Maths Revison Notes :)
- A whole number, which is either positive or negative.
- A fraction, where denominator and numeratior are whole numbers, e.g 1/2, -1/4
- A terminating or reccuring decimal, e.g 0.754, 0.333333333333.
- They are never-ending and non-reating decimal, e.g π (pi).
- A good sourse of irrational numbers is square roots and cube root (surds).
A surd is a square root which cannot be reduced to a whole number. For example, is not a surd, as the answer is a whole number. But is not a whole number. You could use a calculator to find that but instead of this we often leave our answers in the square root form, as a surd.
The Main Rules
- a xb = ab
- a / b = a/b
- ( b) ² = b
- You can't do a +b
- (a +b)² = (a +b) (a +b) = a² + 2ab +bb = a²+ 2ab +b
- (a + b)(a - b) = a² + ab - ab - (b)² = a² - b
- When rationalising the denominator e.g 2/5, you multiply the numerator and denominator by 5 so it becomes 25/55 = 25/5
Upper and Lower bounds
Finding the upper and lower bounds
The rule is the real value as much as half the rounded unit added or subtracted the rounded off value.
If a length is given as 2.4m to the nearest o.1m.
The upper bound is half 0.1 added to 2.4, which is 2.4 +0.05 = 2.45
The lower bound is half 0.1 subtracted from 2.4, which is 2.4 - 0.05 = 0.35
Upper and Lower bounds
Maximum and minimum values of a calculation
- Upper bound ÷ Lower bound
- Upper bound - Lower bound
- Upper bound x Upper bound
- Upper bound + Upper bound
- Lower bound ÷ Upper bound
- Lower bound - Upper bound
- Lower Bound x Lower bound
- Lower bound + Lower bound
Four main things you need to know about Reciprocals are:
- The Reciprocal of a number is one over the number, e.g Reciprocals of 7 is 1/7
- You can find the Reciprocals of a fraction by turning it upside down,e.g Reciprocals of 1/4 is 4/1
- A number multiplied by its Reciprocals gives 1. e.g 7 x 1/7 = 1 and 1/4 x4/1 is 1
- 0 has no Reciprocals
A ratio is a way of comparing the relative magnitude of different quantities. An example of a ratio is 20 : 30 : 50.
Simplfing ratio: Division by common factors reduces the numbers used in a ratio. The ratio 20 : 30 : 50 becomes 2 : 3 : 5 . More examples,8 : 12 becomes 2 : 3, 20 : 55 becomes 4 : 11 by dividing by 5, 39 : 12 becomes 13 : 4 by dividing by 3, 56 : 24 becomes 7: 3 by dividing by 8.
- Add up the parts, e.g if its 2:3 than you add 2 and 3.
- Find amount of one part, e.g say the total amount was 500, once you added 2 and 3 which equals 5, you than divide 500 by five, which will give you one part, which is 100.
- than to work out 2:3, so 2 x 100 = 200 and 3 x 100= 300
- to check if correct make sure when you add them they equal total amount (500)
Multiply top and bottom seperatly, e.g 3/5 x 4/7 = 3 x 4/5 x 7 = 12/35
Turn the 2nd fraction upside down and then multiply, e.g 3/4 ÷ 1/3 = 3/4 x 3/1 = 3x3/4x1 = 9/4
Adding and subtracting
You add or subtract numerator as long as the denominator is the same, e.g 2/6 +1/6 =3/6 and 5/7 - 3/7 =2/7
To making denominator the same you divide or multiply the fraction until both fractions have the same denominator
Finding a fraction of something
Multiply by the numerator and than divide by the denominator, e.g 9/20 of 360 = (9 x 360)÷ 20 = 162
or you can divide first than multiply, e.g 9/20 of 360 = (360÷ 20) x 9
Terminating and Recurring decimals
Recurring decimals which have a patter of numbers which repeat forever, e.g 1/3 = 0.333333333...
Termination decimals are finite, e.g 1/20 = 0.05
The denominator of a fraction tells you if it will be a recuring or terminating decmal when converted. if denominator has a prime number of 2 or 5 its termination, if not then its when converted it will be a Recurring decimals
Recurring Decimals into Fractions
- Find the length of the repeating sequence and multiply by whatever will move it all up past the decimal point by a full repeated sequence (10,100,1000), e.g 0.234234234234..., 234 is repeated so that is one full repeated sequence so you multiply by a 1000. 0.234234234234.. x 1000 = 234.234234234...
- Subtract the original number (r) from the new one, e.g 1000r - r = 234.2342342... - 0.234234234.. which gives 999r=234
- Than divide to leave r: r=234/999 and cancel if possible: r = 26/11
The easy way out
The fraction always has a repeating unit on the numerator and the same number of 9's as the denominator, e.g 0.44444.. = 4/9, 0.124124124... = 124/999 and so on. If you can cancel down, than you make sure you cancel down.
