RIC-6832 Maths Essentials - Number Algebra and Strategies 2 (Ages 11-15)

Patterns
Algebra
Mathematical terms
Checking strategies
Strategies
Mathematical terms
Terms
Algebra Number
Maths Strategies
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
Pascal’s triangle
A triangular array of numbers
where each number is the sum
of the two numbers directly
above it.
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
5
7
9
x 5
45
35
25
Out
1 3 6 10
1 4 9 16
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 subtraction.)
subtraction
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
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
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.
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.
Ask
clarifying
questions
What is the
question asking?
Do I know
anything about
this topic?
What does …
really mean?
Addition
odd + odd = even
odd + even = odd
even + even = even
Subtraction
odd – odd = even
odd – even = odd
even – odd = odd
even – even = even
Multiplication
odd x odd = odd
odd x even = even
even x even = even
Problem-solving strategies
What else
could I find out
about to help
me answer the
question?
Make
assumptions
Using my
knowledge of
… I can assume
…
I think …
because I know
…
If I know …,
then it could
be …
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
So the estimate would be 13 000.
NOTE: you will always end up with an under
estimate.
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 looking for patterns.
Estimate
Can I use my
judgment
to make a
suitable guess?
What
strategies
could I use
to estimate a
solution to the
problem?
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?
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?)
Repeat the
calculation:
• carefully, in exactly the same
way
• using the inverse operation
• using a different method.
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Locate
information
Where could I
find information
to help me solve
the problem?
Internet
Library
Things I
already know
Ask an expert
Friends
Adults
Present
findings
What is the
best way to
show what I
have found?
Diagrams
Tables
Graphs
Calculations
Explanations
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.
commission
Fee based on percentage of the
value of sales.
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.
Special numbers
even number
Whole number exactly divisible by two.
odd number
A number that leaves a remainder of 1 when
divided by 2.
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.
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 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 (3).
equality
Having the same value.
gradient
Slope; the steepness of a line,
calculated vertical rise divided by
vertical run.
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).
loss
When costs exceed returns.
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.
parabola
Curved graph constructed from a
quadratic equation.
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.
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
principal
Amount borrowed or invested.
profit
The gain made (n a fi nancial
transaction).
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.
simple interest
Interest is calculated on the
original amount (the principal) at
the end of set periods.
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 (fi nding a difference)
e.g. What is the difference
between 18 and 13?
18 – 13 = ?
3. Complementary addition
(fi nding 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.
surd
Numbers that can only be
expressed using the root sign ( ).
total
The result when two or more
numbers are added.
whole number
The numbers 0, 1, 2, 3, 4 … are
called whole numbers.
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.
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).
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.
6832RE maths 1 yr9.indd 1
13/10/05 4:20:18 PM

Simplifying terms
Algebra
Collect like terms by rearranging the expression.
Add or subtract like terms.
e.g. 5a + 6b – 2a
= 5a – 2a + 6b
= 3a + 6b
collect like
terms
Difference of two squares
(a – b)(a + b) = a 2 + ab – ab – b 2
= a 2 – b 2
e.g. (x – 7)(x + 7) = x 2 – 7x + 7x – 49
= (x 2 – 7 2 )
= (x 2 – 49)
Expanding
To expand an expression a(b + c) use the:
Distributive law
a(b + c) = ab + ac
(factorised form) (expanded form)
To expand an expression (a + b)(c + d)
(a + b)(c + d) = a x c + a x d + b x c + b x d
= ac + ad + bc + bd
Remember the rules for multiplying directed numbers
(integers)!
Perfect square rule
(a + b)(a – b) = a 2 – ab + ab – b 2
= a 2 – b 2
(a + b) 2 = (a + b)(a + b)
NOT a 2 – b 2
e.g. (3 + m)(3 – m) = 3 2 – 3m + 3m – m 2
= (3 2 – m 2 )
= (9 – m 2 )
= a 2 + ab + ab + b 2
= a 2 + 2ab + b 2
NOT a 2 + b 2
= a 2 – ab – ab + b 2
= a 2 – 2ab + b 2
NOT a 2 – b 2
Factorising
e.g. (m + 3) 2
e.g. (y – 4) 2
so a = m, b = 3
so a = y, b = 4
= m 2 + 2 x m x 3 + 3 2 = y 2 – 2 x y x 4 + 4 2
Look for common factors.
e.g. 5x 2 – 10x
= m 2 + 6m + 9
= y 2 – 8y + 16
common factors: 5, x
Place the common factor outside the brackets.
e.g. 5x(x – 2)
Linear equations
Check your answer by expanding!
e.g. 5x(x – 2) = 5x 2 – 10x
Perform inverse operations on both sides to solve linear
equations. Inverse operations are: + and – and x and ÷
Remember to check if the expressions follow the:
e.g. 2x + 3 = –15
• Difference of Two Squares rule (2 terms)
e.g. x – 5 = 7
• Perfect Square rule (3 terms)
2x + 3 – 3 = –15 – 3 4
x
2x = –18 4 – 5 + 5 = 7 + 5
e.g. (a + b)(a + b) = a 2 + ab + ab + b 2
2x
= a 2 + 2ab + b 2
2
= – 18
x
4 x 4 = 12 x 4
2
x = 48
NOT a 2 + b 2
x = –9
(a – b)(a – b) = a 2 – ab – ab + b 2
= a 2 – 2ab + b 2
Quadratic equations
Quadratics produce graphs called parabolas.
To plot a quadratic:
1. Complete a table of values
by substituting values as x.
