Facts And Terms Flashcards
Factor
A whole number that goes into another number without remainder.
Multiple
Any numbers that are in the times table.
Prime
A number that has exactly two factors; one and itself.
LCM
Lowest Common Multiple.
HCF
Highest Common Factor.
Writing a number as product of primes
Use prime factor trees to find prime factors. Write them in a single line using multiplication (and powers if we choose.
How do you find the HCF when using a venn diagram?
Multiply the middle.
How do you find the LCM when using a venn diagram?
Multiply everything.
Factor decomposition
Product of primes
Dividing decimals
Use bus stop method when we know the times table. Include the decimal point in the same place and add in extra o’s as needed.
Negative numbers: adding and subtracting
5 - 6 = ?, 5 + -6 = ?
8 - -3= ?, 8 + 3 = ?
When there are 2 signs in the middle of the sum the operation changes and the second number becomes positive.
5 - 6 = -1, 5 + -6 = -1
8 - -3= 11, 8 + 3 = 11
Negative numbers: multiplying and dividing
- 4 x 2 =
- 3 x 8 =
- 12 / -2 x -3 =
- 20 / 5 x -2 =
If there is an odd number of negative signs in the sum the answer is negative, if there is an even number the answer is positive.
- 4 x 2 = 8
- 3 x 8 = -24
- 12 / -2 x -3 = -18
- 20 / 5 x -2 = 8
Anything to the power of 0 is…
1.
(Indices) Negative power -
Reciprocal is …
Negative power - reciprocal
Reciprocal is 1/number
Rules for indices - multiplying
a^m x a^m = a^m+n
Rules for indices - dividing
a^m / a^n = a^m-n
Indices (a/b)
A -
B -
A - new power
B - root
Rules for indices - brackets
(a^m)^n = a^mn
Calculating with standard form: multiplication
(4 x 10^6) x (5.1 x 10^3) =
x value + power
(4 x 5.1) x (10^6 x 10^3) = 2.04 x 10^10
Calculating with standard form: addition/subtraction
(4.7 x 10^3) + (2.1 x 10^2) =
Convert + or -, convert to standard form
= 4910 = 4.91 x 10
Calculating with standard form: division
3.6 x 10^4) / (9 x 10^6
(3.6/9) x (10^4 x 10^6) = 0.4 x 10^-2 = 4 x 10^-3
Divide values, - powers
Calculating with standard form: using brackets
(2 x 10^4)^3 =
Do the power to each part
2^3 x (10^4)^3 = 8 x 10^12
Expanding brackets
Times inside by outside.
Factorising
Means put back into brackets.
Interior angle
Angle inside the shape.
Exterior angle
Angle made from extending the side.
Regular shape
All angles and sides are equal.
Interior + exterior
= 180 degrees
Sum of interior
180(n-2)
Sure of exterior angles
360 degrees, no matter how many sides.
Each interior angle of regular shape =
(n - 2) x 180)/ n
Quadratics have…
Two answers
E.g. x = 8 or x = -2
Expression
3x + 3y
Equation
3x + 4 = 12
Identity
3(2x + 3) = 6x + 9
Variable
Y
Formula
r = 3x + 3y
Rearranging formulae
It is just like rearranging an equation. Rather than working to get an answer. We end up with algebra.
Write r terms of a means …
Make r the subject or r =
Percent means out of…
100
Multiplier
The percentage of an amount we want, but as a decimal.
If you want to get to the original amount you…
Divide by the multiplier.
If you want to get to the amount afterwards you …
Multiply by the multiplier.
If you increase something by 10%, then decrease that by 10% would you get the same as decreasing by 10% then increasing by 10%?
Yes.
Reverse percentages
Find the original amount after an increase or decrease. Get back to how much 100% is.
Percentage increase and decrease
Increase - add % to original
Decrease - take % from original
Find % first then + or -.
Simple interest
Interest is calculated from original amount. Same amount added on every time.
Compound interest
Interest is based on what is in there at that time.
Direct proportion
As one value increases the other increases by the same rate (doubled / x 10 / quartered).
Inverse proportion
As one value increases by a rate, the other decreases by the same rate (x2 and /2 or /5 and x5 or x8 and /8)
Other phrases for direct proportion
. Directly proportion to . Is proportional to . Varies directly . Y ∝ x . Y = kx
Other phrases for inverse proportion
. Inversely proportional to
. Varies inversely
. Y ∝ 1/x
. Y = k/x
Types of average
Mean, mode and median.
Mean
Add them all up and divide by however many there are (can be a decimal, doesn’t have to be in the list).
Mode
Most common (can have either no mode, 1 value or 2 values. Must be in list).
Median
The middle number when they’re in order (can be a decimal - in middle of two values on list or value on list).
Quartiles:
Split data up into quarters.
How do you find the median?
