Transcript Module 3 Revision

Module 3
Super Learning Day Revision
Notes November 2012
In any exam
Always
•
•
•
•
Read the question at least twice
Show ALL your working out
Check your units eg. cm, cm² etc.
Read the question again to make sure you
have actually answered the question asked
• Check that your answer is sensible
Revision Notes
Shape and Measurement
Interior and exterior angles of
polygons
•
•
•
•
In a REGULAR polygon
Exterior angles add up to 360°
360 ÷ number of sides = exterior angle
180 – exterior angle = interior angle
Regular polygons have
n lines of symmetry
Rotational symmetry of order n
- where n is the number of sides
Tessellations
A tessellation is a tiling
pattern with no gaps
Learn names of shapes
• Triangles:Equilateral, isosceles, right angled,and scalene
• Quadrilaterals:Square, rectangle, parallelogram, rhombus,
trapezium, kite and arrowhead.
Perimeter
The perimeter is the distance
round the edge of the shape
Area formulas
•
•
•
•
Area of a Triangle = base x height ÷ 2
Area of a Rectangle = length x width
Area of a parallelogram = length x height
Area of a Trapezium = (a + b) ÷ 2 x height,
(where a and b are the lengths of the parallel sides)
• Area of a circle = πr²
Volume Formulas
• Volume of cuboid = length x width x height
• Volume of a prism = Area of cross-section
x length
• Volume of cylinder = area of circle x height
Surface Area
Work out the area of every face separately
then add them together
NOTE
To find the surface area of a cylinder you need to add
together the area of the 2 circles AND the rectangle that
wraps round the cylinder.
The length of the rectangle is equal to the circumference
of the circle and the width is the height of the cylinder
Views
Plan view
Plan
Side view
Side elevation
Front
Elevation
Front view
Conversions
1cm² = 100mm²
1cm = 10mm
1cm = 10mm
1m³ = 1 000 000 cm³
100cm
100cm
1m =100cm
Congruent
Means
Alike in every respect
Similar
Means
Same shape, Different size
( one is an enlargement of the other)
Metric /Imperial conversions
•
•
•
•
1Kg = 2¼ lbs
1m = 1 yard (+10%)
1 litre = 1¾ pints
1 inch = 2.5 cm
• 1gallon = 4.5 litres
• 1 foot = 30 cm
• 1 metric tonne = 1
imperial ton
• 1 mile = 1.6 Km
or 5miles = 8 Km
Calculators and time
• Beware
When using a calculator to work out
questions with time make sure you enter the
minutes correctly
e.g. 30 minutes = 0.5 of an hour
15 minutes = 0.25 of an hour
Density = Mass
Volume
e.g. g/cm³
Speed = Distance
Time
e.g. Km/hour
m/sec
Geometry and Graphs
Angle Facts
• An Acute angle is less than 90°
• A Right angle is equal to 90°
• An Obtuse angle is more than 90°, but less
than 360°
• A straight angle is equal to 180°
• A Reflex angle is more than 180°
Bearings
Always
Measure from the NORTH line
Turn clockwise
Use 3 figures (eg. 30° = 030°)
Drawing Bearings
To measure a bearing of B from A the North
line is drawn at A. This is because the
question says ‘from A’
N
B
A
Angle Rules
•
•
•
•
•
Angles in a triangle add up to 180°
Angles on a straight line add up to 180°
Angles in a quadrilateral add up to 360°
Angles round a point add up to 360°
The two base angles of an isosceles triangle
are equal
Parallel lines
• Look for ‘Z’ angles (Alternate angles)
• Look for ‘F’ angles (corresponding angles)
Alternate angles are equal
Corresponding angles are equal
Transformations
(ask for tracing paper to help you with these)
Reflections
Always reflect at right angles to the
mirror line
Diagonal mirror lines are sometimes
called y = x
or y = -x
Rotations
Always check for (or state)
1. The centre of rotation
2. The amount of turn
3. The direction (either clockwise or anticlockwise)
Enlargements
• Check the scale factor and centre of
enlargement (if there is one)
• Draw construction lines from the centre of
enlargement to help you draw the new
shape
• Remember a scale factor of ½ will make the
shape smaller
Vector translations
A vector translation slides the shape to a
new position
+y
-x
x
y
+x
-y
The top number x moves
the shape right (or left if it
is negative)
The bottom number y
moves the shape up (or
down if it is negative)
Loci
A Locus (more than one are called Loci) is
simply:A path that shows all the points which fit a
given rule
There are only 4 to remember
Locus 1
The locus of points which are
A FIXED DISTANCE from a GIVEN POINT
Is simply a CIRCLE
•
Locus 2
The locus of points which are
A FIXED DISTANCE from a GIVEN LINE
This locus is an oval shape
It has straight sides and ends which are perfect semicircles
Locus 3
The locus of points which are
A EQUIDISTANT from TWO GIVEN
LINES
This is the Angle Bisector (use compasses!)
