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Singapore Math

Singapore Math is a method of teaching maths to students by using model drawing and visualization techniques in order to help solve problems. The benefits include improved confidence in mathematics, improved overall results, and Singapore achieving the highest results in mathematics in 4th and 8th grade maths. This teaching method has recently been tried in America.

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Poverty in Africa


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An inconvenient truth on Math Education


News Report on Singapore Math


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Calendars

Calendar

Numbers

Plain sticky notes

Natural Numbers

Natural numbers start at 1. This is because you start counting (Naturally) at 1.

What types of Number are there

There are many types of number! Natural, Integar, Whole, odd, even and prume numbers.

Whole Numbers

Whole numbers start at 0 and go up to infinity.

Inverse operations

Inverse operations are when a person works backwards in an equation (A formal term for a sum) to work out unknown values. (x+9=87 the inverse is 87-9=x. Now the equation can be solver. x is 78)

Examples of Multiplying Decimals

2.3 x 4.1=9.43 5.5 x 3.3=18.15 12 x 3.1= 37.2 15x1.5=22.5

What is a Sequence?

A sequence of numbers is a list of numbers that increase of decrease following a certain pattern. We call the nembers terms. We call the pattern they're following the term to term rule. An example of a sequence is 2,4,6,8. The term to term rule is add 2.

Integar numbers

Integer numbers start at Negative infinity and go up to infinity.

Diverging or Converging?

If a sequence increases by each term moving closer to infinity, it diverges If a sequence decreases by each term moving closer to 0, it converges

Integars example

-5 x -5= 25 -5 x 5= -25 5 x 5 = 25 5 x -5=-25 -5/5= -1 5/-5= -1 5/5= 1 -5/-5=1

What is a power?

A power is how many times you multiply a number by itself. 2 to the power of three is 8

LCM

LCM stands for lowest common multiple. This is the lowest multiple that two numbers share. eg. the LCM of 2 and 5 is 10. A way we can do this is: 2 4 6 8 10 5 10 15 20 What is the LCM of 5 and 15?

What is an index

An index is how many times you multiply a number by itself to give you an answer. The plural for this is Indicies.

Prime numbers

A prime number is a number that will only divide with no remainder into itself and 1

Info on 1

1 is not a prime number.

Multiplying numbers

Positive x positive=Positive Negative x Negative= Positive Positive x negative= negative Negative x Positive= Negative

Multiplying decimals

Multiply the numbers as if they weren't decimals. (3.4 x 5.3 would be 34 x 53) Then count how many numbers there are behind the decimal point. ( In this case, there are two decimal points.) Then place the decimal point the number of places behind the two numbers. ( So the answer would be 18.02) If you still don't understand, there are more examples.)

Prime Factorisation

Prime Factorisation is finding the prime factors of a number.

Recurring and terminating decimals

A recurring decimal is a decimal that goes on forever. (3.333333333333 recurring is a recurring decimal because it goes on forever. It has no end)3.3333333333 recurring can also be written as 3.3r A terminating decimal is a decimal has an ending. (So 1.2 is a terminating decimal because it stops)

Divide a whole number by a decimal

To divide a decimal by whole number, multiply the decimal to make it a whole number. What you do to one side, you must do to the other. Now do the sum. (If you had 20/0.2, multiply 0.2 by 10 to make it a whole number. Multiply 20 by 10. Now divide 200 by 2 to get your answer of 100)

Divide a decimal by a decimal

Multiply the larger number to make it a whole number. Do the same to the other number. Now you can divide it like an ordinary number. (0.8/0.04. Multiply 0.8 by 10. Do the same to 0.04. Now you have 8/0.4. Multiply 0.4 by 10 to make it a whole number. Do the same to 8. Now do 80/4. Your answer is 20)

Prime Factorisation

To find the Prime Factors of a nurmber, you first write out A multiplication sum that gives you the number. (To find the prime factors of 20, do 4 x 5) If there are any prime numbers put a circle around it and leave that number. Repeat this process for the remaining numbers until you have only prime numbers. ( You should end up with 2x2x5)

Rounding to 2 decimal places

When rounding a number to 2 decimal places (eg 3.1454) only look at the first three decimal places (Only look at 3.145) if the third number is 5 or over round up, if it's below 5, round down. ( 3.1454 becomes 3.15)

Examples of rounding to 2 decimal places

1.1312 becomes 1.13 3.340 becomes 3.34 5.789 becomes 5.79 6.777 becomes 6.78 5.555 becomes 5.56

Rounding to 3 decimal places

The process for rounding to three decimal places is exactly the same as rounding to 2 decimal places except you round look at the fourth decimal place. (eg 5.43219 only look at 5.4321. The one is not over 5 so it is rounded down, making the answer 5.432)

Examples of rounding to 2 decimal places

7.77777 becomes 7.778 9.12345 becomes 9.123 4.56789 becomes 4.568

Odd and even numbers

An even number is a number that is divisable by 2. (It can be divided by 2 and the answer will be a whole number) An odd number is the exact opposite. (It isn't divisable by 2)

Dividing mixed numbers

To divide mixed numbers (A mixture of a number and a fraction, like 2 and 2/9) you need to convert (Change) them to improper fractions. Improper fractions are fractions where the numerator (The top half of the fraction) is bigger than the denominator (The bottom half of the fraction) To do this, multiply the whole number (The number next to the fraction) by the denominator (The bottom half of the fraction). Then add the numerator (The top half of the fraction) to the answer you just worked out. Place this new answer as the numerator (The top half of the fraction) for the improper fraction. The denominator (The bottom part of the fraction) dosen't change. ( to work out the the mixed number 5 and 4/5, do 5x5 and then add 4. Your new numerator is 29 and your denominator will always stay he same, so it will be 5. Your new fraction is 29/5) (To work out the mixed number 4 and 1/3, you 4x3 and then add 1. Your new numerator will be 13 and your denominator will always stay the same, so it will be 3. Your new fraction will be 13/3) (To work out the mixed number 8 and 7/8, do 8x8 and then add 7. Your new numerator will be 71 and your denominator will always stay the same, so it will be 8. Your new fraction will be 71/8) (To work out the mixed number 5 and 9/10, do 5x10 and add 9. Your new numerator will be 59 and your denominator will always stay the same, so it will be 10. Your new fraction will be 59/10)

Multiplying fractions

Multiply both denominators (The bottom parts) and numerators (the top parts) together. You will now have one fraction. If both halves of the fraction are divisable by a certain number, simplify the fraction. (2/4 can be simplified because both halves can be simplified because they're both divisable by 2)

Types of fractions

A proper fraction, is a fraction in which the numerator is smaller than or equal to the denominator. A complex fraction is when a fraction contains another fraction. These smaller fractions can be in either the numerator or denominator. An improper fraction (Also called a top heavy fraction) is a fraction in which the numerator is larger than the denominator. Improper fractions can be written as mixed numbers Mixed numbers are always larger than 1 and contain both fractions and numbers.

Adding fractions

First, have a look at both denominators, if they are different, you need to find the LCM (Also known as lowest commen multiple) of both those numbers. For example, if the numerators are 5 and 9, you would need to work out what is the Lowest common multiple of both those numbers. The LCM would be 45. This number becomes the numerator for both those numbers.

How to work out the gradient of a straight line.

The gradient of a straight line is the steepness of the line on the graph. To work out the gradient on a graph, Pick 2 points on the graph. Make sure they are not too close to each other. You want to try and get from the lower point to the larger pint. First, work out how many spaces up or down you need to travel. If you go up, it is a positive number. If you go down, it is a negative number. Make a note of it. This will be the numerator (The top part) of a fraction you are going to write.

How to work out the gradient of a straight line. (Part 2)

Work out how many spaces left of right to get to the next point. If you go right, it is a positive number, if it goes left, it is a negative number. This number will become the denominator (The bottom part) of the fraction. Remember, a fraction means divide so divide the two numbers together. Don't forget how to divide integars.

How to divide a decimal by a whole number

To divide a decimal by a whole number, use the same process that you would use for multiplying decimals. Divide the two numbers as if they are whole numbers. Then count how many units there are behind the decimal point. Move the decimal point a number of spcesto the left. (So if it was 2.5 divided by 5, you would do 25 divided by 5. There is one unit behind the decimal point. You move the decimal one place to the left. The answer will be 0.5)

How to divide a decimal by a decimal

To divide a decimal by a decimal, make the second number a whole number. (If it was 0.5 divided by 0.25, multiply 0.25 by 100 to get a whole number) What you do to one side, you have to do the other side. (0.25 has become 25 because you multiplied it by 100, you have to multiply 0.5 by 100 you will get 50) Now divide it as you would do a whole number. Remember to add the decimal. Your answer will be 2

Examples of dividing decimals

0.9 divided by 3 would be 0.3 0.6 divided by 2 would be 0.3 0.9 divided by 0.3 would be 3

Subtracting fractions

To subtract fractions, work out the lowest common muliple of the 2 denominators. (Work out the LCM of the bottom halves of the fractions) The LCM becomes the denominator for both fractions. Work out what you need to do to to the old denominators to get the new denominators. (If the LCM of 2 and 3 was 6, find out what you need to do in order to get 6. Work out what you need to do 2 to get 6 etc) (If you wanted to do 3/4-1/7, you need to find the lowest common multiple of 4 and 7 because they are the denominators. The lowest common multiple is 28. 28 becomes the new denominator of both fractions. What did you have to do to 4 to get 28? You multiplied 4 by 7 to get 28. You need to multiply 3 by 7 as well in order to get your new fraction. Your fraction will now be 21/28.) (Now you need to do the same to 1/7. We already know that the denominator is now 28, now we need to work out what the new numerator will be. What did you have to do to 7 to get 28? You multiplied 7 by 4 to get 28. You need to multiply 1 by 4 because it is the numerator. Your new fraction will be 4/28)

Subtracting fractions (Part 2)

The last stage is simple. All you need to do is take away the numerators. If the fraction canbe simplified, simplify it. (Simplify is to make it into a smaller fraction)

Reciporical

If a number is multiplied by another number and the product is 1, we say that the 2 numbers are reciporicals. (2x0.5=1 2 is a reciporical of 0.5)

Increasing numbers by percentages (Part 1)

First, convert (Change) the percentage into a fraction, the denominator being 100. 4% is 4/100 10% is 10/100 50% is 50/100 99% is 99/100 Divide the numerator by the denominator 4/100 means 4 divded by 100 10/100 means 10 divided by 100 50/100 means 50 divided by 100 99/100 means 90 divided by 100 Your answer will be a decimal If the question asks to increase a number by a percentage, add 1 to the decimal. 0.04+1 0.1+1 0.5+1 0.99+1 If the question asks to decrease then number by a percentage, subtract the decimal from 1 1-0.04 1-0.1 1-0.5 1-0.99 Multiply the number you're changing by the decimal 10 increased by 4% means 10x1.04 10 increased by 10% means 10x1.1 10 increased by 50% means 10x1.5 10 increased by 99% means 10x1.99

Increasing numbers by percentages (Examples)

50 increased by 8% 8%=8/100=0.08 1+0.08=1.08 1.08x50=54 3 increased by 10% 10%=10/100=0.1 1+0.1=1.1 1.1x3=3.3 76 increased by 99% 99%=99/100=0.99 0.99+1=1.99 1.99x76=151.24

Finding LCM and HCF (Differently)

To find the LCM using a different method than listing out all the multiples of each number and circling the multiples that appear in both lists and working out which one is the smallest, work out the 2 numbers a products of their prime factors, (Write out the numbers as answers to multiplication sums only using prime numbers) and see which multiples appear in both sums. Multiply these numbers together to get the lowest common multiple eg. 8= 2x2x2 6=3x2 There are 3 2's in the first sum, they make up one set of brackets in the final sum. There is one 3 left over, so that goes in another set of brackets on it' s own. (2x2x2) x (3)= 24 LCM HCF= 2 because it is the only number of appear in both lists.

Detailed explanation of working out the LCM (The simple way)

LCM stands for Lowest common multiple, and is finding out what is the smallest multiple for 2 numbers. (The smallest number that appears in both times tables) (The LCM of 8 and 6 is 24, because it is the smallest number that goes into the 6 times table and the 8 times table) (The LCM of 3 and 2 is 6 because it is the smallest number that goes into the 3 times table and the 2 times table) (The LCM of 9 and 5 is 45, because it is the smallest number that goes into the 9 times table and the 5 times table) (The LCM of 8 and 12 is 24 because it is the smallest number to go into both times tables)

Example #1 of LCM and HCF

Find the LCM of 18 and 4. 18, 36, 54, 72, 90, 108, 126, 144, 162, 180 4,8,12,16,20,24,28,32,36,40 Circle all the multiples that appear in both lists. The only number is 36, this is the LCM. (An easier way to work out the multiples of a number is to type in 0+ n (The number you want) Add n again, and just press equals over and over again.

Example #1 of LCm and HCF (Alternate method)

Find the LCM of 18 and 4. Make 2 factor trees for 18 and 4 to work out the prime factors. 18= 3x3x2 4=2x2 3 x 6 2x2 3 x 2 There are 3 lots of 2, we only need to multiply twice (We don't need the third 2) Multiply all the prime factors in both trees, so 3x3x2x2=36 36 is the lowest common multiple

Example #2 of LCM and HCF

Find the LCM of 16 and 5. 16,32,48,64,80 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80 Circle all the multiples that are in both lists. 80 is the LCM because it's the only number in both lists.

Example #2 of LCM and HCF (Alternate Method)

Find the LCM of 16 and 5. Make 2 factor trees for 16 and 5. 16=2x2x2x2 5 (Is a prime number) 8 x 2 4x2 2x2 Because there are no 2's in the second list for 5, no 2's are gotten rid of. Multiply all these factors together. 2x2x2x2x5=80 You get the same answer no matter what method you use. This method saves time because you don't have to work out every single multiple.

Example #3 of LCM and HCF

Find the LCM of 8 and 14 8,16,24,32,40,48,56,66,72,80 14,28,42,56 The LCM is 56

Example #3 of LCM and HCF (Alternate method)

Find the LCM of 8 and 14 Make 2 factor trees for 8 and 14. 8 14 2 x 4 2x7 2x2 Circle all the common factors The 2 in the 14 tree cancels out. 2x2x2x7=56 The LCM is 56.

How to find the HCF of 2 numbers (Alternate method)

To work out the Highest Common Factor (Greatest Common factor) of 2 numbers, you first make a factor tree for each number so you know their prime factors. See what numbers appear in both trees, these are called common factors. Circle any common factors, and see what different numbers you have. Multiply each different number you have together to give you the HCF.

Example #4 of finding the HCF and LCM

Find the HCF of 63 and 15. list the factors for 63 and 15 63: 1, 63, 3, 21, 7, 9 15: 1, 15, 3, 5 The only number that appears in both lists is 3, so 3 is the HCF of 63 and 15.

Example #4 of finding the LCM and HCF (Alternate method)

Find the HCF of 63 and 15. Make a factor tree for 63. Make a factor tree for 15. 63=3x3x7 15 7 x 9 5 x 3 3 x 3 The only number that is a prime factor of 63 and 15 is 3, there is nothing to multiply 3 by so it is the highest common factor.

Example #5 of finding the LCM and HCF (Alternate method)

Find the HCF of 6 and 24 6 =2x3 24=3x2x2x2 2 x 3 3 x 8 2 x 4 2 x 2 The 2 factors that appear in both lists are 3 and 2. Multiply the 2 together, and the highest common factor of 24 and 6 is 6.