(n+1) / 2 or 1/2n + 0.5
In the formula for finding the median, what is n?
The number of values in the list.
IQR =
UQ - LQ
How do you find LQ?
(n+1) / 4
How do you find UQ?
3((n+1) / 4)
How do you find the mean from a table?
Sum of fx / sum of f
How do you find modal class/ mode from a table?
The one with the most frequency.
How do you find median from a table?
Sum of frequency + 1 / 2
This gives the place of the value
How do you find the estimate of median (When the table is in classes and answer is a number)?
. Find median class and the place of the value. . Amount of numbers in (to get to the place of the value) / frequency of class. . Then times that by the class width . Add that onto the lowest value of the class.
How do you know whether something is terminating or recurring?
A terminating fraction is a fraction that’s denominator has only 2 and 5 as prime factors.
Irrational
Cannot be expressed as a fraction: surds, pi, e.
Rational
Can be expressed as a fraction e.g. any terminating value/ decimal, any recurring decimal
Probability can be written as…
A fraction, % or decimal
In probability with Venn diagrams, for ‘n’, count the section(s) that has…
The most amount of ticks (2 if two circles, 3 if three circles etc.).
In probability with Venn diagrams, for ‘u’, count the section(s) that has …
A tick in (all sections with a tick).
Mutually exclusive
Cannot happen together.
Exhaustive
All outcomes are covered, probabilities add up to 1.
Probability
We use a tree diagram to help find probabilities of combined events. Multiply along branches (and). Add different outcomes if needed (or).
Independent events
Probability of one does not effect the other.
Dependent events (conditional probability)
The outcome of the first event, effects the probability of the second.
Ratio
Allows us to see values in comparison with each other. It is usually in its simplified form. W can also have decimal inside a ratio if required.
How do you calculate percentage profit?
(Profit / cost to make) x 100
How do you calculate to find the percentage increase or decrease?
((Original - now) / original) x 100 ??
Arithmetic sequence
Constant difference from one term to the next
Eg. 4, 7, 10, 13 (+3)
Geometric sequence
The ratio of one number and the next is the same
Eg. 3, 6, 12, 24 (x2)
Fibonacci sequence
Uses previous terms to get the next
Eg. 1, 1, 2, 3, 5
Quadratic sequence
Second difference is constant
Eg. 2, 8, 18, 32, 50
+6 +10 +14 +18
+4 +4 +4
With quadratic sequences, what do you do to the second difference?
Half it, and then write it out, and minus it from the original. Then find 0th term.
Term to term rule
From one term to the next.
Nth term
Any term in the sequence (10th n =100).
Quadratic sequence steps
. Find the common difference (2nd difference is equal, divide by 2)
This gives n^2
. Take of the n^2 part
. Find the nth term of whats left
Fibonacci sequence
Series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … The next number is found by adding up the two numbers before it and so on.
What kind of line does this give: y = 3x - 2?
Straight line
What kind of line does this give: y = x^2 + 1?
‘U’ shape
What kind of line does this give: y = 2x^3 + 1?
Curved shape (like sideways S)
How do you find gradient?
Change in y / change in x or Rise / run
If a line is perpendicular to a line with a gradient of -3/5, what is its gradient?
5/3 —> negative reciprocal
Midpoint
E.g. (x, y) and (x2, y2)
The middle of the line
Mp = (x + x2/2 , y + y2/2)
Parallel lines
Always have the same gradient. Make sure lines are in the same format before comparing (y=mx+c).
Perpendicular lines
Their gradients multiply to make -1. Gradients are negative reciprocal of each other.
Chord of a circle
A line that cuts across the inside (doesn’t cross centre).
Segment of a circle
The area caused by a chord.
How do you find arc length?
(Angle/360) x (pi x diameter)
How do you find sector area?
(Angle/360) x (pi x radius^2)
Which axis is frequency always on?
Y-axis
What is cumulative frequency?
Frequency gets added as it goes along.
Scatter graph
Allows us to compare different variables and shows us if there is any correlation (relationship) between them. Correlation is not the same as relationship.
Positive correlation relationship
As x increases y also increases.
Negative correlation relationship
As x increases y decreases.
Completing the square form
(x+p)^2 - q
Write 2x^2 + 12x - 2 in completing the square form
First, factorise the x^2 and x term.
2(x^2 + 6x) - 2
Now write the x^2 + 6x part in completing the square form.
X^2 + 6 = (x + 3)^2 - 9
So, 2(x^2 + 6x) - 2 = 2[x + 3)^2 - 9] - 2
= 2 (x + 3)^2 - 18 - 2
= 2(x + 3)^2 - 20
Write x^2 + 10x + 3 in completing the square form
(x + 5)^2 - 22
Quadratic formula
x = -b + or - √b^2 - 4ac / 2a
In a quadratic equation, which numbers are the roots?
The two answers (x = … or x = …).