Locus 4
The locus of points which are
A EQUIDISTANT from TWO GIVEN
POINTS
B
This locus is the
perpendicular bisector
of the line AB
A
Pythagoras
Pythagoras
The square on the hypotenuse is equal to the sum of the
squares on the other two sides
Hypotenuse
a
h² = a² + b²
b
Remember
To find the
hypotenuse
Square
Square
Add
Square root
To find a
short side
Square
Square
Subtract
Square root
x and y Coordinates
y
10
9
x negative
y positive
A graph has 4
different regions
x positive
y positive
8
7
6
Always plot the x
value first followed
by the y value
5
•(-4,3)
4
3
•(4,2)
2
1
-10 -9
-8 -7
-6 -5 -4
-3 -2
•(-6,-2)
-1
0
1
2
3
4
5
6
7
8
9
10
x
x goes across
-1
y goes up/down
-2
-3
-4
-5
-6
x negative
y negative
-7
-8
-9
-10
•(7,-5)
x positive
y negative
‘in the house and
up the stairs’
Midpoint of a line
Midpoint is just the middle
of the line!
To find it just add the x
coordinates together and
divide by 2
Then add the y coordinates
together and divide by 2
You have just found the
midpoint
For example
If A is (2,1) and B is (6,3)
Then the x coordinate of the
mid point is
(2 + 6) ÷ 2 = 4
And the y coordinate is
(1 + 3) ÷ 2 = 2
So the mid point is (4,2)
Straight Line Graphs
y
x= a is a vertical line
through ‘a’ on the x axis
10
x = -5
9
y= x
8
7
y = b is a horizontal line
through ‘b’ on the y axis
6
5
4
3
Don’t forget that the y
axis is also the line x = 0
y=2
2
1
-10 -9
-8 -7
-6 -5 -4
-3 -2
-1
0
1
2
3
4
5
6
7
8
-1
9
10
x and the x axis is also the
line y = 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
y= -x
The diagonal line y = x
goes up from left to right
and the line y = -x goes
down from left to right
Straight Line Graph:– y = mx + c
In the equation y = mx + c
The m stands for the gradient and the c is
where the line crosses the y axis
Using y = mx + c to draw a line
1. Get the equation in the form y = mx + c
2. Identify ‘m’ and ‘c’ carefully (eg. In the
equation y = 3x + 2, m is 3 and c is 2)
3. Put a dot on the y axis at the value of c
4. Then go along one unit and up or down by
the value of m and make another dot
5. Repeat the last step
6. Join the three dots with a straight line
Finding the equation of a straight
line
1. Use the formula y = mx + c
2. Find the point where the graph crosses the
y axis. This is the value of c
3. Find the gradient by finding how far up
the graph goes for each unit across. This is
the value of m.
4. Now just put these two values into the
equation
Quadratic Graphs
1. Fill in the table of values
2. Carefully plot the points
3. The points should form a smooth curve. If
they don’t they are wrong!
4. Join the points with a smooth curve
5. The graph should be ‘u’ shaped
Simultaneous equations with
graphs
1.
2.
3.
4.
Do a table of values for each graph
Draw the two graphs
Find the x and y values where they cross
This is the solution to the equations
Numbers and
Algebra
Special Number Sequences
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Even numbers 2, 4, 6, 8, 10,……
Odd numbers
1, 3, 5, 7, 9, 11,…..
Square numbers 1, 4, 9, 16, 25,….
Cube numbers 1, 8, 27, 64, 125,….
Powers of 2
2, 4, 8, 32, 64,…..
Triangle numbers 1, 3, 6, 10, 15, 21,…..