Example #5 of finding the LCM and HCF

Find the HCF of 6 and 24. Write out the factors for 6 and 24. 6: 1, 6, 2, 3 24: 1, 24, 2 , 12, 3, 8, 4, 6 Circle all the factors that appear in both lists. The only one is 6, therefore the HCF of 6 and 24 is 6.

Example #6 of LCM and HCF

Find the HCF and 36 and 18. List the factors of 36 and 18. 36: 1 ,36, 2, 18, 3, 12, 4, 9, 6 18: 1, 18, 9, 2, 3, 6 The highest factor to appear in both lists is 9, so the HCF of 36 and 18 is 9.

Example #6 of finding the LCM and HCF (Alternate method)

Find the HCF of 36 and 18. 36=3x3x2x2 18=3x2x3 6 x 6 3 x 6 3 x 2 3 x 2 3 x 2 Circle all the factors that appear in both lists. The only numbers to appear in both lists are 3 and 2, so the HCF of 36 and 18 is 9. (3x3=9)

Example #7 of finding the LCM and HCF

Find the LCM of 4 and 20 List the multiples of 4 and 20 4: 4, 8 , 12, 16, 20 20: 20, 40 , 60 , 80, 100 Cirlce all the numbers that appear in both lists. The only number to appear in both lists is 20.

Example #7 of finding the LCM and HCF (Alternate method)

Find the LCM of 4 and 20 4=2x2 20=2 x 2 x 5 2 x 2 10 x 2 2 x 5 There are 4 lots of 2, so we only need to multiply twice, giving you the sum: 2x2x5=20 The LCM of 4 and 20 is 20

Absolute value

A number's absolute value is the number of spaces it is away from 0 on a number line. The opposite of a number is the same number of places away from 0, but goes in the opposite direction. eg. The opposite of 5 is -5. Both numbers are 5 spaces away from 0 yet one is negative, the other is positive. eg. The opposite of 19 is -19, because both numbers are 19 spaces away from 0, yet one is negative, and one is positive. eg. The opposite of -6 is 6 because both of them are 6 spaces away from 0, except one is negative and the other is positive. eg. the opposite of 0 is 0. It's 0 spaces away from 0 (It is 0)

Absolute value

There is a way to write the absolute value of a number, and first write the number in a set of vertical lines, and write an exuals sign and it's absolute value: I6I=6 This means 6 is 6 spaces away from 0, the absolute value of 6 is 6. I-6I=6 This means -6 is 6 spaces away from 0, and the absolute value of -6 is 6. I8I=8 This means the absolute value of 8 is 8, because 8 is 8 spaces away from 0. I-8I=8 The absolute value of -8 is 8, because -8 is 8 spaces away from 0. I0I=0 the absolute value of 0 is 0.

Absolute value

The additive identity is any number added to 0, the number will remain the same. Any number to the power of 0 is 1. Any number multiplied by 0 is 0. This is called the 0 factor. Nothing can be divided by 0, and 0 can never be a denomiator. (a calculator would just say 0) A negative number added to a negative number= Negative A positive added to a positive will always be positive. The proper way to write out a subtraction sum is: Minuend-Subtrahend=Difference If you substitute the subtrahend with it's opposite and add it to the minuend, you still get the difference. Implied multiplication is when 2 numbers in brackets are standing next to each other. It still means multiply, but it's written another way.

Introduction to surds

A surd is a number with a square root symbol. The opposite to a surd is an index. (A power) Often, it is easier to write an answer in surd form because you don't have to worry about converting (turning or changing) it into a decimal, more useful is you don't have a calculator. To multiply 2 surds together, multiply the numbers inside the square root symbol. In this example * represents the square root symbol *5 x *8= *5x8= *40 *4 x *4= *4 x 4= *16= 4

The sequences of Square and Odd numbers

Can you see the relationship between the following sequences? 1,3,5,7,9 1,4,9,16,25 The first sequence is the sequence of odd numbers The second sequence is the sequence of square numbers Italian mathematician Leonardo Fibonacci wrote a book in the year 1225 called the book of squares, in which he explains in the first chapter what the relationship is between the Sequences of odd numbers and squares. This is what he said in the very first paragraph: "I thought about the origin of all square numbers, and discovered that they arise out of the increasing sequence of odd numbers" This is what he means; each number in the sequence of square numbers corresponds with each number in it's place in the sequence of odd numbers. (eg the first number in the sequence of square numbers is 1, which corresponds with the first number in the sequence of odd numbers, which is one) 1=1 The first term 1+3=4 The second term 1+3+5=9 The third term 1+3+5+7=16 The fourth term

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Divide a decimal by a decimal video

Data Handling

Plain sticky notes

Mode mean and range

When looking at data, there are three factors that can explain the graph. The mean ( The average result), the mode (The most common result), and the range (The difference between the largest and the smallest result)

How to find out the mean result

To find the mean is to find the average result. To find it, add all the results together and divide by how many results there are. (To find the average result out of the numbers 1,2,3,4,5 add them all together, giving you 15. Then divide by 5 because there are 5 results. The average result should be three)

How to find the mode

To find the mode out of a series of results, find which result is the most common. (The mode of the results 2,4,2,3,5,2,2 would be 2 because 2 is the most common result

What's a corrilation

If there is a corrilation between a set of results, (Eg an IQ test and age group) there will be a pattern in the result. (Age group tends to be higher among students) This result is just an example. The results are not real and do not affect any real tests conducted. There are three types of corrilation, a positive corrilation, a negative corrilation and no corrilation.

Probability

Probability is the likelyhood of an event happening. The probability of something is often shown through either a ratio or a fraction. To work out the probability, work out how many possible events there are. For example, what is the possibility of a red sweet in a bag? First, work out how many sweets there are in total.(In total means how many all together) Now, count how many of each sweet there is. For example, count how many blue sweets there are, how many green, how many red etc.

Our project on statistics

In class, we are doing a project on the birth and death rates in some of the major countries of the world. Japan, China, Russia. Our research indicates that the People's Republic of China's population will remain incredibly high. (14 births to 7 deaths per thousand people)

Statstics project

We expanded our project to other countries: South Africa Germany Switzerland Spain Italy Taiwan (Republic of China) Slovakia Slovenia Czech republic Poland Vietnam

Frequency

7<n<10 means n is bigger than 7, but smaller than 10 7<n>10 means n is bigger than 7 and 10 7<_n means n is bigger or equal to 7

cumulative frequency

An accumlative frequency is the quantity of data below a certain amount eg 4<n<10=10 10 is the frequency 11<n<17= 12 12 is the frequency 22 is the cumulative freuqency because add both frequencies together (10+12) and it's the total frequency (A running frequency)

Fair Probability

A fair dice is a six sided dice. (The probability of rolling a six sided dice is 1 in 6 because each side is the same side) A fair coin is a coin with a heads side and a tales side. (The probability of tossing a heads or a tails is exactly 50:50)

Working out the median in a frequency table with a variable

If the median out of all these probabilities is 2, find the largest value of x. 1 4 2 11 3 6 4 x How many 1's, 2'3 and 3's are there? 1,1,1,1 2,2,2,2,2,2,2,2,2,2,2 x Count how many numbers there are to the left hand side of the end 2.The one next to the x There are 14 numbers next to the 2, meaning that there would have to be 14 numbers next to it for 2 to be the median. We don't know how many of these numbers are 4, but we do know how many 3's there are, (6) so we can take 6 away from 14 for us to know how many of those numbers are 4's. (There are 8 4's) Meaning the largest value of x can be 8.

Working out the mean (Advanced)

To work out the average mass of a collection of data, work out the median of each catagory. Put your data in a table. Category Median Frequency Mean 0<n<2 1 6 6 2<n<4 3 7 21 4<n<6 5 8 40 6<n<8 7 9 63 8<n<10 9 10 90 Cumulative frequency: 40 Total mean: 220 220/40=5.5 Mean=5.5

Level 8/High GCSE Statistics (Part 1)

Look at this problm and try to solve it: The number of cars sold in different colours are listed in the table below: Colour 1 6 Colour 2 3 Colour 3 3 Colour 4 x If the mean is 4, find the value of x. There is no way to solve this problem using regular equations, so you have to use trial and improvement. First consider what we know: (6+3+3+x)/4=4 All the numbers (Including x) added together, then divided by 4, will give us an answer of 4. To use trial and improvement, substitute x for different values and see whether the sum would still equal 4. Consider what the sun would be without the x. (6x1)+(3x2)+(3x3)/12. (The frequencies added together equal 12). 6+6+9=21 21/4= 5.25 Trial 1. If x is 3: (6x1)+(3x2)+(3x3)+(4x3) 6+6+9+12=43 43/15=3 (all the frequencies added together, including 3, equal 15) TOO SMALL Trial 2. If x is 4: (6x1)+(3x2)+(3x3)+(4x4) 6+6+9+16=37 37/16=2 TOO SMALL Trial 3. If x is 7 (6x1)+(3x2)+(3x3)+(4x7) 6+6+9+28=49 49/19=3 Trial 4. If x is 11

Frequency graph terminology

The upper quartile is the section of a frequency graph that goes from 75% to 100%. The lower quartile is the space from 25% to 0%. The halfway point is 25% to 50%. The interquartile range is the upper 25% (75%)- the lower 25% (25%)

How to draw a frequency table

The cumulative frequency is always plotted (Drawn) on the y axis of the frequency graph. The graph is always a line graph The x axis is labelled like any other graph, each point going from 1 to 10. Then they're grouped into uneven classes based on the possibble results. When grouping them, draw a straight line underneath the x axis, with markers indicating each catagory. To make it easier when practicing, draw each class in a different colour, but this doesn't work in an exam.

Cumulative frequency

Label the y axis like you would with any graph. Don't write every number, just place a few key points, such as 5, 10,15 etc. When plotting the points, always put the points at the end of the class. If it's 1-2, place the point on the 2. Join the points with or without a ruler, but the line has to be curved. Examinors won't reward or take marks if you draw it a different way.

Introduction to Probability (Part 1)

Probability is how likely something will happen. A probability can be written as a percentage, a fraction or as a decimal. (The decimal is always smaller than 1) If you add all of the probabilites of all the possible events happening, the total will always be 1. If 2 events do not effect each other, (1 event happening will not stop another event happening) the 2 events are called independant. (A girl in America is late for school on tuesday, and a British weightlifter wins a gold medal in Beijing.) These 2 events are independant because the girl being late for school does not make the weightlifter win.

Introduction of Probability (Part 2)

If 2 events are mutually exclusive, that means that there is no chance that they will happen at the same time. (A fair coin is tossed. The probability of the coin landing on heads and a tails at the same time are mutually exclusive) The 2 events are mutually exclusive because it's impossible that the coin will land on both heads and tails. To work out the probability of something happening when you have all the other probabilities, write out a sum where all the events are added together and equal 1. Add all the other probabilities together. You should have 1 big number add the unknown probabilites. (Gather lke terms) (0.1, 0.2, 0.3, 0.3 are my probabilites) (There are 5 possible events and I know 4 of them, and I need to know probability c) (Write out an equation, where all the probabilities added to c to make 1) (0.1+0.2+0.3+0.3+e=1) (Add together all of the decimals, gather like terms) (0.9+e=1) Using inverse operations, work out the value of e e=0.1 The probability of event e happening is 0.1 or 10% or 1/10

Introduction to Probability

Probability= Number of events wanted/total number of possible events Number of events wanted divided by the number of possible events. (10 sweets in a bag, 7 of them are green. The probability of choosing a green sweet is 7/10 because there are 7 green sweets (The events we want) and there are 10 sweets in total (Total number of possible events). This can be re-written (Written in a different way) as a decimal, (To work it out divide the numerator by the denominator) or a percentage. (Multiply the decimal by 100) On an exam, the question will specify how they want you to write your answer

Tree diagrams

A tree diagram is an easier way to show all the possible probabilities. Start by drawing lines branching off from a certain point. (for each number of possible events. 2 children being born are independant events. The gender of the first child has no effect on the gender of the second child. Use a tree diagram to show all the possible events. Boy1/2 -------- Boy1/4 1/2x1/4=1/8 ---------Girl1/4 1/2x1/4=1/8 Girl1/2 ------------Boy1/4 1/2x1/4=1/8 -------------Girl1/4 1/2x1/4=1/8

Introduction to probability (Part 4)

In the tree diagram above, the probability of the second event happening (The second child being a boy or a girl) is a quarter, because there are 4 possible outcomes: 1st Child. 2nd child Boy,Boy 1/4 Boy, Girl 1/4 Girl Girl 1/4 Girl Body 1/4 We can check that this is correct because all the probabilities added together equal 1

Harder Statistics

The times that students at a school got during a race are recorded below, with how many students ran inbetween that time are recorded as frequency. Time Frequency 60<t<80 19 90<t<110 6 120<t<140 8 150<t<170 10 180<t<200 3 How many students are there? What catagory was the median result? How do you know this? What is the mean? Answers: 46 students 90<t<110 46/2=23 The 23rd pupil is in the second catagory, because if you count using the frequencies, you will get to 23 in the second catagory.

Mathematical processes and applications

Plain sticky notes

Long multiplication

Lay out the sum like this 23 x72 ------- Make sure that it is laid out in columns. Make sure the 100's , 10's and units are in the same column. If you're multiplying numbers with different amounts of units, always put the smaller one on the bottom.

Long multiplication (Part 2)

23 72 Start with the bottom unit. (2) Do 2x3 and 2x2. Write the answer underneath the line. Still make sure that it is layed out in columns. If the answer is over 10, place the 10 next to the next number.

Long multiplication (Part 3)

23 72 ------- 26 Next, multiply each number by the 10 in the bottom row. (7) Start writing on the next line down. When starting a new line, place a 0 in the units. This shows you're multiplying by 10. If you're multiplying by a number in the 100's, write two 0's.

Long multiplication (Part 4)

23 72 ----- 46 1510 -------- 1556 Once you have your two numbers, add them together.

BODMAS (Part 1)

There is a certain order that you must use to work out a sum, this process is called BODMAS (Also known as Bidmas) BODMAS works in the same way that an acrostic poem works. (Each letter in BODMAS stands for a different mathematical process) If you do the sum in a different order, you will get a completely different answer to the correct answer.

BODMAS (Part 2)

B-Brackets. Always do the sum in the brackets first. (In the sum 2+(8x2), do 8x2 because it is the sum in the brackets) (In the sum (5-1)x4, do 5-1 because it is the 1st sum in the brackets) (In the sum (4+2)x(8x3), do 4+2 and 8x3 because they are the sums in the brackets) (In the sum (6-9)x(5+9) do 6-9 and 5+9 because they're the sums in the brackets) If you get a sum like this= (9+[1+1]), always do the sum in the square brackets first, then do the sum in the round brackets. It will be (9+2) (In the sum ([64 divided by 8]-9) do 64 divided by 8 because it is the sum in the brackets. Then do (8-9)) (In the sum (5x[4x4]), do 4x4 because it is the sum in the brackets. Then do (5x16 because it is teh new sum in the brackets))

BODMAS (Part 3)

The O part of it is diffcult to understand because the next order of operation dosen' t start with O. You have to put a p in front of it. order of operation of pOwers. You do any powers (Indices such as squared, cubed) after you do any powers (For the sum 5x(7+2) squared, do 7+2 because it is in brackets, then do 9x9 because 7+2 is 9 and the sum is 9 squared. (Squared means to multiply a number by itself) (For the sum (6-5) cubed, do 6-5 because it is in brackets, and then do 1x1x1.

