RIC-6833 Maths Essentials - Number Algebra and Strategies 3 (Ages 11-15)

Algebra Number
Maths Strategies
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
Remember:
When x = a constant (e.g. x = 3),
a straight line is formed parallel
to the y-axis.
Null factor law
3. Create a table of values and
calculate coordinates.
Table of values
x –3 –2 –1 0 1 2 3
y
When y = a constant
(e.g. y = –5), a straight line is
formed parallel to the x-axis.
For two numbers, m and n, if m x n = 0 then either or both m = 0
or n = 0.
For example 2x(x – 5) = 0. Using the Null Factor Law, then:
2x = 0
x = 0
Check
2(0) (0 – 5) = 0
0 (–5) = 0
0 = 0
Inverse
proportion
Between two variables
a and b;
1
a
b
As an equation
a = K b
K = constant of variation
OR
x – 5 = 0
x = 0
Check
2(5) (5 – 5) = 0
10 (0) = 0
0 = 0
Algebra
Quadratic equations
Quadratics produce graphs called parabolas.
To plot a quadratic: Table of values
1. Complete a table of values x –2 –1 0 1 2 3 4
by substituting values as x.
y 4 –1 –4 –5 –4 –1 4
For example:
y = x 2 – 2x – 4
y = (–2) 2 – 2(–2) – 4
y = 4
so when x = –2, y = 4
gives coordinate (–2, 4)
2. Construct the axes.
3. Plot the points.
4. Join the points with a
smooth curve.
-6 -5 -4 -3 -2
Solving quadratic equations
10
9
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
y axis
1 2 3 4 5 6 7 8
x axis
(1, –5)
y = x 2 – 2x – 4
Turning point (1, –5)
1. Rearrange to make the right
side zero.
2. Make –x positive.
3. Factorise the quadratic
trinomial.
4. Use the Null Factor Law.
Simultaneous equations
Solve using a graphics calculator; or solve algebraically by:
Elimination
1. Number equations (1) and
(2).
2. Eliminate x or y by fi nding
the LCD.
3. Add equations.
4. Solve for x or y.
5. Substitute answer into
equation not used.
6. Check by substitution.
Word problems as equations
Substitution
1. Number equations (1) and
(2).
2. Replace x or y into
equation.
3. Add like terms.
4. Solve for x or y.
5. Substitute answer into
equation not used.
6. Check the solution satisfi es
both equations.
Follow these steps to make solving problems easier.
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.)
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
Checking strategies
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 defi nitely wrong or possibly correct.
Problem-solving strategies
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?
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
Make
assumptions
Using my
knowledge of …
I can assume …
I think …
because I know
…
If I know …,
then it could
be …
Strategies
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
Mathematical terms
addend
Any number which is to be added;
e.g. 2 + 5
(2 and 5 are addends).
addition
A mathematical operation;
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
compound interest
Unlike simple interest, with
compound interest, the interest on
the borrowed or invested amount
is added at regular intervals to the
principal.
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
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.
exponent
A number placed above a base number
to denote what power the base is
raised to.
gradient
Slope; measure of a line’s steepness,
calculated vertical rise divided by
horizontal 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.
inflation
A rise in prices.
Integers (directed numbers)
Numbers which are positive (+8) or
negative (–6).
Terms
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;
e.g. 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.
null factor law
If the product of numbers equals
zero, then one or more of the
numbers must be zero.
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).
scientific notation
(standard form) A way of expressing
very large or very small numbers.
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.
6833RE maths 1 yr10.indd 1
13/10/05 3:56:31 PM

Simplifying terms
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)
(a + b)(a – b) = a 2 – ab + ab – b 2
= a 2 – b 2
e.g. (3 + m)(3 – m) = 3 2 – 3m + 3m – m 2
= (3 2 – m 2 )
= (9 – m 2 )
Factorising
Look for common factors.
e.g. 5x 2 – 10x common factors: 5, x
Place the common factor outside the brackets.
e.g. 5x(x – 2)
Check your answer by expanding!
e.g. 5x(x – 2) = 5x 2 – 10x
Remember to check if the expressions follow the:
• Difference of two squares rule (2 terms)
• Perfect square rule (3 terms)
e.g. (a + b)(a + b) (a – b)(a – b)
= a 2 + ab + ab + b 2 = a 2 – ab – ab + b 2
= a 2 + 2ab + b 2 = a 2 – 2ab + b 2
NOT a 2 + b 2 NOT a 2 – b 2
Algebraic fractions
Multiplying and dividing:
Invert divisions to become
multiplications.
Factorise.
Look for factors in top and
bottom.
Cancel common factors.
Simplify.
Adding and
subtracting:
Factorising quadratic trinomials
Algebra
Find the LCD.
Rewrite as equivalent
fractions with same
denominators.
Add or subtract the
new numerators.
A quadratic (highest power is 2) trinomial (has 3 terms) such as:
ax 2 + bx + c and can be factorised by:
Find two factors that ...
– multiply to give ac
– add to give b
Rewrite x term as 2 terms.
Group the terms and factorise.
Take out a common factor.
Check by expanding.