What happens is you multiply or divide by a negative with an inequality?
The sign flips.
What is the Sine Rule (missing sides)?
a/SinA = b/SinB = c/SinC
What is the Sine Rule (missing angles)?
SinA/a= SinB/b= SinC/c
Area of triangle with angle
1/2 x a x b x Sin C
What is the Cosine Rule (missing sides)?
a^2 = b^2 + c^2 - 2bcCosA
What is the Cosine Rule (missing angles)?
Cos(A) = b^2 + c^2 - a^2 / 2bc
Ambiguous case
Finding obtuse + acute versions in the triangle (subtract from 180).
What are the numbers (from thumb down) that you can use on your fingers for exact trigonometry values?
0, 30, 45, 60, 90
What is the rule for Sin in exact trigonometry values?
√amount of numbers above / 2
What is the rule for Cos in exact trigonometry values?
√amount of numbers below / 2
What is the rule for Tan in exact trigonometry values?
√amount of numbers above / √amount of numbers below
Sin of 0, 30, 45, 60, 90
0, 1/2, √2/2, √3/2, 1
Cos of 0, 30, 45, 60, 90
1, √3/2, √2/2, 1/2, 0
Tan of 0, 30, 45, 60, 90
0, 1/√3, 1, √3, not possible
When finding a turning point from (x+p)^2 + q, what’s the rule?
Opposite sign of inside one, same sign of outside one
-p,q
If vectors can be written as multiples of each other, what does this mean?
They are parallel.
Density formula
Density = mass / volume
Similar shapes
Same shape, different size.
Congruent
Same shape and same size.
Linear scale factor
Side lengths of large shape / corresponding side length of small shape
Volume of square based pyramid.
V = 1/3 x ba x h
Vector
Way of representing movement between two places. We visualise a vector with a straight line, the direction indicated by an arrow. We can use a matrix system to define a vector: x is the movement right, y is the movement up.
Estimate
Round every value to 1sf.
Discrete data
Data counted in a very basic sense.
E.g. number of siblings -3, 6, 2 etc.
Pens in a pencil case - 7, 12, 15, etc.
None of these could be made more accurate
Continuous data
Data that has been measured.
E.g. temperature, length of a pencil, mass of a loaf of bread
Depending on how accurate we want to be, these could be more specific.
How to get bounds.
Add or subtract half the accuracy.
E.g. 1cm / 2 = 0.5
7 - 0.5
7 + 0.5
Sector
Area between two radi and an arc.
Angle on circumference inside semi circle is…
90 degrees
Angle at the centre from a chord is…
Double the angle at the circumference.
Angles in a segment from a chord are…
The same
Cyclic quadrilateral
Opposite angles add to 180 degrees
Tangent meets radius at a…
Right angle
Alternate segment theorem:
Alternate angles are equal
Where two tangents to a circle meet…
The lengths are equal.
When a number has been rounded, we use error intervals to…
Determine what it could have been before.
When to use sohcahtoa
Right angle triangle
When to use sine rule
. 2 angles known + 1 length known + 1 unknown length.
. 2 lengths known + 1 angle known + 1 unknown angle.
When to use cosine rule:
. 3 lengths known + 1 unknown angle
. 2 lengths known + 1 angle known + 1 unknown length
n
And
U
Or
A’
Not in set
When are tree diagrams used in probability?
Used when the probabilities are not equally likely or if the probabilities change part way through.
When are probability space diagrams used?
Used when probabilities of event A and event B are all equally likely (e.g. roll a dice and flip a coin)
Conversion graphs
These allow us to convert between currencies/measurements. They are limited to working with the scales we have on them.
Asymptote
A point/line that the graph approaches but does not meet.
Circle graph
X^2 + y^2 = r^2
No asymptotes
Google picture to check if your right
Linear graph
y = mx + c
Google picture to check if your right
Quadratic graph
x^2 is the biggest power
Google picture to check if your right
Reciprocal graph
y = k/x
Asymptote = y=x-axis/y-axis
Google picture to check if your right
Cubic graph
x^3 is biggest
Google picture to check if your right
Exponential graph
y = k^x
Asymptote = y=0/x-axis
Google picture to check if your right
Speed formula
Speed = distance / time
Average speed = total distance / total time
Completing the square steps
- Identify perfect square
2. Adjust it so expressions are equal
What does gradient line tell you?
In general, the gradient line tells you the rate of change of the y variable in relation to the rate of change of the x-variable.
What does area under the graph tell you?
In general, the area under the graph tells you the product of the two units on the two axis.
Gradient of curved line
. Draw a tangent that hits the line only at the point you are trying to find.
Work out the gradient of the tangent (rise/run)
What does y = tan x look like on a graph?
Google picture to check if your right.
What does y = sin x look like on a graph?
Google picture to check if your right.
What does y = cos x look like on a graph?
Google picture to check if your right.