Number Patterns and Sequences
There are five different types of number sequences
1. ADD or SUBTRACT the SAME NUMBER
e.g. 2 5 8 11 14 ….
30 24 18 12….
+3
+3
+3
+3
-6
-6
-6
The RULE
‘add 3 to the previous term’
‘Subtract 6 from the previous term’
2. ADD or SUBTRACT a CHANGING NUMBER
e.g. 8 11 15 20…….. 53 43 34 26……
+3
+4
+5
The RULE
‘Add 1 extra each time to the previous term’.
-10
-9
-8
‘Subtract 1 extra each time from the previous term’
3.
MULTIPLY by the SAME NUMBER EACH TIME
e.g. 5 10 20 40……
x2
x2
x2
The RULE
‘Multiply the previous term by 2’
4.
DIVIDE by the SAME NUMBER EACH TIME
e.g. 400 200 100 50……
÷2
÷2
÷2
The RULE
‘Divide the previous term by 2’
5.
ADD THE PREVIOUS TWO TERMS
e.g. 1 1 2 3 5 8
+
+
+
+
The RULE
‘Add the previous two terms’
Finding The nth Term
To find the nth term you can use the formula
dn + (a – d)
Where ‘d’ is the difference between the terms
And ‘a’ is the first number in the sequence
e.g. 3 7 11 15 19…
‘d’ is 4 (because you add 4 to get the next term)
and ‘a’ is 3 (that is the first number)
This means that (a – d) is (3 – 4) = -1
So the nth term, dn + (a – d) is 4n - 1
Algebra
Terms
A term is a collection of numbers, letters and brackets, all multiplied
/divided together
 Terms are separated by + and
signs
–
 Terms always have a + or a – sign attached to the front of them
 E.g.
4xy
Invisible + sign xy term
+ 5x²
x² term
- 2y
y term
+ 6y²
y² term
+4
number term
Simplifying (Collecting Like Terms)
EXAMPLE
2x
–4
x terms
Simplify 2x – 4 + 5x + 6
+ 5x
+6
=
+2x +5x -4
number terms
7x
+6
+2
So
2x – 4
1.
Put bubbles round each term, making sure that each bubble has a + or –
sign.
2.
Then move the bubbles so that LIKE TERMS are together
3.
Collect the like terms using the number line to help you
+ 5x + 6
= 7x + 2
Multiplying out Brackets
• The thing outside the bracket multiplies each separate term
INSIDE the bracket
• When letters are multiplied together they are just written
next to each other e.g. pq
• Remember R x R = R²
• Remember , a minus outside the brackets reverses all the
signs when you multiply
Expanding Double Brackets
• Remember to multiply everything in the second
bracket by each term in the first bracket
( 2p – 4 ) ( 3p + 1 )
= (2p x 3p) + (2p x 1) + (-4 x 3p ) + ( -4 x 1)
= 6p² + 2p
- 12p
-4
= 6p² -10p -4
Squared Brackets
Example (3p + 5)²
Write this out as two brackets (3p + 5)(3p + 5)
(3p + 5)(3p + 5) = 9p² +15p +15p +25
= 9p² +30p +25
The usual wrong answer is 9p² +25 !!!
Factorising
• This is the exact opposite of multiplying out brackets
• Take out the biggest NUMBER that goes into all terms
• For each letter in turn take out the highest power that
will go into EVERY term
• Open the brackets and fill with all the bits needed to
reproduce each term
• E.g. 15x4y +20 x²y³z -35x³yz²
5x²y (3x² + 4y²z - 7xz² )
Biggest number that
that will divide into
15, 20, and 35
Highest powers
of x and y that will
go into all three terms
z wasn’t in all terms so
it can’t come out as
a common factor
Writing Formulas
These questions ask you to write an equation
The only things you will have to do are:Multiply x
Divide x
Square x (x²)
Add or subtract a number
Example 1:- to find y, you multiply x by 3 and then subtract 4
Start with
x
Times it by 3
3x – 4
3x
So y = 3x - 4
Subtract 4
Example 2:- to find y, square x, divide it by 3 and then subtract 7
Start with x
x²
x²
x² - 7
So y = x² - 7
3
3
3
square it
divide by 3
subtract 7
Formulas from words
This time you change a sentence into a formula
Example:- Froggatt’s deep-fry CHOCCO- BURGERS (chocolate
covered beef burgers) cost 58 pence each. Write a formula for the
total cost, T, of buying n CHOCCO-BURGERS at 58p each.