BODMAS (Part 4)

The M in BODMAS stands for multiplication. You now need to do all the multiplication sums before you do the addition. (If you had the sum 5+9x3, you need to do 9x3, then work out the answer to the sum. The new sum will be 5+27 because 9x3 is 27) (If you had the sum 7x4+8, you need to do 7x4 and then work out the answer to the new sum. The new sum will be 28+8 because 7x4 is 28) (If you had the sum (8-2)+9x9-8, you need to do the sum in the brackets first. Then, do 9x9 because ti is a mulitplication sum. Now, work out the answer to the new sum. The new sum is 6+81-8)

BODMAS (Part 5)

A stands for addition. It means that you now do any addition sums that aren't in the brackets before you do subtraction. (If you have a sum 3x4+7, do 3x4 first. Your answer is 12, then add that to 7) (If you have a sum 7+8/2, do 8 divided by 2. Your answer is 4. Now do 7+4) (If you have the sum (7+9) squared -2, do 7+9 because it's in brackets. Your answer is 16. Now do 16x16 because it's squared. Your answer is 256. Now subtract 256) (If you have the sum 4-2 cubed, do -2 cubed because it's a negative number and it's an index. Your answer is -8. Now write out the sum again. it's now 4-8)

Bodmas (Part 6)

The last letter stands for Subtraction. It's important that you do subraction after addition. If you follow BODMAS, there is no sum you'll get wrong. (If you had the sum 8x4+9-7, you'd do 8x4. your answer is 32. Then add 9 to 32. Your answer is 41. Then take away 7. Your answer is 25) (If you had the 4+2-1, do 4+2. Your answer is 6. Now take away 1. Your answer is 5) (If you had the sum 69-2+9, do 2+9. Your answer is 11. 11 now takes the place of 2+9. Now do 69-11. You answer is 58)

The Soroban (Japanese calculator)

A soroban is a Japanese (Look under History of maths) abacus, which is used by Asian students to work out very large sums. (Some with 4 digits) When the student has used the soroban for a very long time, they can imagine the soroban and solve it that way. (Solving 6782 x 6547 in their heads!!!!) The soroban is alot better than the western calculator which we're used to.

Anzan

Anzan is the ability to perform addition problems mentally incredibly quickly. It originated in Asia.

Frequency table

25% of the questions on a maths exam could be answered by using a frequency table. A frequency table is good when answering word problems, such as (15 chocolates cost $12, how many would 5 chocolates cost?) Rather than try to figure out how much 1 chocolate costs, and multiplying it by 5, you can arrange it in a table. The table consists of the costs one side, and the amounts the other side. Usingthe "Magic L" (See below), figure out what would go in the amount for 5 chocolates.

Magic L

the magic L is the way of working out a blank square in a frequency table. The order it goes in is Divide, Multiply. You can start this anywhere on the table.

Calculus (Part 1)

In calculus, work out the rate of a graph by dividing the change in distance by the change in time The secant line is the change in the rate between 2 points on a graph. (It's a diagonal/Sloping line that goes through the 2 points) An indeterminant form is when the rate is equivilent to 0/0 Instantaneous rate of change is the instant rate of change. I is the slope of the tangent line. The limit is what is happening on the graph going up to a function. (A point on the graph)

Rich sticky notes

Rich text note

Multiplication

Sequences guide

Anzan (Watch their hands)

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Soroban

Algebra

Bookmarks

Greek alphabet

Plain sticky notes

Greece and Maths

Mot symbols in Algebra come from the Greek alphabet: Alpha Beta Gamma Delta Epsilon Zeta Eta Theta- Used for an unknown angle in Trigonometry Iota Kappa Lambda Mu-Population mean Nu Xi Omicron Pi- Circumference/Diameter Rho Sigma-Sum of Tau Upsilon Phi-Golden Ratio. (1.610 Chi Psi Omega-Resistance in an electrical circuit

Like terms

In Algebra, you can only add, subtract, multiply or divide like terms. Like terms are terms in a sum that have the same (letter) variable. So you could only add 2q to 9q because their variables(letters) are the same. You can't add 5y to 3x because their variables (letters)are not the same

The letter x

The letter x is the most commonly usd letter in algebra. x is written as a backward c standing next to a normal c. This is not to confuse it with the multipliation symbol.

Algebra for Dummies (Part 3)

Variables and constants are not the same. Variables are used to represent (show) any value. Any letter can be used. A constant is a letter or symbol which is used to represent a specific value throughout all of maths. An example of a constant is Pi, which is always used to represent 3.14 Letters used to represent variables are often at the end of the english alphabet. For example x,y,z but letters from other alphabets can be used. Letters used to represent constants are often at the beginning of the english alphabet. For example a,b,c,d but letters from other alphabets can be used. Theta, a zero with a line through it, is a letter from the Greek alphabet, and is used to represent the value of an unknown angle in trigonometry.

Algebra for dummies

A simpler way to look at Algebra is to compare it to another way of solving math problems. Replace each letter with a box or other symbol not used in maths. 5+v=100 5+#=100 Writing letters is a formal way of writing a sum when you don't know what a term of the sum is. (The word term refers to numbers in a sum, like a sequence)

Algebra for Dummies (Part 2)

The letters are called Variables and constants. The letters are from English, but some letters used are from the Greek alphabet. (Beta and Pi are a few) Letters used are always lower case letters of the alphabet often mean different things in other subjects. (n is a variable in maths whilst N means mass in Physics) The most used letter is x.

Another version of Pi??

Pi is basically how many diameters of a circle (Distance across the middle of the circle) fit inside the curcumfrence. (Perimeter of a circle)

Advanced Algebra (Part1)

A quadratic expression is any expression that has a square in it. (Any term, number or letter, to the power of 2) If a number has a cube in it (Any number of term to the power of 3) that is called a cubic sequence and not a quadratic sequence, because a quadratic sequence can only have a square term in it. To factorize a quadratic equation, (Write it as 2 sets of brackets), find the HCF of both numbers eg: Factorize 2x2 (2x squared) and 7x. The HCF is x, because it is the only number that goes into both terms. The outside of the brackets would be x, but to work out what actually goes inside the brackets, divide both sides by x, giving you: x(2x+7) If you expand those brackets, you will find it equals 2x. When writing quadratic expressions, write the squared terms first, then the terms with variables (Alphabetically, of course) and then the numbers. eg. 4x+9+9y squared would be re-written as: 9y2+4x+9 If you have a longer expression to factorize that involves 2 sets of brackets, work out what the square term would be: (If you has p2, each bracket would start with p, and then look at the number, what are it's factors? Which pair of factors added together give you that other number?) eg. x2+9x+20 Write out the factors of 20, (1,20,2,10,4,5) which pair of factors make 9? The answer is 4 and 5; Therefore, our brackets would be (x 4)(x 5), to see what the signs are in the middle are, use trial and error to see which combination of signs would work. Trial 1: A + and a + (x+4)(x+5) Using FOIL (First, Outside,Inside,Last) expand these brackets: 1. x x x is x2 (x times x is x squared) 2. x x 5 is 5x (x times 5 is 5x) 3. 4 x x is 4x (4 times x is 4x) 4. 4 x 5 is 20. Add these together: x2+5x+4x+20 x2+9x+20 This matches our original expression, so we have factorized correctly. Advanced trigonometric ratios: Sec0=1/cos0 (Secant theta= 1 over cosine theta) CSC0=1/Sin0 (Cosecant theta=1 over sine theta) COT0=1/Tan0 (Cotangent theta= 1 over tangent theta)

Multiplication of x

When multiplying by x, do not use the multiplication symbol. Instead, place x in brackets and the number outside: 2(x) 3(x) 8(x)

Sticky note

A solution

A solution is a value (A number) that when substituted with the symbol will make an equation true. A solution set is the seies of all solutions for an equation. A solution set is written like this: x+9=10 Solution= 1 Solution set {1}

Greece and Maths

Greek is a language like Russian, it contains alot of characters that aren't in English. There are many Mathematicians and Philosophers who originated from Greece. Many mathematical processes (Like Pythagoras Theorum) come from Greece. the words shown above are the pronounciation, and links to images of the Greek alphabet are at the bottom

Variables and constants

Any letter can be used to represent a variable. a variable is a letter that can be used to represent any value A constant is a letter that is used to represent a specific value. Pi (Used to represent 3.14) is the most well known constant in maths

What is an equation

If a sum has any letters in it, it is called an equation. Letters are also called variables. eg. 5+y=8 The point of an equation is to find out what y is equal to. If a number is directly next to a letter ( A variable) It means to multiply the letter by the number. (2n means 2xn)

What is an equation (Part 2)

An equation always includes an equals sign. An equation is a statement (A sentence) consisting of 2 sums taht are equal. Equations can be true (The right answer such as 1+1=2) it can be false (The wrong answer such as 1+1=4) or it can be neither (x+4=10 is neither because the value of x is unknown)

Solving Simultanious Equations (Elimination method)

The aim of this method is to take 1 equation away from the other. 2x+3=7 5x+4=14 Label the top equation Equation 1 Label the bottom equation Equation 2 Equation 2-Equation 1 3x+1=7 Take away 1 from both sides 3x=6 Divide both sides by 3 x=2

What is an equation (Part 3)

When you replace a symbol with a number, it is called assigning that number a value or substituting the symbol. By assigning (Or substituting) the symbol with different numbers, you can make the equation true. This is a basic way and the proper way to solve an equation is found on this page.

Implied coefficient

The coefficient of -x is -1 The coefficient of -y is -1 -x is short for -1x, but it's not written that way in maths -y is short for -1y, but's it's not written in maths

Multiplication of x (Part 2)

Another way to write it is placing a decimal point inbetween x and the number: 5.x 4.x 3.x 2.x

What is an equation (Part 4)

To substitute a symbol in an equation write it out like this: 2x+9=19 2(5)+9=19 Write down every step that you do in an equation. Make sure that every equals sign is in line. 2x+9=19 2(5)+9=19 10+9=19 19=19 You are not finished with an equation until you have the same value (Number) on both sides.

Equations are not expressions

Equations and expressions are not the same. An equation has an answer, whislt and expression isn't. 5y+6=31 This means multiply y by 5 and add 6. The answer should be 31 4-x=2 Take away x from 4. Your answer is 2. Work out what x is by subtracting 2 from both sides. Take 2 from 4.

Implied multiplication

To multiply x in an easy way, write the number next to x. The number is the coefficient of x: 8x 3x 2x 4x 5x X should always be to the right of the number, not x2

Equations with fractions

If an equation includes a term that's a fraction, turn every term in the equation into a fraction. 2x+1/2-2=40 To get rid of the fraction, make every term a fraction. -2 is equivilant to -2/1 because -2 is a whole number 2x is equivilant to 2x/1 becuase it's a whole number 40 is equivilant to 40/1 because it's a whole number Find the lowest common multiple (LCM) of the denominators The 2 different denominators are 2 and 1. The LCM is 2 Change each of the fractions to have a denominator of 2 by multiplying each numerator by 2. The equation can be solved now because the fractions are the same. The equation is now: 4x+1-4=80

Simultanious equations (Part 1)

Simultanious equations means trying to solve 2 equations at once. 2x+6=x+11 ------- ------ 3x+1=4x-4 Start by cross multiplying these equations because they're fractions. (2x+6)(4x-4)=(3x+1)(x+11) Using FOIL, expand these brackets. 8x-8x+24x-24=3xsquared+33x+x+1 Simplify this equation 24x-24=3xsquared+34x+11 Subtract 24x from both sides -24=3xsquared+10x+11 Subtract 11 from both sides -35=3xsquared+10x Find the square root of 3xsquared and do the same for-35 -5.91=3x+10x Add both x terms -5.91=13x Divide both sides by 13. -5.91/13=x

Harder Simultanious Equations

3x+y=10 x+8y=11 Make the y variable the same on both equations. Find the Lowest common multiple of 8 and 1, which is 8 Multiply equation 1 by 8 Multiply Equation 2 by1 because the y variables need to be th same 24x+8y=80 x+8y=88 23x=8

Simultanious equations and Line Graphs (Part 1)

What are the values for x and y in these equations x-y=1 x-y=2 This equation cannot be solved, because the values of x and y are constant (The same) in both of these equations, and both equations have the same operation (x-y) so the answer would be the same. To work out the values of x and y, rearrange both of these equations in order to form the name of a line: x-y=1 becomes y=x+1 x-y=2 becomes y=x+2 The numbers in this equation are positive, and the y variable is on one side, and the x variable is positive, so it becomes these new equations. Draw a table of 2 rows. One of x and one of y. you will be given a limit from a negative number to a positive number.

Equation key words

Different equations are said to be the same if they have the same solution set. A contradiction/Inconsistent equation is where the equation has no solution The way to show an inconsistent equation is {}.. An identity is an equation which has a symbol that is a variable. (Has any value)

What's a linear sequence?

A linear sequence is a series of numbers the are increasing or decreasing by the same amount. (It's getting bigger or smaller by the same amount each time) If a sequence is getting bigger by different numbers each time, it is not a linear sequence. (If the term to term rule was +1, +2, +3 etc, it would not be a linear sequence)

Working out nth term (Part 1)

Each number in a sequence is called a term. The amount that the numbers are getting bigger or smaller by is called the term to term rule. (The number that each term is geting bigger or smaller by is called the term to term rule) (The term to term rule of the sequence 1,2,3,4,5,6 would be +1 because each term is getting bigger by 1) (The term to term rule for the sequence 3,6,9,12,15 would be +3 because each term is getting bigger by 3) (The term to term rule for the sequence 4,8,12,16 would be +4 because each term is getting bigger by 4) (The term to term rule for the sequence 5,10,15,20,25,30 would be +5 because each term is getting bigger by 5) (The term to term rule for the sequence 8,6,4,2 would be -2 because each term is getting smaller by 2) (The term to term rule for the sequence 100,90,80,70 would be -10 because each term is getting smaller by 10)

Working out the nth term (Part 2)

When you have worked out the term to term rule, work out the next 2 terms in the sequence. Do this through using the term to term rule. (The next 2 terms for the sequence 1,2,3,4,5,6,7 would be 8,9 because the term to term rule is +1) (The next 2 terms for the sequence 3,6,9,12,15 would be 18,21 because the term to term rule is +3) (The next 2 terms for the sequence 4,8,12,16 would be 20,24 because the term to term rule is +4) (The next 2 terms for the sequence 5,10,15,20,25 would be 30,35 because the term to term rule is +5) (The next two terms for the sequence 8,6,4,2 would be 0,-2 because the term to term rule is -2)

Working out the nth term (Part 3)

Next, work out the previous term. The previous term is the name of the term that comes before the 1st term. To work this out, use the inverse (Do the opposite) of the term to term rule. (The previous term for the sequence 1,2,3,4,5,6 would be 0 because the opposite for the term to term rule is -1) (The previous term for the sequence 3,6,9,12,15 would be 0 because the opposite for the term to term rule is -3) (The previous term for the sequence 15,20,25,30 would be 10 because the opposite for the term to term rule is -5) (The previous term for the sequence 9,11,13,15 would be 7 because the opposite for the term to term rule is -2) (The previous term for the sequence 100,90,80,70,60,50 would be 110 because the opposite for the term to term rule is +10)

Working out the nth term (Part 4)

To finally work out the nth term (emphasis on Finally) multiply the variable (Letter) n by the term to term rule and add the previous term. (If the previous term is a negative number, don't place an add sign, just write the number. This rule also applies to the term to term rule) If the previous term is 0, don't write it (If the nth term was 3n+8, the term to term rule would be +3 and the previous term would be 8) (If the nth term was 6n-9, the term to term rule is +6 and the previous term is -9) (If the nth term is 7n+2, the term to term rule would be +7 and the previous term is +2) (If the nth term is -4n, the term to term rule would be -4 and the previous term would be 0) (If the nth term is 9n, the term to term rule would be +9 and the previous term would be 0)

Working out the nth term (Extended)

Now that you have the nth term you can work out any term in the sequence. All you need to do is substituten for the number. (Swap n for whatever number you want) Remember, if a variable (Letter) is next to a number, it just means multiply the 2 numbers) (If you want the 100th term, swap n for 100) (If you want the 50th term, swap n for 50) (If you want the 10th term, swap n for 10) (If you want the 7th term, swap n for 7)

Another way to write nth term (Part 1)

Expression: T(n)=3n+9 T=Term n=variable 3n= 3xn (Each term is increasing by 3)

Another way to work out nth term (Part 2)

T(8)=3x8+9=33 T(8)= 8th term in sequence 33= 8th term in sequence

Basic algebra

In algebra, the value (Number) next to a symbol is call the coefficient of the symbol. 2n 2 is the coefficient of n n is the symbol

Quadratic sequences

A Quadratic sequence (Not to be confused with a quatratic equation) is a sequence which has an nth term formula containing n2. (n squared) The reason why n is written in lower case is because N is a symbol for mass in physics

examples of quadratic sequences

T(n)=n2 (N squared) T(n)=9+2 squared (Do 2 squared first then add 9) T(n)=3+ n3 (Cubed)

What is a sequence

A sequence is a series of numbers that increase or decrease by a certain value. Each number in the sequence is called a term. The amount that the number is increasing or decreasing by called the term to term rule. (Also called the constant difference)The term to term rule can differ for each term in a sequence. (The constant difference for each term in the sequence 1,2,4,7,12 is not the same)

Working out other terms in a sequence

After working out the nth term for a sequence, substitute the letter n for the number of the term that you want to work out. (The 50th term for 2,4,6,8 would be 100 because 2n qbecause 2n would become 2x50 which is 100)

Differences between Equations and expressions

Equations and expressions are not the same. An equation is a sum involving algebra that has an answer with it. Solving an equation could be substitution (Replacing x with a number) or finding the value of x. (3q+7=10 is an equation) Expressions are sums involving algebra that involves no answer. (5a +2a is an expression)

Calculus intro

Calculus is advanced Geometry and Algebra, which covers functions,limits,integrals and dereriatives.

