Key

Subtitle (#ff0000)

Formula(#f76b07)

Extra Info (#ffff00)

Sub-Subtitle (#28ff00)

example (#008000)

Title (#00e6ff)

Sub-Sub-Subtitle(#0014ff)

Read as… (#4B0082)

Super-Small title(#ff42f8)

Normal (#fffff)

Topics

Useful Tricks

Work out Percentage in your head (easy)

Multiply by 5, 25 and 125 easily

More

https://www.youtube.com/@tecmath

Quadratics

What are Quadratics

Quadratics are mathematical expressions or equations written in the form

ax² + bx + c = 0

note it must be x² not anything else

Solving Monics (a=1)

Monics are Quadratics where a = 1

ax² + bx + c = 0

e.g

x² + 10x + 16 = 0

Their are two steps to solve a quadratic equation:

1. Factorising

2. Using Zero Product property to Solve

……………………………………………………………………………………………………………..

1. Factorising

2. Solving Using Zero Product Property

Now that We have Factorised the equation from x² +10x + 16 = 0 to (x+2)(x+8) we can now solve it using the rule known as ZERO PRODUCT PROPERTY:

The Zero Product Property Rule

The Zero Product Property rule states that if ab = 0 either or both a=0 or b=0

this means that in factorised quadratics such as

(x+5)(x-4)=0

one of the brackets must be equal to 0

For this to happen x must be either -5 or 4 so our final answer is:

x = -5 or x = 4

but…

This only works if one of the sides is equal to 0 so before you factorise any equations make sure one of the sides is 0.

e.g:

x² + 8x = -24

would need to be converted to…

x² + 8x -24 = 0

This Means that x=-2 or x=-8

Solving Non-Monics (a≠1)

Non Monics

The Quadratic Formula

The Quadratic Formula

Perfect square expansion

D.O.T (Difference of Two Squares)

Quadratic graphs

Basics

How The Equation Changes the Shape of The Graph

Plotting Quadratic Graphs

Parabola

Surds

What is a Surd

Surds are the irrational roots (√) of rational numbers

How to simplify Surds

How They Multiply and Divide:

To Know how to Simplify Surds you must first know how they Multiply and Divide:

Adding and Subtracting with Surds:

How to Simplify Surds:

now we Know how to multiply and divide them we can learn how to simplify them:

Example

√20 = √4×5 = √4×√5 = 2×√5 = 2√5

Summary:

• The largest square Factor of 20 is 4 so factorise 20 into 4×5

• Use the square root property √a × b = √a × √b

• √4 = 2, leaving the final answer as 2√5.

How to Simplify Expressions Containing Surds

Rationalising the denominator

There are two different scenarios in which you can do this:

when the denominator contains just 1 surd

when the denominator contains more than just 1 surd

Bounds

Significant Figures

What are bounds

Error Intervals

An Error Interval is the range of values that a number could have taken before being rounded. This can be written as inequalities.

Calculations with bounds

Bounds Formula

The Formula two work out bounds is:

……………………………………………………………………………………………………………………………….

Note: If the question asks to round to a certain decimal place (dp) or significant figures (sf), you use the place value:

• For decimal places (dp): The place value is determined by the position of the last decimal place.

  • For example, rounding to 2 decimal places means the place value D is 0.01 because it's in the hundredths place

• For significant figures (sf): The place value is determined by the position of the last significant figure.

  • For example, rounding to 3 significant figures means the place value D depends on the magnitude of the number, like D would be 1 for 3 significant figures in a number like 456, or 100 for 2 significant figures in a number like 9,326

Basic Example

Example:

A mass of 70 kg, rounded to the nearest 10 kg, has a lower bound of 65 kg, because 65 kg is the smallest mass that rounds to 70 kg. The upper bound is 75 kg * because 75 kg is the largest mass that would round down to 80kg.

I know 75kg rounds up not down but we still consider it the upper bound

How bounds add, multiply, subtract, divide

Complex Example 1 (Multiplication)

Final Answer:

Complex Example 2 (Addition)

Complex Example 3 (Subtraction)

Complex Example 4 (Division)

Complex Example 5 (Algebra)

Standard Form

Intro To standard Form

Intro to Standard Form

How to express numbers in Standard Form

……………………………………………………………………………………………………………………………….

Examples

Multiplying And Dividing numbers in Standard Form

More Complex Examples

Adding and Subtracting in Standard Form

When Adding and Subtracting Standard Form there are 3 Rules:

Using these rules here are 2 examples

……………………………………………………………………………………………………………………………….

Indices

Using an index or power

An index, or a power, is the small floating number that goes next to a number or letter. The plural of index is indices.

Indices show how many times a number or letter has been multiplied by itself.

Here is a number written in index form:

2 is the base number and 4 is the index or power.

2⁴ is a short way of writing 2×2×2×2.

a² (read as 'a squared’) is a short way of writing a×a. a has been multiplied by itself.

a³ (read as ‘a cubed’) is a short way of writing a×a×a. a has been multiplied by itself three times.

a⁴ (read as ‘a to the power of 4') is a short way of writing a×a×a×a. a has been multiplied by itself four times, and so on.

Key fact

a¹ (read as 'a to the power of 1') is written as simply a.