In words the formula is
Total cost = Number of CHOCCO-BURGERS x 58p
Putting letters in, it becomes:T = n x 58
It would be better to write this as
T = 58n
BODMAS
B
Brackets
O
Other (like squaring )
D
Divide
M
Multiply
A
Add
S
Subtract
• Remember this is the order for doing the
sums.
Substitution
To substitute numbers into an expression or a formula all
you need to do is replace the letters with their values and
work out either the solution of the equation or the value of
the expression
(don’t forget to use BODMAS)
Examples of substitution
Example of substituting in a formula
If P = 3Q + 7 what is P when Q is 8?
P = 3Q + 7
P=3x8+7
P = 24 + 7
P = 33
Example of substituting in an expression
If a = 3, b = 5and c = 7 what is the value of a² - b + 2c
a² - b + 2c
(3)² - 5 +2( 7 )
9 - 5 + 14
18
Solving Equations
To solve equations just follow these 3 rules
1.
Always do the same thing to both sides of the equation
2.
To get rid of something just do the opposite.
The opposite of + is – and the opposite of – is +
The opposite of x is ÷ and the opposite of ÷ is x
3.
Keep going until you have a letter on its own
Examples of Solving Equations 1
 Solve 5x = 15
5x means 5 x x so do the
opposite and divide both
sides by 5
x=3
 Solve p/3 = 2
p/3 means p ÷ 3 so do the
opposite and multiply both
sides by 3
p=6
 Solve 4y – 3 = 17
The opposite of –3 is +3 so
add 3 to each side
4y = 20
The opposite of x 4 is ÷ 4 so
divide both sides by 4
y=5
Examples of Solving Equations 2
 Solve 2( x + 3 ) = 11
The opposite of x 2 is ÷ 2 so
divide both sides by 2
x + 3 = 5.5
The opposite of + 3 is –3 so
subtract 3 from both sides
x = 2.5
 Solve 3x + 5 = 5x + 1
There are x’s on both sides so
subtract 3x from both sides
5 = 2x + 1
The opposite of +1 is –1 so
subtract 1 from each side
4 = 2x
The opposite of x 2 is ÷ 2 so
divide both sides by 2
2 = x (or x= 2)
REARRANGING FORMULAE
• You do exactly the same for this as for
solving equations.
• When you are asked to make a letter the
subject of the formula then that letter needs
to be by itself on one side of the equation
REARRANGING FORMULAE
 Example: Rearrange the formula 2(b – 3) = a to make
b the subject of the formula
We want to get rid of the times 2 outside the bracket and
the opposite of times 2 is divide by 2
So b – 3 = a
2
The opposite of –3 is +3 so
b=a+3
2
Inequalities
 There are 4 inequalities and you need to
learn the symbols.
 > means ‘greater than’
 < means ‘less than’
 ≥ means ‘greater than or equal to’
 ≤ means less than or equal to’
Inequalities
 There are 3 types of inequality questions
1.
2.
3.
Solving inequalities ( these are just like solving
equations)
Drawing an inequality – use a number line and
remember to fill in the blob if there is ≤ or ≥
Writing values that satisfy an inequality. Just use
common sense (draw a number line to help you if you
need to)
Trial and Improvement
This is a good method for finding an approximate answer
to equations that don’t have simple whole number
answers
Method
1.
2.
3.
4.
Substitute two initial values into the equation that give one
answer that is too small and one answer that is too large.
Choose your next value in between the previous two, and put
it into the equation.
After only 3 or 4 steps you should have 2 numbers which are
to the right degree of accuracy but differ by the last digit.
Now take the exact middle value to decide which is the
answer you want.
Trial and Improvement
Example
A solution to the equation x³ + 3x = 56 lies between 3 and
4.
x²
x³+3x
Comment
3
27 + 9 = 36
Too small
4
64 +12 = 76
Too large
3.5
42.875 +10.5 = 53.375
Too small
3.6
46.656 +10.8 = 57.456
Too large
3.55
44.738875 + 10.65 = 55.388875
Too small
The solution lies between 3.5 and 3.6 but is nearer to 3.6