Basic Derivatives

Saying sqroot25 is the same as saying 25 1/2power Saying sqroot36 is the same as saying 36 1/2power Saying cuberoot 1000 is the same as saying 100 1/3 power Saying 6 -3 power is the same as saying 1/6cubed Saying 7-2 power is the same as saying 1/7squared For a derivative, if the number in brackets is a whole number, it is equal to 0, no matter what the number is: d/dx (15)=0 It doesn't matter if that is 15 or not, it would still equal 0 If the number in brackets is a power, all you need to do is move the power in front of the number, subtract one from the power and make that the new number: d/dx(x3)= 3x2 (3x squared) because the cube is moved in front of the number, 3-1=2, which makes 2 the new power If the number in brackets is the square root of a number (not a variable) it still is a number, so it equals 0 d/dx (sqroot20)=0 sqroot 20 is still a number, the only time at which this rule changes is if it is the sqroot of a variable (like x, y etc) d/dx(sqrootx)= d/dx (x1/2 power) Remember, a square root is the same as raising the number to the half power. Now that we have an index, we can use another rule to apply, where the index is moved in front of the number, and the new index is 1/2x-1=-1/2x

Questions with no answrs?

Simplify these expressions: 2y+5w x2+y3 3e+9-x None of these can be simplified because you can only add like terms.

Working out the height of a triangle using algebra

A triangle has a base of 10cm, and an area of 30cm squared. What is it's height? The formula for the area of a triangle is a=bh/2 Substitute the values. 30=10h/2 H represents the height To get rid of the fraction in this equation, multiply both sides by 2. Now you have the equation: 60=10h Divide both sides by 10 to give you h on it's own 10h means 10 x h, but we don't want 10 lots of h, we only want 1 h, and what you do to one side of the equation you have to do to the other side. 60/10=10h/10 6=h The height of this triangle is 6cm. But how can we be certain? 30= 10 x 6/2 10x6 equals 60, divided by 2 is 30, so we have the right answer

Introduction to Enlarging shapes

A scale factor of enlargement is the ratio or rule that mathematicians use when enlarging (making bigger) a shape. To work out the length of each side after it has been enlarged, multiply each side of the smaller sides by the scale factor. eg. The scale factor is 4. Enlarge an equilateral triangle with sides of 3cm by the scale factor of 4. Each side is 3. Multiply it by the scale factor 4. 3x4=12. Each side of the enlarged shape will be 12cm When drawing an enlarged shape, there will be a center of enlargement. This is the point which you will use to draw the enlarged shape. The length of the construction lines you draw will depend on the scale factor. Measure half of the length of the construction line that travels from the center of enlargement to the shape. Multiply that length by the scale factor. To work out the area of the enlarged shape, square the scale factor and multiply by the area

Inequalities (Version 2)

Inequalities can be solved like ordinary equations, the only difference between inequalities and equations is that an equation always has an equals sign (This can be remembered through the word equal being in the word equation) and inequalities always have a smaller than, greater than, smaller than or equal to or greater than or equal to (An inequality can be remembered that it would never be any certainty that the variable would always be equal to the number) x<4 This means x is always smaller than 4. This can be shown on a number line through drawing a circle above 4 and drawing an arrow going towards the left hand side of the line (All the numbers smaller than 4) x>4 This means x is always larger than 4. This can be shown on a number line by drawing a circle above 4 and drawing an arrow towards the right hand side of the line (All the numbers larger than 4) x<_4 This means that x will always be a number which is 4 or smaller. This can be shown on a number line through drawing a solid dot above 4 and drawing an arrow towards the left hand side of the line, towards all the numbers smaller than 4 x_>4 This means that x will always be a number which is smaller than 4 or equal to 4. This can be shown on a number line by drawing a solid dot above 4 and drawing an arrow towards the right hand side of the line, towards all the numbers larger than 4

Completing the square (Part 1)

Completing the square is a method for solving an equation. The first step is to make sure it is in the form of ax2+bx+c=0 The second step is to half the coefficient of x (or in other words, divide b by 2) Draw a set of brackets and write an x Write the number you got for b/2. (write the number you got when divided b by 2) Finish off by writing a + sign if b is positive, or a - sign if it is negative. Also, write a squared sign outside the brackets. (you will now have the brackets all squared) (A posh way of saying squared is that "You've raised something to the power of 2") It should look like this: (x+a)2 That number you wrote in the brackets, the one for b/2, now square it. What do you get? The number you get will always be positive, because (a positive x a positive= a positive number), whilst (a negative x a negative=a positive number) You do not want this number in your equation, so you have to subtract it on the end of your new sum. This is called "making a correction". It should look like this: (x+a)2-a2 Don't forget the +c term you had on your original equation. Bring that number down into your new equation. Remember the rule on collecting like terms? (if not see the top of this category) Well now you can collect like terms on the numbers you've made. You should now have an equation that looks something like this: (x+a)2 +c

Completing the square when the coefficient of x2 is greater than 1

When completing the square but the coefficient (the big number next to) x2 is great than (bigger than) 1, you can still use completing the square: The first step is to take a common factor out of all three terms. The factor to take out is the coefficient of x2. (for example, if the coefficient of x2 is 5, it would be written as 5x2 and you would take a common factor of 5 out of all three terms. (To take out a common factor, you write the number outside a set of brackets and then divide each term by that number) For example, when the equation is 5x2+10x+25=0 (remember that a quadratic equation has to be written in the form ax2+bx+c=0, because on a graph, where the graph will touch the x axis twice, where y would be 0) (the graph of x2 is called a parabola, and is symmetric around the y axis, meaning a reflection of f(-x) a reflection in the y axis would be exactly the same, which makes sense because any number squared, whether it is positive or negative, would still be positive.

The quadratic formula

The quadratic formula is a formula that will give you the solutions for any quadratic equation- It is often used when the quadratic will not factorize. Often you will get long irrational numbers, so they will normally only ask you to use the formula in an exam question. x=-b+-Rt(b2-4ac/2a The +- sign means "plus or minus," which means you can use the formula in 2 ways: To add or to subtract. To get your two solutions, you need to do the formula adding, and then the formula subtracting

Asymptotes

When a graph, (eg. y=1/x or 1/x2) gets closer and closer to touching the axis but never actually touches is called an asymptote, because the graph will be undefined if x=0 because you can't divide anything by 0, or you can't divide 1 by anything to give you 0, thus meaning it will never touch. Whenever (in calculus) there is a graph that is undefined (eg. gives you a math error on a calculator) at a point, you put a circle at that point.

Quadratic inequalities

To solve a quadratic inequality, first change the inequality to an equals sign. (or just treat it like an equation) Factorize it, and then solve it. (By just changing the signs of the integers in the brackets) This gives you the points where the parabola will cross the x axis. To make it easier to visualize, draw a parabola and mark the points where it crosses the x axis. The inequality now has three parts to it, the letter x, and the 2 points where it crosses, but you need to work out the inequality symbols. To do this, just substitute numbers back into the original inequality , and see if it satisfies the inequality. Substitute the 2 integers into the quadratic to see if it's including them or not.

Geometry And measure

Bookmarks

Article on dimensions

Plain sticky notes

Angles in a triangle

The angles in a triangle add up to 180 degrees.

Co interior Angles

Co interior angles are supplementry.(They both add up to 180 degrees)

Corresponding Angles

Corresponding angles are complentary. (They make an F shape.)

Alternate angles

Alternate angles are equal

Introduction to radians

There are two ways to measure an angle. In either degrees or radians. 360 degrees (One complete turn/revolution) is the equivalent of writing 2PiRadians. Some mathematicians believe that Measuring in radians is not as accurate as measuring in degrees because of Pi being an Irrational number, (Cannot be expressed exactly as a ratio or fraction) but that is largely trivial.

Vertically opposite angles

Vertically opposite angles are complementry.

Right angled triangle

A right angled triangle contains an angle of 90 degrees (A vertical line) and the other 2 angles add up to 90 degrees. Supplementary

Supplementry angles

Supplementry angles are 2 different angles that add up to 180 degrees.

Exterior angles of a polygon

The exterior angles (Angles around the outside of a polygon) of a polygon (Any shape with 3 or more sides) always add up to 360 degrees,

Angles converted to Radians

Here are some angles expressed as radians; to work them out, you really need to know what 360 degrees is and then how the other numbers are connected to it 360= 2PiRaidans 180=PiRadians (180 is half of 360, and 360 is 2PiRadians, so half of 2 x Pi Radians must be PiRadians) 60= Pi/3 Radians (60 is one third of 180, and 180 is PiRadians, so one third of PiRadians must be Pi/3 radians) or (2PiRadians is 360, 60 is one sixth of 360, so dividing 2PiRadians by 6 gives you 2PiRadians/6, which can be simplified to Pi/3Radians) 45= Pi/4 Radians (180 is PiRadians, 45 is one quarter of 180, so PiRadians divided by 4 must be Pi/4Radians 90=Pi/2Radians (180 degrees is PiRadians, 90 degrees is half of 180 degrees, so in radians we can rewrite that, using algrebra as Pi/2Radians, to express PiRadians <180 degrees> being divided by 2 270= 3Pi/2 Radians (270 is 3 x 90, and 90 is Pi/2Radians, we can simply multiply that by 3, giving us 3Pi/2Radians 1= 2Pi/360Radians (If 360 degrees is 2PiRadians, we can work out what 1 degree is by dividing both sides by 360, because we know what 360 is, but we only want one) 72= 72 x2Pi/360Radians (If we know that formula for one degree, than all we need to do is multiply any number by that formula to get a final degree)

Degrees and Radians Continued

Another way to think about the relationship between degrees and radians is to consider the circumference of the circle and it's formula: C=2PiR 360= 2PiRadians/2Pi One complete turn of the circumference is 360 or 2PiRadians. A Unit circle is a circle with a radius of 1, and it's four points would be: (1,0)(0,1)(-1,0)(0,-1)

Congruent triangles

If 2 triangles are congruent, then that means that they're identical. (In length and angle) When asked to find out how 2 different triangles are congruent, look at the lengths of the sides and the size of the angles. Often,only two sides or angles are specified. Each (Vertex)corner would be given a letter to name it. If two lines are the same, then a line is marked on the sides that are equal to show that they're equal.

Height

Another word for height is altitude

Interior angles of a polygon

The interior angles (Angles on the inside of a polygon) of a polygon) added together follow the expression (n-2)x360. The n stands for number of sides

Complementry angles

Complementary angles are 2 angles that add up to 90 degrees

Rotational symmetry

Rotational symmetry is how many times a shape would fit into itself when rotated. If it never fits into itself when rotated, it has rotational symmetry order 1. (An example is a pair of glasses with a curved rim) An equilateral triangle (A triangle with 3 equal sides and angles) has rotational symmetry order 3.

Rotational symmetry (Part 2)

Write the rotational symmetry as: Rotaional symmetry= Order 1

Angles in a square

The angles in a square add up to 360 degrees

Sloping lines

The general equation for a sloping line (Not a straight line) is: y=mx+b The b stands for the y intercept. If the gradient is a fraction, start from the y intercept and move up the graph by the numerator, and go across (To the right if the denominator is positive) by the denominator. If the gradient was 3/4, move up 3 spaces from the y intercept, and move 4 spaces across If the gradient was 1/2, move up 1 space and move acrss 2 spaces

Angles in a polygon

Angles in a polygon add up to 360 degrees.

Angles on a straight line

Angles on a straight line add up to 180 degrees

Overview of Trigonometry

Cosine0= ajd/hyp sine0=opps/hyp tan0=opps/adj When solving a trigonometry problem, always substitute 0 with the value of the given angle first!!!!!!!! Substitute the values of the triangle into the equation, and you should find you get a fraction. Make both sides of the equation factions. Place the side that isn't a fraction to begin with over a 1. (The value will not change) Cross multiply the 2 factions, giving you x on it's own (One side of the triangle may be a variable) and theta on the other side.

Baseline of a triangle

The bottom side of a triangle is called the base. When constructing a triangle, always start by drawing the base. Place the compass on the end of either vertex of the line, and draw an arc . (Part of a circle)

Area, Height and Base of a triangle

To work out the area of a triangle, multiply the base by the height (How tall the triangle is) and divide the answer by 2. If you're given the area and the height, you have to work out the base. Multiply the area by 2 and divide the answer by the height. If you're given the base and the area, you have to work out the height. Multiply the area by 2 and divide the answer by the base. If the final answer contains decimals, round the number to 2 decimal places

Sticky note

In a geometry problem, you will be given the radius of the circle, and the size of the angle of the sector. Find the area of the sector The radius is 5cm. The angle is 60 degrees. Another way to write the angle is 60/360 (as a fraction) The full way to write the formula will be: (60/360) x(5x5) x Pi We can simplify that: (1.6) x (25)

General equation for graphs

The general equation for a line is y=mx+c or y=mx+k. (y is the actual name for the line) (m is the gradient of the line) (x is the variable) (k is the y intercept)

Chords of a circle

A chord of a circle is what divides a circle into 2 halves. These halves are called segments. The larger segment is called the major segment, and the smaller segment is called the minor segment.

Y intercept

The y intercept is the point which the line crosses the y axis. The name of the co-ordinate will always start with 0 as the point is on the y axis.