A percentage is basically over a 100
To find out x% of y, you divide y by a 100 and multiply by x, e.g Find 15% of £46, you divide 46 by 100 and multiply by 15 = £6.90
To Express x as a percentage of y, you put it in a fraction x/y and multiply that fraction by 100, e.g give 40p as a percentage of £3.34, 40/334 than multiply by hundred = 12%
To identify the original value
Example: A house increases in a value of 20% to £72,ooo,Find the original price. Method: 72,000 is 120%, so you divide 72,ooo by 120 to work out 1% which is £600 and then multiply by a 100 to find out 100% which is £60,000 which is the original price. The reaon why 72,000 is 120 persent is beacuse it has increased by 20% on top of the 100% = 120%
A polygon is a shape with many sides. A regular polygon is a shape where all sides and angles are the same.
They are the angles the shape has after the sides are extended. Polygons exterior angles always add up to 360 degrees so, 360 ÷ the number of sides = the exterior angle. On a triangle the exterior angle is the sum of the interior angles of the other two vertices.
If you know the exterior angle of the polygon then the interior angle is 180 - the exterior angle because exterior angle & the interior angle are on a straight line. To find the sum of the angles inside a polygon we cut the polygon into triangles. The cutting lines must not cross e.g. if we cut the pentagon into 3 triangles. The angle sum in each triangle is 180 degrees. Therefore the angle sum of the pentagon is 3 x 180 = 540 degrees
- Area = π x r²
- Circumference = 2πr or π x d d = diameter, r =radius
- Area of sector= (Angle/360) ÷ Area of full circle
- Length of Arc = (Angle/360) x circumferance of whole circle
- Area of segment: you have to find out the area of the sector as above than you find area of triangle and than subtract the area of triangle from area of sector
There are six main rules that your ment to know...
· An angle in a semi circle = 90°
· Angles on the same segment are equal
· Angle on circumference is twice angle in centre
· Opposite angles in a quadrilateral (where all sides are touching circumference) add up to 180°
· Angles in opposite segments are equal
· A chord bisector is a diameter
· Volume of sphere = 4/3π r³
· Volume of a prism = cross section area x length
· Volume of a pyramid = 1/3 x Base area x height
· Volume of cone = 1/3 x π r²
· Volume of Frustum = Volume of original cone - Volume of the removed cone
Surface area of 3D is basically the total area of all the outer surfaces add together. for shapes like; prisms,cubes,cuboid and pyramids you can work out surface area with the help of the net of the shape. 3D shapes like spheres, cones and cylinders nets are difficult to draw, so you need to learn these formulae:
- Cylinder (Curved) surface area: 2πrh
- Cone (Curved) surface area: πrl (l is slant height)
- Sphere surface area: 4πr 2
If you want to know surface area for prisms,cubes,cuboid and pyramids, here they are:
Surface Area of a Cube = 6 a 2 (a is the length of the side). Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac (a,b,c are lengths). Surface Area of a prism = (perimeter of shape cross section) x L+ 2 x (Area of shape cross section). Surface Area of a pyramid = Add the area of the base to the sum of the areas of all of the triangular faces
Just looking at a formulae you should be able to find out if its for length, area or volume.
The rule is:
- Areas always have lengths square (L²)
- Volumes always have Lengths cubed (L3)
- Lengths always are on their own (L)
4πr 2 + 6d² ( Area) Lwh + 6r²L (Volume) 4πr + 15 L (Length)
Converting Area and Volume measurements
1 m2 = 100cm x 100cm = 10,000 cm2
To change area measurements fom m2 to cm2 we multiply area in m2 by 10,000 and to change cm2 to m2 we divide area in cm2 by 10,000.
1 m³ = 100cm x 100cm x 100cm = 1,000,000 cm³
To change area measurements from m³ to cm³ we multiply area in m³by 1,000,000 and to change cm³ to m³ we divide area in cm2 by 1,000,000.
You need to know three main points for Distance-time graphs:
- At any point, Gradeint = Speed, But watch out for units!
- The steeper the graph, the faster its going.
- Flat sections is where it is stopped
Example of a Distance-time graphs
Example of Distance-time graphs
You need to know four main points for velocity-time graphs:
· At any point Gradient = Acceleration (m/s²)
· Negative slopes (going down) means deceleration
· Flat sections mean steady speed
· Area under graph = distance travelled
Example of a Velocity-time graph
Loci and Construction
Equidistant line from a fixed point(a circle)
All the points P lie on a locus of points the same distance from O
Loci and Construction
The Locus of points which are a fixed distance from a given line, would be an oval type shape.
Loci and Construction
The Locus of points that are equidistant from two given points A and B below, would be a perpendicular straight line.