4x 2 + 12x + 5
ac = 20, b = 12
the factors are 10, 2
4x 2 +10x + 2x + 5
2x(2x +1) + 5(2x +1)
(2x +1)(2x + 5)
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) 2 = (a + b)(a + b)
= a 2 + ab + ab + b 2
= a 2 + 2ab + b 2
NOT a 2 + b 2
e.g. (m + 3) 2
so a = m, b = 3
= m 2 + 2 x m x 3 + 3 2
= m 2 + 6m + 9
Linear equations
(a – b) 2 = (a – b)(a – b)
= a 2 – ab – ab + b 2
= a 2 – 2ab + b 2
NOT a 2 – b 2
e.g. (y – 4) 2
so a = y, b = 4
= y 2 – 2 x y x 4 + 4 2
= y 2 – 8y + 16
Perform inverse operations on both sides to solve linear equations.
Inverse operations are: + and – and x and ÷
e.g. 2x + 3 = –15
2x + 3 – 3 = –15 – 3
2x = –18
2x
2
= – 18
2
x = –9
Patterns
e.g. x – 5 = 7 4
x
4 – 5 + 5 = 7 + 5
x
4 = 12
x
4 x 4 = 12 x 4
x = 48
Patterns are repeated designs and always follow a rule.
Square numbers
Triangular numbers
Pascal’s triangle
1 4 9 16
1 3 6 10
Fibonacci numbers
1, 1, 2, 3, 5, 8, 13, 21
and so on
Index laws
a m x a n = a m + n
a m ÷ a n = am
a n = a m – n
a 0 = 1
a 1 = a
(a m ) n = a mn
(ab) m = a m b m
[ a b ]m = am
b m
a –m = 1
a m and 1
a
Fractions
3
4
–m = am
Fractional
powers
a 1 2
= a
(when a is a positive number or 0)
a 1 n
= m a
a 1 3
= 3 a
a m n
= (a 1 n )
m
a m n
= n a m
A fraction is a number that
describes part of a group.
numerator
vinculum
denominator
Working with fractions
Number
Number
Scientific
notation…
or standard form, is a way of
expressing very large or very
small numbers.
The number is written as a
product of:
a number between 1 and 10; and
a power of 10.
For example:
76 000 000 = 7.6 x 10 000 000
= 7.6 x 10 7
Very small numbers can be
expressed using negative
powers.
0.000 048 2 = 4.82 x 10 –5
Use the exponent button on
your calculator to enter powers
of 10.
EXP
proper fractions
1 The value of the numerator is
e.g. 2 smaller than the denominator.
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.
10
12 + 3 12 = 13
12 or 1 1 12
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 proper
2 fraction.
Multiplication table
There are different types of fractions
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
multiplying fractions
Fractions do not need to have
the same denominator to
multiply them.
For example:
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
Exponential growth
In mathematics, a quantity that grows
exponentially is one that grows at a rate
proportional to its size. This means that
for any exponentially growing quantity,
the larger the quantity gets, the faster
it grows; for example, anything that
grows by the same percentage every day
(or hour, month, year etc.) is growing
exponentially, such as cell multiplication
or population growth.
6
7 x 3 12
1. =
6
7 x 12
3 2
2. =
1
7 x 3 2
3. =
3
14
LCD: Lowest Common Denominator – the lowest multiple common to each denominator.
A = P(1 + i) n
A = P(1 – i) n
A = final amount
P = initial amount
i = growth rate/time
period
n = number of time
periods
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
1.
6
7 ÷ 1 3
2. =
6
7 x 3 1
3. =
18
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
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
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
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
=
=
=
15 x 75 3
100 4 x 1
15 x 3
4 x 1
45
4
= 11 1 4
Surds are numbers which can only be expressed using
the root symbol.
For example:
Simplify a surd if the number
under the square root sign has
9 = 3 so 9 not a surd a factor that is a perfect square.
7 2.646 so 7 is a surd For example:
a x b = ab
80 = 16 x 5
a a =
b b
= 4 5
Use the distributive law for surds with brackets:
The distributive law is:
a(b + c) = ab + ac, (a + b)(c + d) = ac + ad + bc + bd
Number
Number
profit and loss
The percentage profi t can be
calculated using:
% profit = profit
CP x 100
CP = cost price
If the % is a negative amount it is
known as a loss.
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
Ratios
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.
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)
compound interest
A = P(1 + i) n
P = principal
A = total amount owing—not just interest ($)
i = rate of interest per period (decimal)
n = number of interest periods
(e.g. if paid quarterly = 4 x years of loan)
inflation
Inflation can be calculated using the
compound interest formula:
A = P(1 + i) n
A = value after infl ation ($)
i = infl ation rate (decimal)
P = initial value ($)
n = number of periods (n = 1)
Powers
(Indices)
base
5 3 index form
5 x 5 x 5 expanded form
(The power tells you how
many times to multiply the base
number.)
Use the power button on your
calculator to help you.
For example:
3 4
5 3
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 –
+ ÷ + give +
– ÷ – gives +
Like signs give +
– ÷ + gives –
+ ÷ – gives –
Unlike signs give –
Maths
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RIC–6833
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ISBN 1-74126-288-7
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