Sine (Part 1)

a/sin20=5/sin100 Side a and angle a is 20 degrees Side b 5cm and angle b is 100 degrees First make this a linear equation by cross multiplying. a sin100=5sin 20 Make a the subject of the equation Divide both sides by sin100 a sin100/sin100= 5sin20/sin100 a=5sin20/sin100 USing a calculator, multiply sin by 20. Then multiply the answer by 5

Equipment for constructions

The following equipment is needed for constructing a triangle: A ruler (30cm) A protractor (A seethrough protractor) A pair of compasses (A compass) An eraser (A rubber) A Pencil A pencil sharpener Do not rub out any construction lines as examiners award marks for seeing this.

Constructions

Constructions are when a shape is made by following a set of instructions that tell you the length of the sides (How long the sides have to be) how wide the angles are etc. To construct a shape, you need: A ruler (For measuring the length) a protractor (For measuring angles), A pair of compasses (For drawing circles and arcs), A pencil, (Not a pen as most of geometry involves rubbing out) An eraser and A hard surface in order to draw clearly.

Intorduction to sine

Sine is a rule in Geometry that links sides of a triangle to the angle in a triangle. The rule works for a right angled triangle. a/sinA b/sinB c/sinC The hypotemuse of a triangle is always opposite the right angle Side a is opposite angle a, Side a has a sine of a degrees.

Multiplying by Sine on a calculator

To multiply a number by sine on a calculator, press the sine button (Which has been shortened to Sin) and type in the number you want. (Enter it as= Sin62)

Rule of Pythagoras

The Pythagoras theorem only applies to right angled triangles

Angles in a circle

Angles in a circle add up to 360 degrees.

Calculator

In Geometry, the 3 most commonly used buttons are Sin (Short for Sine) Cos (Short for Cosine) and Tan (Short for tangent)

Inscirbed Angles

Inscribed angles are made when a polygon is drawn inside a circle. The information given includes 2 arcs (Two parts of the circle with an angle) and angles a and b. add the 2 arcs together and divide the answer by 2. This gives you the value of angle a. Angles a and b are supplementry, meaning that they equal 180 when they're added. Subtract angle a from 180 degress, giving you the value of b.

Area of a circle

Radius x radius xPi The radius is the distance from the centre of the circle(The origin) to the outside.

Diameter of a circle

The diameter is how wide the circle is, from one side to the other. This is drawn as a straight line. It is not the same as the radius, which is half the diameter. (If the diameter is 5cm, then the radius is 2.5cm) (If the radius is 10cm, than the diameter is 20cm)

Perimeter of a circle

The perimeter of a circle is called the circumfrence. (The distance around the outside of the circle) (How long a line would be if the circle did a complete turn) The circumfrence divided by diameter=PI

A tangent

The tangent is a straight line that touches the circle at a point. It is a type of chord

Lines and line segment

In geometry, a line goes on forever. A line segment is a line which has a specific beginning and end. (eg. Starts at point A and ends at point B)

Dimensions

A 2-D shape has 2 dimensions. It is flat and volume does not apply to it. A 3-D shape has 3 dimensions. Volume and Area apply to it

Dimensions (Part 2)

If a sphere was cut in two, the new shapes are called hemispheres. The number of dimensions in an object is the smallest number of straight lines needed to go through every corner (Vertex) on a shape. (A triangle has two dimensions becuase two striaght lines are needed to needed to cover all the Corners (Vertecies) on a triangle)

Volume

Volume is how much space is in a 3-D shape. An easy way to think about it is how much liquid the 3-D shape can hold. Volume is measured in cm cubed (written as cm3)

Area of a trapezium (Trapezoid)

A trapezium (Also called a trapezoid) is a quadrilateral (Four sided shape) that has one pair of parallel lines. To work out the area, label the parallel lines sides a and b. Add these 2 sides together and work out the mean to the total by dividing by 2. (Add them together and divide by 2) Then multiply your answer by the perpendicular height of the trapezium.

Edge

An edge is a line segment which joins two corners in a polygon

Volume of a pyramid

To work out the volume of a pyramid, you have to work out the area of the base. (The square face of the pyramid) Then you need to multiply that by the perpendicular height of the pyramid. Divide you answer by 3.

A Vertex

A vertex is a type of point which is a geometric term for a corner. (An intercection of two line segments)

Volume of a cylinder

To work out the volume of a cylinder, you will be given measurments. The radius of the top face, (It will be a circle) and the height. First, work out the area of the top face. Then multiply the area by the height of the cylinder. Imagine volume as how much water the cylinder can hold. (If the radius was 3cm and the height was 6, work out the area of the circle. 28.26. Imagine the cylinder is in slices, you've just worked out the area of 1 slice. Imagine there are 6 slices. You need to work out the area of all the slices put together.Multiply this by 6.

Prism definition

A prism is a 3-d shape that has the same face at each end. If you cut it through it, the new shapes would be identical

Trigonometry

Trigonomery is the study of triangles and angles.

A sphere?

A sphere has 1 face, no corners, no edges.

Golden Rectangle

Draw a rectangle with each side being 2cm. Bisect (Cut in half) the base of the square, so each half is 1cm. Draw a line from the point that bisects to the top right hand corner. You now have a triangle. Draw a curved line from the top right hand corner to being parallel with the bottom right hand corner. Join the bottom of the curved line to the bottom right hand corner. On your right angled triangle, work out the hypotemuse of the triangle. (Add 1 squared to 2 squared and find the square root)

The Golden Ratio

Al-Samawal was an Islamic Mathematitician who lived in Baghdad, Iraq around1150 and was one of the Mathematiticians who worked on the Golden Ratio. Mathematiticians use a simplified version to explain the golden ratio

Example #1 of trigonometry

Find the length of side h. The hypotemuse is 10cm You're given the lengths of the hypotemuse and length h. length h is the adjacent side to the angle you are given. (Which is 25 degrees) The formula you need to use is CAH. Cosine0=Adjacent/Hypotemuse Substitute the values for theta, adjacent and hypotemuse. Cosine25= h/10 Re-write the cosine so it is a fraction. Cosine 25/1=h/10 Cross multiply the fractions to get h on it's own. 10Cosine25=h Type it in on the calculator h=9.06307787cm Round it off to 2dp h=9.06cm

Example #2 of trigonometry

Find the size of angle theta with a hypotemuse of 20cm and an opposite side of 5cm. The opposite side is the side of the triangle that is opposite the angle you want to find. The sides involved are Opposite and hypotemuse. The formula you need is SOH Sine0=Opposite/Hypotemuse Substitute the values Opposite and hypotemuse. Sine0=5/20 Sine0=0.25 Use the shift (Or inverse key) to give you the answer 0=Sin-1(0.25 0=14.47751219 Angle= 14.5degrees Whenever workiing out an angle, always use the inverse key.

Example #3 of trigonometry

Find the length of angle theta with an adjacent side of 7cm and an opposite side of 6cm. An adjacent side is the side next to the angle you want to measure. The opposite is the side that is opposite the angle you're looking for. The formula you're going to use is TOA, because you know the opposite and the adjactent, and the function you need is tangent Tangent0=Opposite/Adjacent Tangent0=6/7 Tangent0=0.8571428571 Tangent0=0.9 Using the inverse key, work out the size of the angle. Tangent-1(0.9=41.9872125 Angle0=42 degrees

Advanced Pythagoras (Part 1)

The hypotenuse of a diagonal of a square is 4cm. The 2 legs, which would be equal because it is a square., are length i. Find the length of i. a2+b2=c2 i2+i2=h2 Square 4 to give you the value of h2. Divide it by 2 to give you the squareof each side. Square root the answer to give you the length of each side. 4x4=16 16/2=8 sqroot8=2.8 to 1dp. i=2.8cm

Advanced Pythagoras (Part 2)

Solve this word problem: A plane takes of from terminal 5 at 11pm at a height of 10m and it travelled for 15.6m. How long was the runway it travelled on? Draw a right angled triangle with the hypotemuse (The distance it travelled) as 15.6m (Obviously not to scale) and with a height of 10m. We don't know the length of the base, (The length of the runway) so that is the length of y. Using pythagoras's theorum, it can be worked out by using: a2+b2=c2 Substitute the values into that formula a2+10 squared=15.6 squared Square the numbers a2+100=243.36 Using inverse operations, (doing the opposite to what the formula says) work out what a2 is, like solving any other equation. a2+100-100=243.36-100 a2=143.36 You don't want a squared, so by doing the inverse of squaring, we find the square root of 143.36, we can work out a. a=11.97 to 2dp The runway was 11.7m.

Circle theorems (Part 1: The Arrow Theorem)

The first circle theorem is the Arrow Theorem. It is a circle which has a sector drawn and then two lines from the circumference which become diameters. (Very difficult to explain without the use of a diagram. What this theorem says is that angles subtended (basically located) at the centrepoint of the circle are double that of the angles at the circumference. The way to prove this is to first bisect (cut into 2) the quadrilateral with a line going from one side of the circle to another (In other words, a diameter) Through drawing this one diameter, you have created 4 new radii, all of which are equal lengths. This line also cuts our quadrilateral into 2 isosceles and right angled triangles, and this means that there is one right angle and the other two angles are equal because of base angles in an isosceles triangle, and due to the fact it is also a right angled triangle, then the other two angles would have to be 45 degrees. We label these angles as a and b, a for the left hand side, b for the right hand side. Looking at one triangle, we can also notice that this triangle has an exterior angle, which would be equal to either 2a or 2b (depending on which side you're looking at) because the exterior angle of a triangle is always equal to 2 interior angles of the triangle, because of angles in a straight line. Looking back at the angles in the whole shape, then we can see that the exterior angle is 2a+2b, which is double what the angle at the circumference is, which is a+b Watch the video below created by a youtube user called maths247

Trig graphs (Part 2)

The trig functions can be plotted on a graph. The x axis is different angles in a circle, in radians. (Pi/2, Pi, 3Pi/2,2Pi) This is an applied area in both Physics and engineering. (eg. The amplitude of a wave is how much energy the wave has) The period of a line is how long it is before the graph starts repeating the same pattern. For the graphs where the frequency hasn't been altered, the graph will always be 2Pi, because it starts repeating the pattern at 2Pi. The period is changed when the b value in this formula is altered in y=Sinbx; the b simply tells us how many times the pattern will repeat itself before reaching 2Pi. Eg. is the graph was y=Sin5x, the graph would repeat itself 5 times between 0 and 2Pi. To find the period of any line, the formula is 2Pi/b. (The absolute value of b, meaning it is always positive) If the b term is positive, the graph will be compressed, repeating itself more often. If the b term is negative, the graph is stretched out, meaning it will repeat itself less often. If the amplitude of a graph is negative, the line is simply reflected in the x axis

Graphing trig functions

When graphing trig functions, if there is a number between the equals sign and the function, it means that the line graph moves that number of units higher up the graph y=a sinx (The line graph of sinx will extend a units up the y axis) (The line stays in the same place, still goes through the y axis at the same point, etc but will go to point a on the y axis instead of 1, which is typically what a trig graph is) y=sinx+a (The line graph will move up the y axis a spaces, however, stays in the same shape) f(x+a) means the line graph has moved to the left by a spaces f(x-a) means the line graph has moved to the right by a spaces y=sinax (The graph has been squashed together by a spaces, and the period/cycle is repeating itself a times more often) If you're using a graphic calculator (being sold for between £20-£80) then you can see the comparison between 2 different trig graphs

Advanced Trigometric Ratios

Sin2 0=sqrt 1-Cos2 0 Cos2 0= sqrt 1-Sin2 0 Sec0=1/Cos0 Csc0=1/Sin0 Cot0=1/Tan0

The Sine Rule (Part 1)

Trigonometry can also work for non right-angled triangles, but SOHCAHTOA does not work. The first rule needed is The Sine Rule, which is used when you know 2 sides of the triangle and 2 angles in the triangle. The rule is: a/sinA=b/SinB=c/SinC The small letters (a,b,c) represent the sides. The big letters (SinA,SinB,SinC) represent the angles. What side is labeled what depends on what each angle has been labeled. (If one angle is labeled A, then the side opposite it is called a) (If one angle is labeled B, then the side opposite it is called b) (if one side is labeled C, then the side opposite it called c) It doesn't matter what the angles are labeled, because the equation would just be the other way around.

Important note about circle theorems

Angles in the same segment are equal, which can be proved by drawing a right angled triangle using the segment as a base. If we were to move the other 2 lines, we would still have a right angle. Remember, segments are formed through a chord which divides the circle into 2 uneven parts

The Sine Rule (Part 2)

a/sinA=b/SinB=c/SinC a/SinA=b/SinB=c/SinC This can only be used if you are trying to find out the length of an unknown side. The upper case letters represent angles The lower case angles represent sides. If you've done labeling correctly, you should end up with one part of the formula that has no substituted values in it. This part can then be ignored. For example, if I had a triangle with sides of 6cm and 7cm, and angles of 50 degrees and 30 degrees, I can substitute these numbers into the formula: 7/Sin50=6/sin30=c/sinC The C part of the formula has not been used, therefore it can be ignored. If you're looking for an angle and not a side, then you use the reciprocal (the "opposite"): SinA/a=SinB/b=SinC/c The other formula can still be used, but it is much easier to use the angle formula.

Phase shift/Phase Angle

To work out the phase shift (not the phase angle) of a trig graph transformation, use the formula: -c/b This tells you how many units you need to move the trig graph horizontally y=a sin(bx+c)

Asymptotes

An asymtote is when a line gets really really close to the x and y axis, yet will never actually touch them, often because they involve division in some way, and you can't have something divided by 0. (In maths, when something cannot happen, it is Undefined)

Circle theorems

When a diameter (Has to pass through the center point) comes into contact with a tangent, a right angle is made. Any angle in a semicircle is 90 degrees. (A point on a diameter) A cyclic quadrilateral is a quadrilateral inside a circle in which all verticies touch the circumference

Sinusoidal graphs

The Cosine graph and the Sine graph are sinusoidal. This means that they have the same shape as each other. The only difference is that the Cosine graph is a PHASE SHIFT of the Sine graph. (Phase shift just means that it's been translated horizontally) Cosx= Sin (x+90)

Heron's theorem

What if you wanted to find the area of a triangle but didn't know any of the angles? (We can't use the formula 1/2abSinc, because we don't know an angle, and the formula ab/2 isn't very good because you would need to know the perpendicular height, yet we need a formula which doesn't need the perpendicular height) Heron's theorem is used to find the area of a triangle when we know all 3 sides. The formula is this: square root( s(s-a)(s-b)(s-c) The letter s is a variable. Start by labeling your sides a,b,c. It doesn't matter which sides you label, because you're still going to be subtracting it from s anyway, so you'd be doing the sum inside the brackets anyway before multiplying everything together. It's a bit like saying that "is 2x 5 different from 5x2?" S can be worked out by using the following formula: S=a+b+c/2

F(x)

When you don't know the actual equation of a graph and you want to apply certain transformations to it, you called is f(x). (or "F of x")

Basic Vectors

Vectors have direction and magnitude. Direction is whether it is going left or right. (If you're going right, you say the direction is positive) (If you're going left, you say the direction is negative) Magnitude is the size of the Vector. Vectors are a key topic in Mathematics, Physics and Computing. A Vector is a way of showing how to get from one point to another. (The magnitude is how large the movement is) A Position Vector is when you move from the origin of the graph to another point on the graph. A Direction Vector is from one point to another. You cannot multiply Vectors together. (You cannot multiply a Vector by another Vector) You can multiply a Vector by a whole number, which is easy to visualize: Vector 2a= Vector a+Vector a Vector 5b= Vector b+Vector b+Vector b+Vector b+Vector b Whenever you write a Vector, you always underline the letter. (On a computer, you always write a Vector in bold print, but you can't hand-write bold print. Vectors can be shown as lines. (eg. if the Vector was (5/2) you draw a line that goes right 5 and up 2. Vectors are parallel if they have the same direction. (One will be a scalar of the other) Vectors are notated (written) as: (a/b)

Sticky note

When transforming shapes, there are four transformations that you need to remember, which can be remembered by using the acronym TERRy. T-Translation- The shape moving from one place to another, but no change in orientation or size E-Enlargement-The shape is staying in the same spot but is becoming larger by a scale factor. (eg. If the scale factor was 2, each side would be twice as big.

