar
Number
The comparison of one number to another by division, e.g. the
ratio of 3 to 4 can be expressed as 3 4 or as 3:4
For example: 3 weeks:4 weeks
They may be simplified by multiplying or dividing each number
by the same value.
For example:
0.25:1 may be simplified by multiplying each
number by 4 to give 1:4.
5:25 may be simplified by dividing each number
by 5 to give 1:5.
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.
Numbers ending in 0, 1, 2, 3 or 4 are rounded DOWN.
Numbers ending in 6, 7, 8 or 9 are rounded UP.
Numbers ending in 5 may be rounded UP or DOWN
depending on context.
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 number
Whole number exactly divisible by two.
odd number
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).
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; 6 x 4; 3 x 8; 2 x 12; 1 x 24.
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 the product of that number multiplied
by other whole numbers; e.g. the multiples of 5 are 5, 10, 15, 20,
25 and so on.
Square root
The inverse operation
of squaring a number.
(4 2 = 4 x 4 = 16)
It looks like this:
16 = 4
Use the
button on your
calculator.
Place value
Roman
numerals
©R.I.C. Publications
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1 I 8 VIII
2 II 9 IX
3 III 10 X
4 IV 50 L
5 V 100 C
6 VI 500 D
7 VII 1000 M
Place value indicates the position of a numeral;
e.g. 1 354 032.87
1 million, 3 hundred thousands,
5 ten thousands, 4 thousands, 0 hundreds,
3 tens, 2 ones, 8 tenths, 7 hundredths
M HTh TTh Th H T O • Tths Hths
1 3 5 4 0 3 2 • 8 7
Patterns
Patterns are repeated sequences; e.g.
Triangular numbers
A triangular number is a
number that can form the
shape of a triangle.
Square numbers
A square number is a
number that can form
the shape of a square.
Fibonacci numbers
Beginning with 1, add these to make the third number. Add
the second and third number to make the fourth number
and so on. See the pattern below.
1, 1, 2, 3, 5, 8, 13, 21 and so on
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:
A ‘finite’ sequence is a list made up of a limited number of
items;
e.g. 9, 0, 9, 0, 9, 0 is the sequence of 3 alternating 9s and 0s.
An ‘infinite’ sequence is a list that continues without end;
e.g. 2, 4, 6, 8 … is the sequence of even whole numbers.
the 50th number in this sequence is 100.
Function
machines
Function machines are
used to demonstrate
that the same operation
applies to each number.
For example:
operation = x 5
In
Pascal’s triangle
A triangular array of numbers
where each number is the sum
of the two numbers directly
above it.
5
7
9
x 5
45
35
25
Out
1 3 6 10
1 4 9 16
Algebra
Mathematical terms
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.
They have an equal sign.
expressions
An expression is formed by
adding or subtracting terms;
e.g. 2x + 5y.
formula
A rule expressed in symbols;
e.g. x = y + 7.
It is a shortened way of
writing a set of instructions
to calculate a problem.
Word problems as equations
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.
inverse operations
Opposite operations; addition
and subtraction are inverse
operations; multiplication and
division are inverse operations;
halving and doubling are also
inverse operations.
pronumerals
Letters used to represent
numbers in algebra;
e.g. a, b, x, y.
Any letter can be used.
repeating
The pattern uses the same
symbols or pictures over and
over;
e.g. 3, 4, 3, 4, 3 …
terms
A term may be a number, a
pronumeral or a combination
of numbers and pronumerals;
e.g. x, 9, 2y.
Brackets ( )
Index notation 2 3
Multiplication x
Division ÷
Addition +
Subtraction –
Note:
Multiplication and division are equally powerful operations,
completed left to right in order as they appear, as are addition
and subtraction.
A word problem needs to be changed into an equation before it can be solved. ‘Keywords’ 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 keywords and phrases, together with the correct operation
needed to solve the problem.
Operation Keyword Word problem Equation
sum The sum of my age and 15 equals 32. a + 15 = 32
total The total of my pocket change and $10.00 is $12.85. a + $10.00 = $12.85
addition
Fifteen more than my age equals 32.
more than
a + 15 = 32
(Can also be subtracting.)
The difference between my age and my younger
difference
subtraction
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
multiplication
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
division shared 132 lollies shared equally among a number of
equally children is 11 lollies each.
132 ÷ a = 11
Maths
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RIC–6831
Copyright Information
ISBN 978-1-74126-286-5
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