Download Algebra - Crawshaw Academy

To achieve level 4 you need to
Algebra
Understand –
To achieve level 3 you need to
Number Patterns
Can you continue the number pattern
1, 2, 1, 2, 1, 2, . . .
2, 4, 6, 8 . . .
5, 10, 15, 20, . . .
1, 2, 1, 1, 2, 2, 1, 1, 1, 2, . . .
Start to use simple formulae expressed in words.
N
understand about –
NW
Formulae
NE
A formula gives the relationship between two or more things.
Using Formulae
W
E
Directions
Understand Clockwise and Anticlockwise
SW
Compass Directions
SE
S
N is North, E is East, S is South, W is West
Example
Use the formula to work out the total cost when you buy 17 pens at 25p each
Cost of pens = Cost of one pen x number of pens
Answer
C = 25 x 17
C = 425p = £4.25
Using Formulae
North East South and West are called compass directions.
Example - When the Gaelic singers went on tour in Canada they stayed in motels. The formula for the cost of these motels is
“£35 for one person plus £12 for each extra person”
Reading Maps
Find the cost of a motel in which
We often show where something is on a map by using a letter and
a number
The letter is usually given first
a. 4 people stayed
b. 5 people stayed
Coordinates - A way of describing the position of a point is by using two numbered lines called axes. These are
8
the lines labelled x and y on the diagram
Example
The School is at D7. The school is in the
Use and describe coordinates in the first quadrant
School
X
7
Compass Directions
NE is North-East and is halfway between North and East
NW is North-West and is halfway between North and West
SE is South-East and is halfway between South and East
SW is South-West and is halfway between South and West
square above D and along from 7.
The school could be anywhere in this
square and still be at D7.
6
Explore and describe number patterns and relationships
5
Reading from real life graphs
C
D
E
F
Sequences – Find the next term in a simple linear sequence
Y
5
4
3
2
1
0
y-axis
origin
1 2 3 4 5 6 7
Examples
1.
What is the next number in the sequence 2, 4, 6, 8, ___
2.
Draw the next diagram in the pattern
x-axis
Prime Numbers
A number which is divisible by only two numbers itself and 1 is called a prime number.
The first few prime numbers are 2, 3, 5, 7, 11.
1 is not a prime number since it is not divisible by two numbers.
Multiples
The multiples of a number are found by multiplying the number by each of 1, 2, 3, 4, 5 . .
Example
The first six multiples of 4 are → 4, 8, 12, 16, 20, 24
The calculator can be used to find multiples.
Factors
A factor of a given number is a whole number that divides exactly into the given number.
Example
Find all the factors of 12
Answer
Since 12 is divisible by 1, 2, 3, 4, 6, and 12 then the factors of 12 are 1, 2, 3, 4, 6, and 12.
A prime factor is a factor that is a prime number.
Example
The factors of 12 are 1, 2, 3, 4, 6 and 12
Of these factors only 2 and 3 are prime numbers therefore the prime factors of 12 are 2 and 3
Square Numbers
When a number is multiplied by itself the answer is called a square number
For instance since 7 x 7 = 49, 49 is a square number.
Level 5
Construct and use simple formulae involving one or two operations.
Writing Expressions and Formulae
Example
A piece of Blackpool rock is x cm long, Dale breaks off a bit that is 2 cm long
a.
Write an expression for the length that is left
b.
Write a formula for the length that is left.
Answer
a.
x–2
b.
L=x–2
Using Formulae
Replacing a letter with a number is called substituting into a formula
Example
S = N + 7. Find the value of S when N = 3
Answer
S = 3 + 7 = 10
Use brackets appropriately
Multiplying
Just as 2 x a is written as 2a, a x b is written as ab.
a x a is written as a2, just as 3 x 3 is written as 32.
Adding and Subtracting
Examples
5n + 6n = 11n
1.
5n – n = 5n – 1n = 4n
2.
3a + 4m + 2a + 5m
= 3a + 2a + 4m + 5m
= 5a + 9m
3.
5a + 3m – a + 2m
= 5a – a + 3m + 2m
= 4a + 5m
Removing Brackets
Everything inside the brackets must be multiplied by the number outside.
Examples
1.
2(a + 5) = 2a + 10
2.
2(a - 5) = 2a - 10
3.
2(a + 5) + 3(2a + 1)
= 2a + 10 + 6a + 3
= 2a + 6a + 10 + 3
= 8a + 13
Use coordinates in all four quadrants
Sequences – Finding terms in a simple linear sequence and a non-linear
number sequence
Level 6
Conversion graphs
Sequences - A sequence is a list of numbers. There is usually a relationship between the
numbers
Find and describe the rule for the next term or the nth term of a linear sequence
Writing Sequences from Instructions
The first number in a sequence is called the first term, the second number is called the second
term, the third number is called the third term and so on.
Example
For the sequence 1, 4, 9, 16, 25, . . . the first term is 1, the second terms is 4, and so on.
Finding Rules for Generating Sequences.
By finding the pattern in a sequence, we can continue the sequence or find missing terms.
Sequences are often based on these special numbers: odd numbers, even numbers, multiples,
squares, cubes
Formulae and Equations
In list 1 we have formulae.
We can find many different
values for S, L, H or P.
In list 2 we have equations.
Only one value can be found
for x or n or h or c.
List 1
S=N+1
L=a+4
H = 2h
P = 2(l + b)
List 2
2x = 8
n + 2 = 11
3h - 1 = 17
c/2 = 4
Writing Equations
Example - Take a number multiply it by 2. The answer is 8. 2x = 8
Linear tables and graphs
Example
Draw the graph of y = x - 2
X
-3
-2
-1
0
1
2
3
y=x-2
-5
-4
-3
-2
-1
0
1
Plot the points
(-3, -5), (-2, -4), (-1, -3), (0, -2), (1, -1), (2, 0), (3, 1), and then draw a line through the points, (the
line can be extended beyond the points plotted). Label the line and check that other points on the
line fit the rule
Derive equations from diagrams and solve.
Solving Equations –
Solving an equation means we find the value of the unknown
Checking Solutions - Always check your solution by substituting it back into the equation
There are many methods of solving equations.
Flowchart Method
Balance Method
Trial and Improvement
An equation remains true if the same number is added to (or subtracted from) both sides or if both
sides are multiplied (or divided) by the same number.
Solve linear equations
Example
Find the value of the letter in this equation
x+3=7
x=4
Solve equations such as x2 = 20, using trial and improvement methods.
2
3
Finding the value of x in equations involving x or x by using trial and improvement.
Simplify simple algebra
Examples
3a + 4b – 2a + 6b = a + 10b
2a x 4b = 8ab
b x b x b = b3
Simple algebraic rearrangement
Area of a rectangle = length x width
A=lxw
Length of a rectangle = Area
width
l=A
w
Substituting negative numbers into algebraic expressions
Example
Using the formula v = u + at, find the value of v when u = -5, a = 10, t = 3
v = -5 + 10 x 3 = 25