Rich sticky notes

Polygons

A polygon is only made up of straight lines.

A complex polygon is a polygon that crosses itself.

A polygon has to be a closed shape. If it has any gap in it, it  is not a polygon.

A polygon is a shape with three or more sides

The exterior angles of any polygon add up to 360 degrees.

A perpendicular bisector is a line segment that cuts a line in half at 90 degrees

To draw one, make an arc using a compass starting at opposite ends of the line.

The arc should be made above the line, and should cross each other at one point

Draw a line from the line to the arc. 

 

Constructing a triangle

Pythagoras

Pythagoras was born in Greece, living around 2600 years ago.

He discovered Pythagoras theorum.

It says that to work out the longest side of a right angled triangle, you have to work out the sum (total) of both sides squared.

The longest side is called the hypotenuse.

 

If the sides were 2cm and 3cm, convert these to 4cm and 9cm. Add 4 and 9. Your answer is 11cm.

The numbers of Pi

Inscribed angles

Circle Theorems Video 1 (Created by a user called Maths247)

Common Mistakes

Plain sticky notes

Common MIstake 1

A common mistake when working out the mean of a series of numbers is typing this on a calculator: 5,9,8,4,2,1 5+9+8+4+2+1/6=30 and 1/6 30 1/6 is not the mean because Scientific calculators are programmed with BODMAS, meaning it will do the division operation first, (1/6, then it would add 1 divided by 6 to the rest of the operation) when you should put the addition in brackets, then the calculator would do that first: (5+9+8+4+2+1)/6 The answer is 5 (5+9+8+4+2+1)/6=5 5 is the correct average

Common MIstake 2

When working out the area of an icosoles triangle, people often are given the length of the base and another side, but multiply the base by the side and divide by 2, but that is not the area, because the other side is the sloping height, and to work out the area of a triangle , you need to know the perpendicular height. Draw a line down the middle of the triangle, dividing it into 2 right angled triangles. The base for each triangle is half the base of the original triangle, and you know the length of each hypotemuse, so (Using Pythagoras) square the hypotemuse and subtract the base squared, to give you the perpendicular height. Multiply the perpendicular height by the original base and divide by 2. The base is 5cm The 2 identical sides are 10cm. Draw a line down the triangle You now have to identical right angled triangles Each base is 2.5cm (10x10)-(2.5x2.5)=100-6.25=93.75 Find the square root of 93.75=9.7cm 9.7x5(The length of the original base)=48.5 Divide 48.5 by 2=24.25 The area is 24.25cm squared

Common MIstake 3

Word problem for an equation: I think of a number, I add 3 and multiply it by 2 my answer is 20 What was my number? Call the number x Common mistake: 2x+3=20 2x=16 x=8.5 x isn't 8.5 because you have to add 3 to x then multiply the answer by 2: 2(x+3)=20 Expand the brackets 2x+6=20 Subtract 6 from both sides 2x=14 x=7

Common Mistake 4

a2+b2=c2 In Pythagoras, thinking that is a=5 and b=6, c would be 11 without squaring 5 and 6 first. You need to square each number to give you c2, then square root the total to give you c.

Common Mistake 5

Work out the average time a child held their breath for in a class Time Frequency 60<t<80 19 90<t<110 6 120<t<140 8 150<t<170 10 180<t<200 3 Common Mistake: Add up each frequency: 19+6+8+10+3=46 Divide by 5 (There are 5 frequencies) 9.2. That would mean that an average time is 9 seconds!! What should be done is draw another column on the frequency table, with the title fx. This is going to be the frequency multiplied by the number the first column. Add up all the numbers you get in the fx column and divide by the total of the numbers in the f (or frequency) column.

Common Mistake 5

When using a formula or doing some sort of calculation, some people think that the square root cancels out the indices, which it doesn't. eg 1. rt( a2+b2 is not the same as a+b This is because indices and roots only cancel when you're multiplying or dividing, not adding

Miscellaneous

Plain sticky notes

Pointless trivia

There are more bacteria in a person's mouth then there are people in the USA and Canada combined!

Pointless trivia 8

The 10 richest countries in Europe are: Lietchenstien Luxembourg Norway Switzerland Denmark Netherlands Germany Ireland United Kingdom Iceland

Pointless Trivia 2

Russia Spans across 11 time zones, 2 continents and speaks 28 different languages!

Pointless Trivia 9

8% of Iceland's workforce are teachers. That's more than anywhere else in the world.

Pointless Trivia 3

Istanbul (Turkey) is the only city in the world to be both bordering Europe and Asia! The European half is similar to western culture, whilst the asian half is full of Islamic tradition

Pointless Trivia 10

Popular US cartoon The Simpsons has been banned in Russia, China and Venezuela

Pointless Trivia 4

8% of Iceland's work force are teachers. That's more than anywhere else in the world

Pointless trivia 11

The 5 largest websites in the world are: Yahoo.com Google.com Youtube.com Google.co.jp (Japanese Google) Facebook.com

Pointless Trivia 5

A Frill Shark (A species of shark that looks like an eel) was found off the coast of Japan. It was moved to a theme park and was labeled as a living fossil. (It was thought to be extincted)

Pointless Trivia 12

In Japan, McDonald's have rebranded their mascot Ronald McDonald as Donald McDonald because in the Japanese language, there is no actual r sound, and it is hard to pronounce, so they changed it so it would be easier to say.

Pointless trivia 6

The oldest writing system that is still used today is Mandarin Chinese

Pointless Trivia 13

There are only 2 countries in the World that have a population of over a billion: China and India

Pointless trivia 7

The country with the highest suicide rates is Lithuania

Pointless Trivia 14

The largest country in the World is Russia, whilst the second largest country (Canada) has a population 5 times smaller than Russia's

Pointless Trivia 16

There are 2 places in America called Washington. A state called Washington, and Washington, the city the the District of Columbia

Pointless trivia 17

Despite Hong Kong being a city in China, politically it's a separate country. Hong Kong dollars are not legal tender in Mainland China, and Yuan are not used in Hong Kong; Hong Kong has it's own Government, media network, sports teams, legal system (Democracy, not communist)(You can do things in Hong Kong that aren't allowed in the rest of China) and language (Cantonese, not Mandarin as the Majority language) It is known as the Special Administrative Region of the People's Republic of China. (Even the internet domains are different, Mainland china's is .cn whilst Hong Kong's is .hk)

Pointless Trivia 18

The most densely populated country in the World is Bangladesh, whilst the least densely populated country in the world is Greenland (Ruled by Denmark)

Pointless Trivia 18

An average person will spend six months of their life waiting in queues

Pointless Trivia 19

75% of the human body is made up of water. (A human can survive a month without food, but less than a week without water)

Poinltess Trivia 20

The longest running TV series that is a spin off of another series is The Simpsons, which is a spin off of The Tracy Ullman show.

Pointless Trivia 20

The only US State which is entirely made up of Islands is Hawaii, which has an alphabet of only 12 letters!

Pointless trivia 21

The most peaceful country is New Zealand. (The least peaceful countries are Iraq, Somalia, Afghanistan, Pakistan and The Democratic Republic of Congo)

Pointless Trivia 22

The country which gets the highest exam results is Finland

Pointless trivia 23

Antidaephobia is the fear of being watched somehow, somewhere by a duck

Pointless trivia 24

Aquaphobic and Hydrophobic are not the same; Hydrophobia is the name given to chemicals that do not react with water, whilst Aquaphobic is the name given to somebody who is afraid of water.

Pointless trivia 25

Popular comic book character The Incredible Hulk was supposed to be grey, but an error with printing meant that he was printed as green and not grey, but the creators at Marvel comics decided that it looked better than grey, so the Hulk was green. The Grey hulk was eventually recycled as a separate villain.

Rich sticky notes

Mr Bean Maths Exam

Statistics GCSE

Plain sticky notes

Introduction

In some schools, Statistics is offered as a separate GCSE to GCSE Mathematics. This course supports other subjects besides maths, such as Science, Psychology, Sociology, Economics, Accounting, Retail, Business Studies, Finance, Computing and Geography. This page shows information from the GCSE Statistics and goes into much more detail than the data handling category.

Types of Data

Qualitative data is non numerical data (Data that does not contain numbers) eg. Car colour, Titles of books in a shop

Types of Data

Quantitative data is numerical data (Data that contains numbers) eg. Height, Age, or weight

Types of Data

Discrete data is data that can be counted (Data that you count) For example, the number of people in a shop because you're counting something

Different types of Statistics

The two main types of statistics are Inferential Statistics and Descriptive Statistics

Types of Data

Bi-variate data is "2 variable" data, which is data with 2 variables involved, such as height and weight, age and shoe size

Data sources

Secondary data is data that somebody else has collected; an advantage is that the data is readily available, easy to access, FOC (Free Of Charge), and can be used for large values, yet you don't know how reliable or accurate the data is (may be biased) don't know how relevant the data is to your investigation, and you don't know if the data is outdated or not. Examples of secondary data include books, newspapers, the internet and reference material

Central Tendency

Another word for the average is the central tendency

Population

A population is the group of people, objects or events you're interested in finding out about.

Sample

Sometimes, a census is extremely difficult or impossible to do. Luckily, you can use a sample in order to research the population. A sample must reflect the population though.

Types of data

Continuous data is data that can be measured (data that you measure) It can take any value, (eg. 3 decimal places) and usually has a scale.(eg. cm, m km) eg. height, distance travelled in a car, etc

A census

A census is when everybody/everything in the population is surveyed

Some types of sample

Random sample Convenience sample Quota sample Systematic sample Stratified Sample Cluster sample

A sampling frame

A sampling frame is a list of the population. (or where you would get a list of everyone in the population)

Data Sources

Primary data is data that you collect yourself; an advantage is that you know how the data was collected, how accurate the data is, the data is completely relevant to your hypothesis and investigation, yet is very expensive and time consuming.

Stratified sampling

A stratified sample is when the population is sorted into groups called strata, and equal proportions of people are chosen from each strata. (NOT equal numbers of people chosen from each strata, as this could mean that you're choosing only a small percentage of people from one strata, yet choosing loads of people from another strata) The way to work out a stratified sample is: Size of strata (How many people are in the strata)/How many people in the population x the sample size. people are then chosen from each strata at random.

Biased

If a sample does not represent the entire population, it is said to be biased. Biased can also mean that data (although it can be unbiased) is represented in a way which can make it favorable to a certain way. (eg. A company could collect data which makes their company look good)

Systematic sample

A systematic sample is when every n/s person of the population is chosen to be surveyed. n is the number of people in the population, and s is the sample size. (A sample size is how many people are going to be surveyed in the sample) A starting point is then chosen at random. A disadvantage of this is that (depending on how the people are ordered, eg. Alphabetically) the sample may be biased.

Replication

If an experiment is only conducted once, then the results may be due to chance or due to human error. By repeating these results, you have eliminated error. This is called replication.

Convenience sampling

Convenience sampling is when people are chosen by a person completely out of convenience. (A person surveys whoever they see first) A problem with this is that it is prone to chance (eg. depends on the time of day the survey was conducted, whoever was in the area at the time etc) Another problem is that it is very likely to become biased through the surveyor choosing one person over another. An advantage of this sample though is that it is very easy to do

A control group

When an experiment is conducted, a control group is often used. This group is similar to the groups being tested, but is not treated in the same way. The results are then compared to that of the control group.

Quota sampling

Quota sampling is when people are chosen by convenience, but the people have to have certain characteristics. (For example have to have dark hair) This sampling method is very similar to convenience sampling, and is often used by market research companies. (eg. 50 people with dark hair have to be chosen) However, this method is prone to chance again, characteristics may be based on opinion and needs to represent the population as much as possible.

Random Response

The random response method of surveying is when another factor is affecting the result. (eg. Roll a dice, if you get a six, tick yes) This is useful for embarrassing questions, (Eg. Do you watch pirate DVD's?) and people are more likely to answer the question truthfully if random response is used.

Cluster sampling

Cluster sampling is when the population is put into large groups called clusters, (eg. All the students in the country are put into schools as clusters) and then the clusters are chosen at random and then people within those clusters are surveyed. This can give biased results if the clusters chosen are very similar.

Probability (Part 1)

Probability is the likelihood of an event occurring. (eg. The probability of it snowing in winter is how likely it will snow in winter) All possible probabilities for an event happening add up to one. (eg. The probability of it snowing and the probability of it not snowing add up to 1) An event is impossible if the probability is 0. An event is certainly going to happen if the probability is 1

Random Sampling

A random sample is when people from the population are chosen purely at random. One way to do it is to assign each person a number between 001 and 999, and then use the Ran# button on a calculator to give you a 3 digit number; if you do this a certain number of times, then you can end up with a sample. The sample can only be random if every member of the population has the same chance of being chosen.

How to make a good question

A question should be easy to understand. (Written in basic English, so that anybody can understand it, not using any slang or text language) Be clear (Unambiguous, it should be clear what the question is asking and what kind of answer to give) Be unbiased (Not lean towards giving any answer or put the person under any pressure to given a particular answer) Be straight to the point (The person will get very bored if they are having to read through each and every question taking 5 minutes each to answer) Should have an "Other column" if a closed question, in order to cover all possible answers. A good questionnaire should contain both open and closed questions. Using answer boxes can also make a survey more effective

Probability (Part 2)

Probabilities can be shown on a diagram called a probability tree. Each branch of the tree shows a different possible outcome, and the probabilities of each event happening are also shown, which should add up to 1.

Different forms of questionnaire

There are many different forms of questionnaire, including: Postal survey (Sending the survey by post) Internet survey (Sending the survey through the internet) Phone survey (Doing the survey by phone) Personal survey (Face-to-face survey)

Internet survey

An internet survey can be VERY quick, cheap and easy to send to a large number of people; it is also likely to have a larger response rate than a Postal survey because it is much easier to fill it out whilst using a computer. A disadvantage is that it can't be sent to people who don't use the internet, or people may not want to open a survey because they don't trust sending information to someone they don't know.

Postal survey

A postal survey can be sent to loads of people very quickly, but may have a poor response rate. (Very few people send the surveys back) It can also cost alot of money to send lots of surveys. It also can't access people who don't have a permanent address or access to the post

Stem and Leaf diagrams

Stem and leaf diagrams are made up of two parts: The Stem and The leaf. A stem and leaf can be used to work out the mode,median and range of a set of data. An advantage of using Stem and Leaf diagrams is that information can be displayed clearly and efficiently. A disadvantage is that they can look Ugly, and can be difficult for dyslexic learners to understand.

Histograms

A histogram is a way of showing data where the AREA of each bar is important, not the height of the bar. Frequency density always goes up the y-axis. Frequency density=Frequency/Class width The bars touch each other (No gap between the bars) if the data is continuous, but if the data is discrete, than the bars never touch. (There is a gap between the bars)

Sample Mean vs Population Mean

The x with the dash is the symbol for the sample mean The Greek letter Mu is the symbol for population mean. The sample mean is the mean of a sample of the population, whilst the population mean is the mean of the whole. population

Step cumulative frequency

Cumulative frequency can be used for both continuous and discrete data; an Ogive Curve is used for continuous data (a smooth curve) whilst a Step Cumulative frequency diagram is used for discrete data (where all the points are connected with straight lines which look like a set of steps) if you do a Step CF diagram, you need a horizontal straight line at the end of your graph (as if it were an actual set of steps)

Deciles and percentiles

When using a Step CF diagram, Deciles and Percentiles can be found. To find Deciles, divide the total cumulative frequency by 10, which gives you a percentile of 10%. Draw a line from the deciles on the y axis to the step line, and then draw a line down to the x axis to find the corresponding amount.