The 6 laws of indices

Law 1: multiplying indices

When the bases are the same

When multiplying indices, with the same base add the powers.

a^b × a^c = a^b+c

1 Add the powers.

2 Multiply any coefficients.

………………………………………………………………………………………………………………………………

Example

Simplify 4a³ × 7a²

1. Add together the indices (3 and 2).

3+2=5

2. Multiply 4 and 7 together.

4×7=28

3. Now put that together

28a⁵

When the bases are different

When multiplying indices, with different bases there are two methods.

method 1-works for indices when the base and the powers are numbers:

2³ × 3² = 8 × 9 =72

1. convert the indices into regular numbers

2. multiply

method 2-works for indices when the base or/and the powers are different and letters:

You Can’t do It!

Law 2: dividing indices

when the bases are the same

When dividing indices, with the same base subtract the powers.

a^b ÷ a^c = a^b-c

1 subtract the powers.

2 divide any coefficients.

………………………………………………………………………………………………………………………………

Example

Simplify 8a⁷ ÷ 8a⁴

1. subtract the indices (7 and 4).

7- 4=3

2. divide 8 and 8 .

8÷8=1

3. Now put that together

a³

when the bases are different

When dividing indices, with different bases there are two methods.

method 1-works for indices when the base and the powers are numbers:

2³ ÷ 3² = 8 ÷ 9 = 8/9

1. convert the powers to regular numbers

2. divide

method 2-works for indices when the base or/and the powers are different:

You Can’t do It!

law 3: raising powers

law 4: power of 0

law 5: negative indices

law 6: fractional indices

Probability

What is Probability

The Probability Scale

Probability Notation

All Possible Outcomes add up to 1

Example

How to Calculate Probability

Example

Relative Frequency

What is Relative Frequency

Are Relative Frequency and Probability the Same

Theoretical and Experimental Probability

Experimental Probability

Theoretical Probability

The More Trials The better

Listing Outcomes

Terms to Know

Sample space diagrams

I know the image says that sample space diagrams are tables but they can actually be a:

• List

• Table

• any other organised diagram

The Product Rule

Venn diagrams

What are Venn diagrams

Universal Set (ξ)

• The universal set is a rectangle outlining the space in which all values within the smaller sets are found. The universal set is denoted using the symbol ξ

• 

Constructing Venn diagrams

Set Notation

Set notation is mathematical notation that is used in probability.

A set can be a list of items known as elements.

A subset would be a selection of these elements. The elements of a set could be a set of integers, shapes, people etc.

∈ means is an element of e.g 5∈A (Meaning 5 is an element of set A)

∉ means is not an element of e.g 5∉A (Meaning 5 is not an element of set A)

When we list elements within a set, we use these curly brackets { } and separate each element in the list with commas (,).

The universal set, ξ, is the list of every element that there is available to choose from.
(ξ is the Greek letter epsilon.)

For example,

Universal set, ξ, of integers from 1 to 12 is written as
ξ={1,2,3,4,5,6,7,8,9,10,11,12}.This set contains 12 elements.

If we wrote a subset of this set, for example A={prime numbers}, we would have the set A={2,3,5,7,11}. This set contains 5 elements. We use capital letters to define a subset.

The complement of a set is denoted to an apostrophe ‘. It would be the remaining elements in the universal set that are not part of that set.

The complement of A={prime number} from 1 to 12 would be the following set,
A′={1,4,6,8,9,10,12}. We can also say not set A.

We also have the empty set (or null set), ϕ, which contains no elements.

For example, from the universal set ξ={1,2,3,4,5,6,7,8,9,10,11,12}, the set
B={numbers greater than 12} = ϕ , because there are no numbers greater than 12 in the universal set.

There are other symbols that are used to describe set relationships.

The union, (∪), of the set of even numbers E={2,4,6,8,10,12}and multiples of 5,F={5,10} is the set E∪F={2,4,5,6,8,10,12}. Note that 10 appeared in both sets but we did not write it twice in the union. So the Union basically means ‘or’

The intersection, ∩, of the set of odd numbers O={1,3,5,7,9,11}and the set of prime numbers P={2,3,5,7,11} is the set O∩P={3,5,7,11}. To find the intersection of two or more sets, we find the elements that are contained within both/all of the sets only. So the Intersection basically means ‘and’

The cardinality of a set, n(A), is the number of elements in a set.

E.g If A={2,3,5,7,11}, n(A)=5.

The ‘And’ + ‘Or’ Rules

Independent and Dependent events

The ‘And’ Rule

The ‘Or’ Rule

Frequency trees

Frequency Trees

Tree Diagrams

Tree Diagrams-1

Tree Diagrams-2

Ratios + Proportion

Intro + how to simplify

Simplifying harder ratios

Converting ratios

to convert a ration to a fraction, decimal or percentage it must first be converted to a fraction and then into a decimal percentage so this video only covers how to convert a ratio into a fraction

Sharing in Ratio

Combining Ratios

Using Ratios

Using Ratios

Problem Solving Using Ratios

Problem Solving Using Ratios

More Problem Solving Using Ratios

Direct Proportion

Intro

Writing Equations With it

Inverse Proportion

Multiplicative Relationships

Part 1

Part 2

Compound Measures

Intro

Average Speed

Converting Compound Units

Converting Compound Units

Useful Links

Intro

Using Ratios

Problem Solving Using Ratios

More Problem Solving Using Ratios

Direct Proportion

Inverse Proportion 1

Inverse Proportion 2

Multiplicative Relationships

Compound Measures

Harder Direct Proportion

Harder Direct and Inverse Proportion