All the points along the straight line are equidistant from A and B.
Let each point be the centre of a circle, the distance from A to B the radii. Draw circles note the intersections. Draw a line connecting the intersections, this line is equidistant.
Note: Draw a line connecting A to B, this line is perpendicular to the line connecting the intersections.
Loci and Construction
The locus of points that are equidistant from two given lines X and Y, would be a straight line bisecting the angle of the lines.
Using a fixed radius start where the lines meet draw an arc that intersects both lines. from these new points 2nd and 3rd, draw arcs that intersect with each other.
Connect this new point to the 1st point for the equidistant line. The angle between the two lines is now halved.
Loci and Construction
Constructing Accurate Angles: accurate angles can be achieved with just a compass and a ruler. The radius of all the circles have to be the same.
Draw the first circle at the end of the line.
Draw the second circle with the centre at the point where the first circle intersects the straight line.
Where both circles intersect with each other a line drawn from the first point will be exactly 60.
Congruent and similar
- Congruent - same size and same shape.
- Similar - same shape but diffrent size.
For a triangle to be congruent, at least one of these rules must apply:
- ASS - All 3 sides are same,
- AAS - 2 Angles and a side are the same,
- SAS - 2 sides are equal and so is the angle between them,
- PHS - A right angle, hypotenuse and another side are the same.
For a shape that is similar its shape is diffrent but all the angles stay the same.
- Translation, Rotation and Reflection are all congruent
- Enlargements are similar
o if the scale factor is larger than 1, shape gets bigger
o If the scale factor is less then 1, the shape gets smaller
o If the scale factor is negative, shape is rotated 180
o The scale factor tells you the distance between the original point and the new point form centre of enlargement e.g. point Original point a is 3 cm away from centre of enlargement and it is enlarged by a scale factor of 3 so you do 3 x 3 = 9, so the new point is 9 cm away from centre of enlargement.
Area and volume of enlagement, For scale factor n:
- Sides - n times bigger
- Area - n² times bigger
- volume - n³ times bigger
To find or plot a bearing you must rememeber the 3 key words:
- FROM - Find the word from in the question, and put your pencil on the diagram at the point your going from.
- NORTH LINE - A the point you are going from, draw a north line.
- CLOCKWISE - Draw an angle clock wise from the northline to the pont or line. this is the angle your meant to measure - bearing
All bearing are meant to have 3 figures, e.g 018, 360, 034
- Label three sides of the triangle's sides O (opposite), A (adjacent) and H (hypotenuse)
- You need to memorise SOH CAH TOA and write it down, the way i memorise it is : Silly Old Homework Can Always Help Trignometry Out Alot
- Descide which two sides are involved,O,H A,H or O,A
- And then using the prevoius step decide which formulae to use: Sinθ = O/H , Cosθ= A/H and Tanθ =O/A
- Translate into number and work it out and make sure answers are sensible.
- H is the longest side. Oppisite is side oppisite the angle being used (θ ) and adjasent is the other side that is next to the angle being used (θ ).
- When working out the angle - use reverse sin,cos,tan.(make sure your calculator is on DEG mode)
- you only use trigonometry in right- angled triangles
A vector quantity has both direction and magnitude (size).
(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)
For example this arrow represents a vector. The direction is given by the arrow, while the length of the line represents the magnitude.This vector can be written as: , a, or .In print, a is written in bold type. In handwriting, the vector is indicated by putting a squiggle underneath the letter:
If two vectors have the same magnitude and direction, then they are equal
Adding vectors: Look at the graph below to see the movements between PQ, QR and PR.. Vector followed by vector represents a movement from P to R. Written out the vector addition looks like this
Subtracting a vector is the same as adding a negative version of the vector. (Remember that making a vector negative means reversing its direction.)
Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors and ? You could say it is vector followed by a backwards movement along . So we can write the path from X to Z as
Written out in numbers it looks like this:
3D Pythagoras and Trigonometry
- On the diagram, make a right-angled triangle using the line, a line in the plane and a line betwwen them 2.
- Draw the right-angled triangle again so its clear to see. Label the sides. you might have to use Pythagoras to work out a legth.
- If question askes for angle, use Trigonometry to calculate angles.
Drawing Straght Line Graphs
- Get the equation in to the form y = mx + c
- Put a dot on the y axis at the value of c
- Than go along one unit and up or down by the value of m (step) and make another dot
- Repeat the same 'step' in both directions
- check gradient is correct
Finding the equation of a straight line.
- From the axes identify 2 variables
- Find the gradient and y-intercept from the graph
- Use this information to write the equation
Graphs to Learn
y= ax²+bx+c (n shape when negative, u when positive)
y = ax³+bx²+cx+d
Graphs to Learn (continuation...)
x² + y² = r²
Quadratic graphs are always symmetrical u shapes or n shapes if negative.