Population Pyramids

A population pyramid is essentially 2 back-to-back bar charts, used to compare data. There are three types of distribution on a population pyramid: A Barrel shape , where the data is virtually identical (or in the same proportions) on both sides, a pyramid shape, where the modal classes are at the of the bottom of the graph, which can be used to make conclusions (eg. Are there more old or young people?) and the Inverted Pyramid, where the highest values are at the top of the graph

Histogram distributions

When studying histograms, there are three ways that the data can be distributed. (Spread out) Symetric, where most of the data is in the middle (and evenly spread out) a positive skew, where the highest values are at the start of the histogram (the left hand side) and a negative skew, where the highest values are at the end of the histogram (the right hand side)

Geometric mean

When doing the geometric mean when doing interest rates, or any kind of percentage change, the key is to convert the numbers into decimals based on what the overall result is, which can be confusing based on what there is: an increase of 5% would be written as 1.05 a decrease of 10% would be 0.9 an increase of 40% would be 1.4 a decrease of 90% would be 0.1 an increase of 1% would be 1.01

Different types of mean

There are two types of mean: The Arithmetic Mean (The mean you already know as the sum of x/n) and The Geometric mean, which is the mean used for working out percentage changes. (which is the nth root of all the numbers multiplied together) eg. the geometric mean of 5,6 and 7 can be cube rt(5x6x7)

Comparitive Pie charts

A comparative pie chart is when you use information from one pie chart to find out information of another pie chart. You will be given a radius of one of the pie charts. Work out the area as you would normally. (Pi x r x r) Work out the "Area per person." (this can be done by dividing the area of the pie chart by the population of the pie chart) Multiply this number by the population of the other pie chart, and you get the area of that pie chart. You can then work out the radius of the other pie chart by remembering that this area equals Pi x r x r. Divide by Pi and square root the answer

Physics

Plain sticky notes

Basic Waves

Waves transfer energy, not matter. (There is no net movement of the particles, the energy is transferred from one particle to the next) There are two types of Wave: Transverse and longitudinal. Waves travel in oscillations. (Vibrations) Transverse waves are at a 90 degree angle (perpendicular) to the direction they're traveling in. (Side to Side wave) Longitudinal waves travel in the same direction they're traveling in. (Parallel) (Back and forth wave) All electromagnetic waves (Radio,Microwave, Infra-red, Visible, Ultra-violet, X-ray, Gamma) are transverse waves. The only EM wave that can be seen is Visible. All EM waves have the same speed through a vacuum. Microwaves have the longest wavelength and the lowest frequency. Gamma waves have the shortest wavelength and the largest frequency. Frequency is the number of waves (complete periods/cycles) that pass a point in one second, and is measured in Hertz. (Hz) The Peak/Crest is the highest point that a wave reaches. A trough is the lowest point that a wave reaches. The wavelength is the distance from one peak to another. Amplitude is how high a wave reaches from the x axis that a wave reaches. Transverse waves cannot travel through liquids, which is how scientists have used Seismic waves (P-waves/Primary Waves and S-waves/Secondary waves) to work out what the Earth looks like, because S waves are transverse waves

Refraction

When a wave passes from one medium to another, and the materials have different densities, the wave changes speed and direction; this is called refraction, and occurs when the incident ray (the ray traveling into the medium. A medium is simply a material that a wave passes through) is at an angle smaller than the critical angle, which is about 48 degrees. The critical angle is when the angle of refraction is 90 degrees. When the angle of incidence is larger than the critical angle, Total Internal Reflection occurs. When a wave passes from a less dense to a dense medium, it slows down and bends towards the normal. (an imaginary line at 90 degrees) When a wave passes from a dense to a less dense medium, it speeds up and bends away from the normal

UV light

There are three types of UV light: UV A, UV B and UV C. UV C is the most dangerous, UV A is the least dangerous. Because UV light is dangerous, your skin darkens in order to prevent the light going to the deeper layers of skin, other than the top layer of skin. This is because darker skin absorbs lighter better than lighter skin, meaning the top layer of skin absorbs it. (Giving a tan, but can also cause severe skin damage) UV light is also used to detect forged banknotes, because ordinary paper is covered in chemicals which fluoresce under UV light, whilst genuine banknotes have markings only visible under UV light.

Uses of microwaves

Microwaves are used to predict the weather; microwaves are sent into space, and the water particles from clouds will either absorb or reflect the microwaves, meaning a picture can be built up of where clouds are.

Ohm's law

In Ohm's law (named after the German scientist Georg Ohm) Voltage (V/Volts/Potential Difference) and Current (I/Amps/Ampere) are directly proportional, where Resistance (Ohms/Omega) is the constant of proportionality. (Current and Voltage increase in the same proportion to each other) A conductor conducts electricity and heat because of delocalised electrons, which are electrons that have escaped the general attraction of the lattice positive ions (Protons left behind) and are free to move, which is why heat and electricity (energy) is carried by the electrons. A conductor that follows Ohm's Law is called an Ohmic conductor. When electricity is flowing through electric overhead cables, a large current will mean that most of the energy being carried is being lost as heat, which means that a large voltage will be needed to allow a small current (voltage is the driving force through the object, which is needed to make the current flow) making the overhead cables more efficient. (efficiency=useful energy output/total energy input x 100) Efficiency of something can be shown in a Sankey energy flow diagram, which involves one large arrow with smaller branching arrows coming off it to show how the energy is being transferred.

Fiber Optics

Fiber Optics work through being made of a material such as glass. It works by transmitting either light energy or infra-red energy. It worked on the principle of total internal reflection, where the wave can be reflected without losing strength or being interfered by noise up to 20km. (copper only really works for about 2km before it loses strength) There are two ways that the information can be transmitted: Analogue or Digital Analogue is constantly varying, but can also pick up interference. (Like crackling or other signals) Digital only has two signals: On (1) and Off (0)

Atomic structure

Everything in existence is made up of atoms. The word atom comes from the Greek word "Uncutable," and the way to visualize an atom is that if you took an apple, and started cutting it smaller and smaller, until you got to a point where you couldn't cut it any more, you would say it is an atom. (or it's Uncutable) Atoms are so small that they cannot be seen, even with a powerful microscope. Atoms are made up of smaller parts called Sub-atomic particles. ("Below Atomic" particles) (Particles smaller than an atom) There are three kinds of sub-atomic particle: Protons, Neutrons and Electrons. Protons and Neutrons can be found in the Nucleus of the Atom. The Nucleus is the Center-point of the atom, where most of the mass of the atom is located. However, the Nucleus is actually very small in comparison to the size of the atom. (If an atom were the size of a football pitch, the Nucleus would be the size of a marble!!) Electrons are found in the Orbital of the atom. (An Orbital is not the same as an orbit) In an orbit, you would be able to determine factors like velocity at a given point or where it would be etc. An Orbital is slightly different; an Orbital is essentially a probability function, a way that Scientists can show where an electron is likely to be around an atom. However, (this might sound contradictory) the electrons can be grouped into shells. Shells are rings where the electrons are going to be.

Comets

Comets have highly elliptical orbits, and are balls of ice and dust that, when flying close to the Earth's atmosphere, leave a trail where the ice has melted, making a shooting star. There are about 1000 million galaxies in the universe.

Space travel

Space travel can be dangerous, because of the cosmic radiation (which is very dangerous) that comes from the sun, the fact there is a lack of gravity means that the heart doesn't have to work as hard to pump blood around the body, meaning that when the astronaut comes back to Earth, their heart is weak and will have trouble becoming accustomed to the Earth's gravity again. (The astronaut is very weak because they don't have to work as hard to move around, meaning their muscles won't work as well) There is also a psychological affect, because the astronaut needs to keep stimulated, and may suffer from severe stress if they have to remain in constant contact with the same people. The first man in space was Yuri Gagarin.

SETI

SETI stands for the Search For Extra Terrestrial intelligence, and involves scientists sending information out into space and hoping that someone or something will intercept them and respond.

Red Shift (The Doppler Effect)

The Doppler effect is what Red shift is about. It proves that galaxies are moving away from Earth, reinforcing the big band theory. If an object is still and is sending out waves, then all the waves are symmetrical and distributed evenly, but if the object is moving, then the waves get squashed, and there is a change in wavelength and the sound changes pitch depending on how far away the object is from you. For planets further away from Earth, we see them as red, because the frequency of the light coming from the planet is decreased, shifting to the red end of the spectrum. For planets closer to Earth, the light is shifted to the blue end of the spectrum because the light has less distance to travel. Red shift is used to prove the big bang theory

Newton's laws of motion

Newton had 3 laws of motion: (He rephrased and proved them, but the main person to work on them was Galileo Galilee) An object will remain at rest unless acted upon by an unbalanced force. An object in motion will stay in the same speed and motion unless acted upon by an unbalanced force. (the law of inertia) (an object will continue doing what it's doing, moving or not, unless acted upon by an unbalanced force) The more mass an object has, the more force is needed to make the object accelerate. (eg. You could push over somebody light easier than somebody heavy) (force=Mass x acceleration) Every action has an equal and opposite reaction. (Forces work in pairs, and for every force there is another force, that is just as big, but in the opposite direction)

Scalar and vector quantities

The difference between a scalar and vector quantity is that a scalar quantity only has magnitude. (size) eg. 5m 10kg 4N A Vector quantity has magnitude and direction. eg. 5m left 10kg down 4N right A vector is usually denoted by an arrow above the letter. Some examples of scalar/vector quantities: Distance/Displacement Speed/Velocity (denoted with a v) Momentum (denoted with an italic p) The letter delta (triangle) means "Change in"

How to convert from km/h to m/s

If you have a measurement in km/h and want to convert to m/s, your measurement is going to be getting much smaller. To convert 5km/h into m/s, you first convert into metres per hour. There are 100m in a km, so multiply 5 by 1000 5000m/h There are 3600 seconds per hour. (60 secondsx60 minutes) 5000m in one hour, so you need to work out how much of that 5000m is covered in 1 second, so you need to divide by 3600, which gives you 1.4 to 1dp

Sticky note

How to work out the distance of a velocity/time graph

On a Velocity time graph, the total distance travelled can be worked out by working out the area underneath the graph. (this can be done by dividing it into 2d shapes, taking the measurements on the axis, and finding the areas of the 2d shapes and adding them together.

Rich sticky notes

Formulae

Weight= Mass x Gravity

Gravity= Weight/Mass

Mass= Weight/Gravity

Moment (In N/m squared)= Distance from pivot/ weight being applied in Newtons

Speed= Distance/Time

Voltage=Current x Resistance   (V=IR)
Electrical energy= Current x time x voltage    (Ee=ItV)  (Ee=Pt)

Power= Current x Voltage   (P=IV)

Displacement=Distance from starting point

Gravitational potential energy= mass x gravity x height (GPE=MGH)

Price for electricity= Time used (Hours) x Power rating (kilowatts) x cost of one unit (0.1)

Wave speed=Frequency x Wavelength

Work=Force x Distance

Charge=Current x Time

Electric Charge of a proton= (roughly) 1.602 x 10 (to the power of -19) coulombs

Electric Charge of an electron= (roughly) -1.602 x 10 (to the power of -19) coulombs

Charge of a coulomb= 6.2 x 10 (to the power of 18) protons/electrons

Current in a semiconductor= The flow of particles called holes

Acceleration=Change in Velocity/Time                               or

Acceleration=Final Velocity-Starting velocity/time

Quiz

Plain sticky notes

Question 1

What is -7 x 4

Sticky note

Answer 1

-28

Question 2

Is a Circle a polygon?

Answer 2

No. A polygon is a shape with 3 or more sides

Question 3

Find the LCM of 60 and 2.

Answer 3

60

Question 4

What does supplementry mean?

Answer 4

Supplementry means that 2 different angles add up to 180 degrees.

Question 5

Are corresponding angles supplementry?

Answer 5

No.

Question 6

Find the HCF of 20 and 5

Answer 6

5

Question 7

Is a polygon ever an open shape?

Answer 7

No. It is always a closed shape

Question 8

Find the prime factors of 30

Answer 8

2,3 and 5

Question 9

10 cubed

Answer 9

1000

Question 10

What do the angles on a straight line add up to?

Answer 10

180 degrees

Question 11

Angles d and e are alternate angles. Angle d is 150 degrees. How much is angle e?

Answer 11

Angle e is 150 degrees.

Question 12

-5 x -3

Answer 12

15

Question 13

30 / -10 / Means divide

Answer 13

-3

Question 14

What is a complex polygon?

Answer 14

A polygon that crosses over itself

Question 15

-2x-3

Answer 15

6

Question 16

Simplify this expression 5x+3y+2x+2y+2a

Answer 16

7x+5y+2a

Question 17

Simplify this expression 2 3 2a + 3a+2b+3a

Answer 17

2 3 2a +3a+2b+3a

Question 18

Is a googlegon a polygon?

Answer 18

Yes

Question 19

What do the angles in a square add up to?

Answer 19

360 degrees

Question 20

Jane puts her hand in a bag of 12 marbles. 1 is red, 9 are blue, and 2 are green. What colour is she most likely to pick?

Answer 20

Blue.

Question 21

Using the approximation of PI is 3.14 and the raduis is 5cm, find the area of the circle

Answer 21

78.5cm squared

Question 22

3.4 x 5.7

Answer 22

19.38

Question 23

Find the larger of 2/5 or 1/10

Answer 23

2/5 >1/10

Question 24

3.2 x 0.7

Answer 24

2.24

Question 25

4.8 x 3.3

Answer 25

15.84

Question 27

23 x 2.4

Answer 27

55.2

Question 28

-5 x 2

Answer 28

-10

Question 29

using the equation of a straight line y= 2x+1, find the gradient of the line and the point which it crosses the y axis

Answer 29

Gradient= 2 Point that it crosses the y axis= (0,1)

Question 30

Round 2.146 to 1decimal place

Answer 30

2.1

Question 31

Reduce £42 by 30%

Answer 31

£29.40

Question 32

Round 3.2345432 to 3 decimal places

Answer 32

3.235

Question 33

Using the approximation of pi being 3.14, and the raduis being 2cm , work out the area of this circle.

Answer 33

12.56 cm squared

Question 34

Is a triangle a polygon?

Answer 34

Yes. A poloygon is a shape with three or more sides.

Question 35

Find the larger out of these 2 fractions. 2/3 or 3/4

Answer 35

3/4

Question 36

Billy has 25 marbles. 10 are red, three are blue and 12 are various colours. What is the probability of choosing a blue marble at random?