For example:
y = x 2 – 4x + 4
y = (–1) 2 – 4(–1) + 4
so when x = –1, y = 9
2. Construct the axes.
3. Plot the points.
4. Join the points with a
smooth curve.
Table of values
x –1 0 1 2 3 4
y 9 4 1 0 1 4
Linear relationships
A linear relationship is a relationship between two variables
which produces a straight line when graphed.
The gradient is the slope of a straight line.
Gradient (m) of line AB
vertical rise
m =
horizontal run or
where A(x 1
, x 2
),
and B(y 1
, y 2
)
y 2
– y 1
m = x 2
– x 1
You can sketch a linear graph using:
1. x- and y-intercepts.
At x-intercept, y = 0
At y-intercept, x = 0
2. y = mx + c
m = gradient (rise/run)
c = y-intercept
3. Create a table of values and
calculate coordinates.
Table of values
x –3 –2 –1 0 1 2 3
y
Symbols
+ addition
– subtraction
x
multiplication
÷ division
> greater than
greater than or equal to
< less than
less than or equal to
= equal to
Place value
not equal to
approximately equal to
° degree
% per cent
. decimal point
: ratio
c
cent
$ dollar
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
Fractions
A fraction is a number that
describes part of a group.
adding and subtracting fractions
Add or subtract the numerators when the denominators are the
same:
1
e.g.
4 + 2 4 = 3 4 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:
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.
3
4
numerator
vinculum
denominator
Working with fractions
10
12 + 3 12 = 13
12 or 1 1 12
Number
proper fractions
1 The value of the numerator is
e.g. 2 smaller than the denominator.
improper fractions
4 The numerator is larger than the
e.g. 3 denominator.
mixed numerals
e.g.1 1 Both a whole number and a
2 proper fraction.
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
Multiplication table
There are different types of fractions
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
multiplying fractions
Fractions do not need to
have the same denominator
to multiply them.
For example:
6
7 x 3 12
1
6
1. =
7 x 12
3 2
1
2. =
7 x 3 2
3
3. =
14
LCD: Lowest Common Denominator – the lowest multiple common to each denominator.
equivalent fractions
Fractions that name the same numerical value
even though the numerals are different;
1 2 3 4
e.g. 2 , 4 , 6 , 8 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.
4
e.g.
8 in its simplest form is 1 2
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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:
6
7 ÷ 1 3
6
1.
7 ÷ 1 3
6
2. =
7 x 3 1
18
3. =
7
4. = 2 4 7
inverse
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;
8
e.g.
10 x 100
1 = 800
10 = 80%
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
1
10 10% 0.1
Decimals
This system is based on multiples of ten, when a whole number
is divided into tenths, hundredths, thousandths … and so on.
recurring decimals
decimal numbers that continue forever in a repeating pattern.
e.g. 3. 333 333 333....
This can be written as 3.3
The digit below the dot is the repeating digit.
or 2.232 323.....
Can be written as 2.23
The digits below the bar are the repeating digits.
terminating decimals
are decimal numbers that do end
e.g. 3.125
finding percentages
of whole numbers
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
rounding
The rules for rounding decimals are the same as for whole
numbers.
0, 1, 2, 3 and 4 – round down
6, 7, 8, 9 – round up
5 – may round up or down depending on context
Round to 2 decimal places (2 d.p.)
e.g. 1.832539 = 1.83 (round down)
Round to 3 decimal places (3 d.p.)
e.g. 1.832539 = 1.833 (round up)
Surds
Surds are numbers which
can only be expressed
using the root symbol.
For example:
9 = 3 so 9 not a surd
7 2.646 so 7 is a surd
a x b = ab
a a =
b b
Number
Simplify a surd if
the number under
the square root sign
has a factor that is a
perfect square.
For example:
80 = 16 x 5
= 4 5
3
Ratios
profit and loss
The percentage profit
can be calculated using:
% profit = profit
CP x 100
CP = cost price
If the % is a negative
amount it is known as
a loss.
simple interest
SI = PRT or SI = PrT
100
SI = simple interest ($)
P = principal ($)
R = rate per annum (decimal)
r = rate per annum (%)
T = time (years)
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 simplifi ed
by multiplying or dividing
each number by the same
value.
For example:
0.25:1 may be simplifi ed by
multiplying each number by 4
to give 1:4.
5:25 may be simplifi ed by
dividing each number by 5 to
give 1:5.
Powers
(Indices)
base
2 3
2 3 index form
2 x 2 x 2 expanded form
(The power tells you how
many times to multiply the
base number.)
Use the power button on
your calculator to help you.
x y or y x or a b or a x
e.g. 3 4 3 x y 4 = 81
(3 x 3 x 3 x 3)
Multiplying and dividing
power
Directed numbers (Integers)
+ x + gives +
– x – gives +
Like signs give +
+ x – gives –
– x + gives –
Unlike signs give –
commission
A percentage of the
value of goods sold
C = R x SP
C = commission to be paid
R = rate of commission
SP = selling price
discounts
Whenever you
buy something at a
discounted price the
following applies:
SP = MP – D
(where D = MP x D%)
MP = market price
D% = discount percentage
+ ÷ + give +
– ÷ – gives +
Like signs give +
– ÷ + gives –
+ ÷ – gives –
Unlike signs give –
Maths
No part of this publication may be reproduced in any form or
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or recording, or by any information storage and retrieval
system, without written permission from the publisher.
RIC–6832
ISBN 1-74126-287-9
9!BMFBI