Answer 36

3/25

Question 37

Find the LCM and HCF of 3 and 5

Answer 37

LCM= 15 HCF= 1

Question 38

Find the larger out of these pairs of fractions 3/4 or 9/10

Answer 38

3/4<9/10 9/10

Question 39

Increase 88 by 10%

Answer 39

96.8

Question 40

Work out the equation of this line if the gradient is 4 and the y intercept is (0,5)

Answer 40

y=4x+5

Question 41

Find the gradient and the y intercept of this line. y=8x+2

Answer 41

Gradient= 2 Y intercept= (0,2)

Question 42

-2 x -3

Answer 42

6

Question 43

Expand these brakets 5(x-1)

Answer 43

5x-5

Question 44

Expand these brackets 2(4x+3)

Answer 44

8x+6

Question 45

-5 x -7

Answer 45

35

Question 46

Find the inverse to 2x-5

Answer 46

x-5 x-5/2 ------ 2

Question 47

Find the TSA (Total suface area) of a cube in which each face is 2X6. (Two by six)

Answer 47

72cm squared

Question 48

27x92

Answer 48

2484

Question 49

9.7x5.1

Answer 49

49.47

Question 50

Expand these brackets 4(x-2)

Answer 50

4x-8

Question 51

What is the plan view of a 3-D shape?

Answer 51

The plan view is a bird's eye view of the 3-D shape. It should always be drawn as 2-D.

Question 52

-8x-7

Answer 52

56

Question 53

What is the front elevation of a 3-D shape?

Answer 53

A 2-D sketch of what a 3-D shape looks like from the front. You must put line to show how many shapes there are.

Question 54

What is Isometric paper?

Answer 54

A special type of paper which is used to sketch 3D shapes.

Question 55

There are 20 M &M's in a packet. 10 are red, 5 are green, 4 are yellow and 1 is black. What's the most likely colour to be pulled out?

Answer 55

The most likely colour to be pulled out is red

Question 56

A survey was conducted over people's favourite food. there are 1500 people surveyed. 80% of those people voted pizza. How many people voted pizza?

Answer 56

1200 people voted that their favourite food was pizza. Divide 1500 by 10. This will give you 10%. Mulitply this number by 8. This gives you 80%.

Question 57

900 people are living in a tower of apartments. 30% of those poeple are children. How many children are living in the tower?

Answer 57

270 Divide 900 by 10. This is 10%. Multiply it by 3 to get 30%.

Question 58

In a DVD store, there is a sale. 55% off all DVD's. I buy a DVD for $10.00. This is before the price is reduced. What do I have to pay now?

Answer 58

$4.50 Divide $10.00 by 2 in order to get 50%. To work out 10%, divide $10.00 by 10. Divide 10% by 2 in order to get 5%. Add 50% and 5% together in order to get 55%. Take that away from $10.00

Quesiton 59

Find the term to term rule for this sequence 1,5,9,14,19.

Answer 59

Add 4

Question 60

A boy takes out a book from his library. He returns the book 100 days overdue. The library fines him 19 times the original price of the book. The book is represented as y. He is also charged 5 euros extra for a DVD. His bill comes to 100 euros. Write this as an equation.

Answer 60

19y+5=100

Question 61

What do the following things have in common? A Pineapple The Mona Lisa The UN Building in New York The Great Pyramids of Egypt?

Answer 61

They all incorporate the number Phi in some way. (1.614) This is also known as The Golden Ratio

Question 62

Expand and simplify this binomial (7+6)(7y-1)

Answer 61

y=5

Question 63

Find the larger out of these 2 fractions 7/8 or 3/4

Answer 62

Because the range can be distorted by extreme values. (It only takes one value to be dramatically higher than the others to give inaccurate results) Whilst the inter-quartile range uses the data around the median, which gives a representation of the majority of the data

Question 64

9+8x2

Answer 62

91y-13

Question 65

A train from Brussels bound for Berlin leaves at 05:30, the journey takes 7 and a half hours, what will the time be when it arrives?

Answer 63

7/8 > 3/4

Question 66

Tickets for a plane journey from Shanghai bound for Tokyo cost 299 yen. How much change would I get from 500 yen?

Answer 64

25 (Use BODMAS)

Question 67

10/0.5

Answer 65

12:30

Question 68

What's the y intercept for: y=4x5+2

Answer 66

201 yen. (Round 299 up to 300, take that away form 500 and add one extra on)

Question 69

If a flight that travels from Shanghai to Tokyo lasts 50 minutes, how long would 2 return flights cost. (Return would mean from Shanghai to Tokyo, then back to Shanghai)

Answer 67

20

Question 70

A shop sells boxes of erasers containing 400 erasers in each box. How many erasers would be in 20 boxes

Answer 68

(0,5) Remember when working out coordinates, do "along the corridor and up the stairs" Horizontal then vertical.

Question 71

-8+9-2

Answer 69

200 minutes

Question 72

Peter can write 50 words a minute. After taking a course at the college, he can write 90 words a minute. By what percentage has his skill increased

Answer 70

8000 erasers Use frequency tables

Question 72

In trigonometry, what does SOHCAHTOA stand for?

Answer 71

-1

Question 73

In trigonometry, if you know the 2 sides, but need to find out the angle, what function do you use on a calculator?

Answer 72

SOH: Sine0=Opps/Hyp CAH: Cosine0=Adj/Hyp TOA: Tangent0=Opps/Adj

Qustion 74

What do the following variables have in common? Alpha Beta, Pi Theta Xi

Answer 73

Press the shift (Or inverse key) and then press the sine, cosine, or tangent button and type in the rest of the equation. (This only works on scientific calculators) This will give you sine-1(, cosine-1(, or Tangent-1(

Question 75

Work out this indices problem: 9999999 (To the power of 0) +7

Answer 74

The variables all come from the Greek alphabet. (A guide to the Greek alphabet can be found in the algebra category. This guide is just to provide a further understanding of algebra, and is not deigned for somebody who wants to learn Greek as a language.

Question 73

Write 70 as a sum of it's prime factors

Answer 75

8. Anything to the power of 0 is 1. 1+7=8

Question 74

Find the larger of these 2 fractions 1/7 1/8

Answer 72

80% 50 words is 100% 10 words is 20% The difference between 50 and 90 is 40 40 would be the equivilent of 80%

Question 75

1/9+2/3

Answer 73

70=2x5x7

Question 76

How do you work out Pi?

Answer 74

1/7>1/8 1/7 is larger

Question 77

What's a polygon?

Answer 75

7/9

Question 78

Where do natural numbers start?

Answer 76

Circumfrence/Diameter

Question 79

How many F's are in this sentence? Francis, from Finland, invited his friend Fred of California to have French Fancies and have a conversation about the Duke of France

Answer 77

The name for a shape which has 3 of more sides

Question 80

There are 1,000 grams in a Kilogram, how many grams are ther in 25 kilograms?

Answer 78

1 because that's where you'd start counting naturally

Question 81

Work out the area of a doughnut using the following figures. Area of outer circle= 100cm squared Radius of inner circle= 2cm

Answer 79

There are 10 F's in this sentence

Question 82

What's a variable in Algebra?

Answer 80

25,000 kilograms

Question 83

What term is 18 in this sequence? How much are the terms changing by? T(2)=9x2=18

Answer 81

87.44cm squared

Question 84

A person travels 5 miles from their home in the morning, and comes home in the evening. What's their displacement?

Answer 82

A symbol (A letter) used to represent an unknown value. This value could be anything and the letters are not specific values The letters used are typically at the end of the English alphabet.

Question 85

A baseball player hits 3 balls: Number 1 traveled 2 metres Number 2 taveled 2.5 metres Number 3 traveled 3 metres What's the average distance?

Answer 83

It's the second term in the sequence The term to term rule is +9

Question 86

9x+3=48 Find the value of x

Answer 84

0 miles. Displacement is distance traveled from a starting point, and the house was the starting point.

Question 87

Work out the first 4 terms in the sequence with the nth term: T(n)=2+n2 (n squared)

Answer 85

2.5 metres 2.5+2+3=7.5/3=2.5

Question 88

What side of the triangle is the hypotemuse?

Answer 86

x=5

Question 89

What country did Pythagoras come from?

Answer 87

3, 6, 11, 18

Question 90

How do you work out the angle of a right angled when you know 2 of the sides?

Answer 88

The longest side of a right angled triangle which is always opposite the right angle

Question 91

Which pair of sides of you need to know in order to use the Sine, Cosine and Tangent function on the calculator

Answer 89

greece

Question 92

What is the acronym for the 3 trig ratios?

Answer 90

Use the inverse function. (sin-1, cos-1 tan-1)

Question 93

How is the Fibonacci sequence related to the Golden Ratio/Phi? (1.618)

Answer 91

Sine: Opposite and Hypotenuse Cosine: Adjacent and Hypotenuse Tangent: Opposite and Adjacent Remember of acronym for remembering the 3 trig ratios?

Question 94

Why is the inter-quartile range better than the range when using cumulative frequency data? (Which can be used to represent data in groups)

Answer 92

SOHCAHTOA

Question 95

What is the general form that all quadratic expressions should take?

Answer 94

Because the range can become distorted by an extreme value (Just one) And give a completely inaccurate result. By using the inter-quartile range, we can get a range from around the median, giving a more accurate reading

Question 96

Factorize this expression: 6x2+9x+3

Answer 95

ax2+bx+c

Question 97

Increase 70 by 12%

Question 98

If you're solving an inequality by a negative number, what should you do to the bridge? (Inequality sign)

Question 99

Simplify 16x2-25y2

Question 100

Simplify 64x2-81y2

Additional Mathematics

Plain sticky notes

Long division of Polynomials

Divide x2+2x-7 by x-2 Start by laying your division out like you would a normal long division x-2/ x2+2x-7 Only focus on the (x) in (x-2). Work out what you would need to multiply (x) by to get x2. (need to multiply it by x) Multiply (x-2) by (x) Write x(x-2) underneath (x-2), and write the expanded form of x(x-2) underneath (x2+2x) x (x-2) / x2+2x-7 x(x-2) /x2-2x Take (x2-2x) away from (x2+2x) and write your answer underneath x (x-2) /x2+2x-7 x(x-2) /x2-2x 4x Bring the (-7) term down to (4x) to make (4x-7) x (x-2) /x2+2x-7 x(x-2) /x2-2x 4x-7 Work out what you need to multiply (x) by to get (4x). (You need 4) Multiply (x-2) by 4. (If you multiply the (x) by something, you need to multiply it by (-2) as well. Write 4(x-2) underneath (x-2) and x(x-2) and write the expansion underneath 4x x+4 (x-2) /x2+2x-7 x(x-2) /x2-2x 4x-7 4(x-2) 4x-8 Subtract (4x-8) from (4x-7) x+4 (x-2) /x2+2x-7 x(x-2) /x2-2x 4x-7 4(x-2) 4x-8 1 Your answer is x+4+1/x-2

Binomial distribution

The only kind of probability you will come across in additional maths is binomial distribution. There would be only two outcomes, success or failure. The easy way to write this would be x-Bin(probability of achieving success, number of trials) eg. X-Bin(0.6, 10) The event X is binomially distributed, where there are two outcomes, the probability of achieving success is 0.6, and the event X will happen 10 times What is the probability of achieving success of achieving success 5 times after 10 trials? 10C5- This takes into consideration the number of different ways that you can achieve success 5 times. (0.6)^5- The probability of achieving success 5 times. (it's ^5 because of the and rule in probability, where if two events happen one after another, you multiply their probabilities. (0.4)^5- The probability of not achieving success 5 times. Working out the probability of failure is 1-the probability of success, which is 1-0.6=0.4. To work out the power that it is raised to, you do total number of trials-number of times achieving success

nCr

To find how many different combinations of a series of events happening in Pascal's triangle there are, two methods can be used: nCr or n!/r!(n-r)! The nCr method means "how many different ways in the nth row of pascal's triangle can you choose r" For example, 3C2 would tell you how many different combinations of achieving success twice for three trials. (Pascal's triangle only works for events which are binomially distributed) When you're expanding brackets, you can quickly find all the coefficients instead of having to draw out all the rows of pascal's triangle. nCr n- Which power you're expanding. (the highest order of the polynomial) C- Performs n!/r!(n-r)! r- The power of the term you're trying to find the coefficient of. (eg. the x^2 term in (x+1)^7 would be 7C2 To work out all the coefficients of (x+1)^7, see below 7C0- 1 7C1-7 7C2- 21 7C3- 35 7C4- 35 7C5-21 7C6-7 7C7-1 Notice how pascal's triangle becomes symmetrical around the middle terms.

Binomial expansion (Part 1)

Pascal's triangle was discovered as a way to expand long brackets quickly, and is part of a topic called binomial expansion. For example, expand (x+y)5, which would take some time if you did (x+y)(x+y)(x+y)(x+y)(x+y), but Pascal's triangle lets you find the coefficients really quickly. A binomial is an expression that has 2 terms in it. eg. (a+b) (x+1) (y+z) (x-2) Pascal's triangle looks like this: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Each row represents a certain power. (eg. Power of 1, Power of 2, Power of 3 etc) The first row is called row 0, because it shows the coefficients if you raised a binomial to the power of 0. (anything to the power of 0 is 1) The second row shows you the coefficients if you raised a binomial to the power of 1. Pascal's triangle also tells you how many terms you should get if you expanded a binomial. Each row of Pascal's triangle starts and ends with a 1.

Introduction to calculus

There are two halves to calculus; differentiation and integration, which are exact opposites of each other. Differentiation is where you find an expression for the gradient of a curve, whilst integration would be to find the area underneath the curve. In terms of mechanics, differentiating certain quantities will give you other quantities. Distance If you differentiate distance, you get velocity. (or how much distance is changing per second) If you differentiate velocity, you get acceleration, which is how much the velocity is changing per second.

Example of Binomial Expansion

When you expand brackets, the x terms need to be ascending and the constant (number) terms need to be descending so that they add up to the power that you're expanding by. eg. 1 (x+5)^4 x^4 means you need the x^4 row of pascal's triangle. (The 5th row) This gives you the coefficients of each of the terms 1 4 6 4 1 Start by placing descending powers of x from left to right. 1(x^4)+4(x^3)+6(x^2)+4(x^1)+1(x^0) Now we need to add the expansion of the 5. The powers need to add up to 4, so we need to do ascending powers of 5. 1(x^4)(5^0)+4(x^3)(5^1)+6(x^2)(5^2)+4(x^1)(5^3)+1(x^0)(5^4) Now we can tidy this all up. x^4+16x^3+150x^2+500x+625 It was much quicker and less tedious than doing manual expansion.

Calculus (differentiation) Part 1

The general expression for differentiation is dy/dx. This is the same as change in y/change in x. dy/dx basically reads as differentiate y, with respect to x. To differentiate an expression, you first have to multiply the coefficient at the front of the x term by its power. Then, you decrease the power by 1. eg 1: Differentiate x^3 Multiply the coefficient of x (which is 1) by 3 3x^3 Then move the power (order) of the x^3 term down by 1. 3x^2 eg. 2: Differentiate 4x^2 Multiply the coefficient of x (which is 4) by 2 8x^2 Decrease the power if the x^2 term by 1 8x

Calculus (differentiation) Part 2

If you differentiate a curve, you can use it to find the gradient of the curve at any point. eg 1: Find the gradient at x=5 for x^3+2x^2+8x+3 STEP 1: Differentiate each term 3x^2+4x+8 The 3 disappears because it was a constant term and it had no x term originally. STEP 2: Substitute x=5 into the dy/dx function 3(5)^2+4(5)+8 75+20+8 103 The gradient of the curve at x=5 is 103

Alternate ways of asking to differentiate

Find f'(x) for f(x) f(x) [said as "f of x"] Find the derivative of y=.... Find the gradient of y=....

Differentiation example 1

Find dy/dx for y=x^2+8x+3 Multiply the coefficient of each term by the order of the term 2x^2+8x+3(x^0) decrease each of the coefficients by 1 2x+8 dy/dx=2x+8

Rich sticky notes

Long division of Polynomials (PatrickJMT)

Binomial Expansion (m4ths.com)

Cool maths tricks

Rich sticky notes

Cool Maths Trick

Learn the 9 times table easily